Integrals of Jacobi Elliptic Functions Raised to the Power 4

We use combinations of the following to evaluate integrals in this post:

All results can be found in section 310 – 321 of Handbook of Elliptic Integrals for Engineers and Scientists by Byrd and Friedman. We use their equation numbering scheme.

BF 310.04

\begin{align}
\int \mathrm{sn}^{4} &= \int \mathrm{sn}^{2} (1 \,-\, \mathrm{cn}^{2}) = \int \mathrm{sn}^{2} \,-\, \int \mathrm{sn}^{2} \mathrm{cn}^{2} \\
&= \frac{1}{3k^{4}}\left[(k^{2} + 2)u \,-\, 2(k^{2} + 1)E + k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} \right]
\end{align}
We used GR 5.134.1 and BF 361.01.

BF 312.04

\begin{align}
\int \mathrm{cn}^{4} &= \int \mathrm{cn}^{2} (1 \,-\, \mathrm{sn}^{2}) = \int \mathrm{cn}^{2} \,-\, \int \mathrm{sn}^{2} \mathrm{cn}^{2} \\
&= \frac{1}{3k^{4}}[{k^{\prime}}^{2}(2 \,-\, 3k^{2})u + 2(2k^{2} \,-\, 1)E + k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} ]
\end{align}
We used GR 5.134.2 and BF 361.01.

BF 314.04

\begin{align}
\int \mathrm{dn}^{4} &= \int \mathrm{dn}^{2} (1 \,-\, k^{2}\mathrm{sn}^{2}) = \int \mathrm{dn}^{2} \,-\, k^{2}\int \mathrm{sn}^{2} \mathrm{dn}^{2} \\
&= \frac{1}{3}[2(1 + {k^{\prime}}^{2})E \,-\, {k^{\prime}}^{2}u + k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} ]
\end{align}
We used GR 5.134.3 and BF 361.02.

BF 311.04

\begin{align}
\int \mathrm{ns}^{4} &= \int \mathrm{ns}^{2} (\mathrm{ds}^{2} + k^{2}) = \int \mathrm{ns}^{2} \mathrm{ds}^{2} + k^{2} \int \mathrm{ns}^{2} \\
&= \frac{1}{3}[(2 + k^{2})u \,-\, 2(1 + k^{2})E \,-\, \mathrm{dn}\,\mathrm{cs} (\mathrm{ns}^{2} + 2 + 2k^{2})]
\end{align}
We used BF 311.02 and BF 361.22.

BF 317.04

\begin{align}
\int \mathrm{cs}^{4} &= \int \mathrm{cs}^{2} (\mathrm{ns}^{2} \,-\, 1) = \int \mathrm{cs}^{2} \mathrm{ns}^{2} \,-\, \int \mathrm{cs}^{2} \\
&= \frac{1}{3}[2(1 + {k^{\prime}}^{2})E \,-\, {k^{\prime}}^{2}u + \mathrm{dn}\,\mathrm{cs} (2 + 2{k^{\prime}}^{2} \,-\, \mathrm{ns}^{2})]
\end{align}
We used BF 317.02 and BF 361.21.

BF 319.04

\begin{align}
\int \mathrm{ds}^{4} &= \int \mathrm{ds}^{2} (\mathrm{ns}^{2} \,-\, k^{2}) = \int \mathrm{ds}^{2} \mathrm{ns}^{2} \,-\, \int k^{2}\mathrm{ds}^{2} \\
&= \frac{1}{3}[(2 \,-\, 3k^{2}){k^{\prime}}^{2}u + 2(2k^{2} \,-\, 1)E \,-\, \mathrm{dn}\,\mathrm{cs} (\mathrm{ns}^{2} + 2 \,-\, 4k^{2})]
\end{align}
We used BF 319.02 and BF 361.22.

BF 313.04

\begin{align}
\int \mathrm{nc}^{4} &= \int \mathrm{nc}^{2} (\mathrm{sc}^{2} + 1) = \int \mathrm{nc}^{2} \mathrm{sc}^{2} + \int \mathrm{nc}^{2} \\
&= \frac{1}{3{k^{\prime}}^{4}}[{k^{\prime}}^{2}(2{k^{\prime}}^{2} \,-\, k^{2})u + 2(2k^{2} \,-\, 1)E + \mathrm{dn}\,\mathrm{sc} ({k^{\prime}}^{2}\mathrm{nc}^{2} + 2 \,-\, 4k^{2})]
\end{align}
We used BF 313.02 and BF 361.07.

BF 316.04

\begin{align}
\int \mathrm{sc}^{4} &= \int \mathrm{sc}^{2} (\mathrm{nc}^{2} \,-\, 1) = \int \mathrm{sc}^{2} \mathrm{nc}^{2} \,-\, \int \mathrm{sc}^{2} \\
&= \frac{1}{3{k^{\prime}}^{4}}[2(1 + {k^{\prime}}^{2})E \,-\, {k^{\prime}}^{2}u + \mathrm{dn}\,\mathrm{sc} ({k^{\prime}}^{2}\mathrm{nc}^{2} \,-\, 2 \,-\, 2{k^{\prime}}^{2})]
\end{align}
We used BF 316.02 and BF 361.07.

BF 321.04

\begin{align}
\int \mathrm{dc}^{4} &= \int \mathrm{dc}^{2} ({k^{\prime}}^{2}\mathrm{nc}^{2} + k^{2}) = {k^{\prime}}^{2}\int \mathrm{dc}^{2}\mathrm{nc}^{2} + k^{2}\int \mathrm{dc}^{2} \\
&= \frac{1}{3}[(2 + k^{2})u \,-\, 2(1 + k^{2})E + \mathrm{dn}\,\mathrm{sc} ({k^{\prime}}^{2}\mathrm{nc}^{2} + 2 + 2k^{2})]
\end{align}
We used BF 321.02 and BF 361.13.

BF 315.04

\begin{align}
\int \mathrm{nd}^{4} &= \int \mathrm{nd}^{2} (k^{2}\mathrm{sd}^{2} + 1) = k^{2}\int \mathrm{nd}^{2} \mathrm{sd}^{2} + \int \mathrm{nd}^{2} \\
&= \frac{1}{3{k^{\prime}}^{4}}[2(2 \,-\, k^{2})E \,-\, {k^{\prime}}^{2}u \,-\, k^{2}\mathrm{sn}\,\mathrm{cd} ({k^{\prime}}^{2}\mathrm{nd}^{2} + 4 \,-\, 2k^{2})]
\end{align}
We used BF 315.02 and BF 361.19.

BF 318.04

\begin{align}
\int \mathrm{sd}^{4} &= \int \mathrm{sd}^{2} (\mathrm{nd}^{2} \,-\, \mathrm{cd}^{2}) = \int \mathrm{sd}^{2} \mathrm{nd}^{2} \,-\, \int \mathrm{sd}^{2} \mathrm{cd}^{2} \\
&= \frac{1}{3{k^{\prime}}^{4}k^{4}}[2(2k^{2} \,-\, 1)E + {k^{\prime}}^{2}(2 \,-\, 3k^{2})u \,-\, k^{2}\mathrm{sn}\,\mathrm{cd} ({k^{\prime}}^{2}\mathrm{nd}^{2} + 4k^{2} \,-\, 2)]
\end{align}
We used BF 318.02 and BF 361.19.

BF 320.04

\begin{align}
\int \mathrm{cd}^{4} &= \int \mathrm{cd}^{2} (1 \,-\, {k^{\prime}}^{2} \mathrm{sd}^{2}) = \int \mathrm{cd}^{2} \,-\, {k^{\prime}}^{2} \int \mathrm{cd}^{2} \mathrm{sd}^{2} \\
&= \frac{1}{3k^{4}}[(2 + k^{2})u \,-\, 2(1 + k^{2})E + k^{2}\mathrm{sn}\,\mathrm{cd} (2 \,-\, {k^{\prime}}^{2}\mathrm{nd}^{2} + 2k^{2})]
\end{align}
We used BF 320.02 and BF 361.27.