Integrals of Gradshteyn and Ryzhik: 5.136 – 5.139 – Integrals of Combinations of Jacobi Elliptic Functions (Part 1)

5.136.1

\begin{equation}
\int \mathrm{sn}\,\mathrm{cn} = -\frac{1}{k^2}\mathrm{dn}
\end{equation}

5.136.2

\begin{equation}
\int \mathrm{sn}\,\mathrm{dn} = -\mathrm{cn}
\end{equation}

5.136.3

\begin{equation}
\int \mathrm{cn}\,\mathrm{dn} = \mathrm{sn}
\end{equation}
The three integrals above follow from the derivatives of the Jacobi elliptic functions.

5.137.1

\begin{equation}
\int \frac{\mathrm{sn}}{\mathrm{cn}^2} = \frac{1}{{k^{\prime}}^2} \frac{\mathrm{dn}}{\mathrm{cn}} = \frac{1}{{k^{\prime}}^2} \mathrm{dc}
\end{equation}

5.137.2

\begin{equation}
\int \frac{\mathrm{sn}}{\mathrm{dn}^2} = -\frac{1}{{k^{\prime}}^2} \frac{\mathrm{cn}}{\mathrm{dn}}
= -\frac{1}{{k^{\prime}}^2} \mathrm{cd}
\end{equation}

5.137.3

\begin{equation}
\int \frac{\mathrm{cn}}{\mathrm{sn}^2} = -\frac{\mathrm{dn}}{\mathrm{sn}} = -\mathrm{ds}
\end{equation}

5.137.4

\begin{equation}
\int \frac{\mathrm{cn}}{\mathrm{dn}^2} = \frac{\mathrm{sn}}{\mathrm{dn}} = \mathrm{sd}
\end{equation}

5.137.5

\begin{equation}
\int \frac{\mathrm{dn}}{\mathrm{sn}^2} = -\frac{\mathrm{cn}}{\mathrm{sn}} = -\mathrm{cs}
\end{equation}

5.137.6

\begin{equation}
\int \frac{\mathrm{dn}}{\mathrm{cn}^2} = \frac{\mathrm{sn}}{\mathrm{cn}} = \mathrm{sc}
\end{equation}
The six integrals above follow from the derivatives of the Jacobi elliptic functions.

5.138.1

\begin{align}
\int \frac{\mathrm{cn}}{\mathrm{sn}\,\mathrm{dn}} &= \int \frac{\mathrm{cn}}{\mathrm{sn}\,\mathrm{dn}} \frac{\mathrm{dn}}{\mathrm{dn}} = \int \frac{\mathrm{cn}}{\mathrm{dn}^2} \frac{1}{\mathrm{sd}} \\
&= \int \frac{1}{w} = \ln(\mathrm{sd}) = \ln\left(\frac{\mathrm{sn}}{\mathrm{dn}}\right)
\end{align}
We used the substitution \(w=\mathrm{sd}\).

5.138.2

\begin{align}
\int \frac{\mathrm{sn}}{\mathrm{cn}\,\mathrm{dn}} &= \int \frac{\mathrm{sn}}{\mathrm{cn}\,\mathrm{dn}} \frac{\mathrm{cn}}{\mathrm{cn}} = \int \frac{\mathrm{sn}}{\mathrm{cn}^2} \frac{1}{\mathrm{dc}} \\
&= \frac{1}{{k^{\prime}}^2} \int \frac{1}{w} = \frac{1}{{k^{\prime}}^2} \ln(\mathrm{dc}) = \frac{1}{{k^{\prime}}^2} \ln\left(\frac{\mathrm{dn}}{\mathrm{cn}}\right)
\end{align}
We used the substitution \(w=\mathrm{dc}\).

5.138.3

\begin{align}
\int \frac{\mathrm{dn}}{\mathrm{sn}\,\mathrm{cn}} &= \int \frac{\mathrm{dn}}{\mathrm{sn}\,\mathrm{cn}} \frac{\mathrm{cn}}{\mathrm{cn}} = \int \frac{\mathrm{dn}}{\mathrm{cn}^2} \frac{1}{\mathrm{sc}} \\
&= \int \frac{1}{w} = \ln(\mathrm{sc}) = \ln\left(\frac{\mathrm{sn}}{\mathrm{cn}}\right)
\end{align}
We used the substitution \(w=\mathrm{sc}\).

5.139.1

\begin{equation}
\int \frac{\mathrm{cn}\,\mathrm{dn}}{\mathrm{sn}} = \int \frac{1}{w} = \ln(\mathrm{sn})
\end{equation}

5.139.2

\begin{align}
\int \frac{\mathrm{sn}\,\mathrm{dn}}{\mathrm{cn}} &= \int \frac{\mathrm{sn}\,\mathrm{dn}}{\mathrm{cn}} \frac{\mathrm{cn}}{\mathrm{cn}} = \int \frac{\mathrm{sn}\,\mathrm{dn}}{\mathrm{cn}^2} \frac{1}{\mathrm{nc}} \\
&= \int \frac{1}{w} = \ln(\mathrm{nc}) = \ln\left(\frac{1}{\mathrm{cn}}\right)
\end{align}
We used the substitution \(w=\mathrm{nc}\).

5.139.3

\begin{equation}
\int \frac{\mathrm{cn}\,\mathrm{sn}}{\mathrm{dn}} = -\frac{1}{k^2} \int \frac{1}{w} = -\frac{1}{k^2} \ln(\mathrm{dn})
\end{equation}

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