Optimal. Leaf size=81 \[ -e^{-x} \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};-e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-e^{2 n x}\right )+\frac{e^{-x}}{2}+\frac{e^x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0696508, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5601, 2194, 2251} \[ -e^{-x} \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};-e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-e^{2 n x}\right )+\frac{e^{-x}}{2}+\frac{e^x}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5601
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int \sinh (x) \tanh (n x) \, dx &=\int \left (-\frac{e^{-x}}{2}+\frac{e^x}{2}+\frac{e^{-x}}{1+e^{2 n x}}-\frac{e^x}{1+e^{2 n x}}\right ) \, dx\\ &=-\left (\frac{1}{2} \int e^{-x} \, dx\right )+\frac{\int e^x \, dx}{2}+\int \frac{e^{-x}}{1+e^{2 n x}} \, dx-\int \frac{e^x}{1+e^{2 n x}} \, dx\\ &=\frac{e^{-x}}{2}+\frac{e^x}{2}-e^{-x} \, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};-e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-e^{2 n x}\right )\\ \end{align*}
Mathematica [B] time = 0.170531, size = 164, normalized size = 2.02 \[ \frac{1}{2} e^{-2 x} \left (-\frac{e^{2 n x+x} \, _2F_1\left (1,1-\frac{1}{2 n};2-\frac{1}{2 n};-e^{2 n x}\right )}{2 n-1}+\frac{e^{(2 n+3) x} \, _2F_1\left (1,1+\frac{1}{2 n};2+\frac{1}{2 n};-e^{2 n x}\right )}{2 n+1}-e^x \left (\, _2F_1\left (1,-\frac{1}{2 n};1-\frac{1}{2 n};-e^{2 n x}\right )+e^{2 x} \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-e^{2 n x}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int \sinh \left ( x \right ) \tanh \left ( nx \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} - \frac{1}{2} \, \int \frac{2 \,{\left (e^{\left (2 \, x\right )} - 1\right )}}{e^{\left (2 \, n x + x\right )} + e^{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sinh \left (x\right ) \tanh \left (n x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \tanh{\left (n x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (x\right ) \tanh \left (n x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]