Optimal. Leaf size=74 \[ \frac{b \cosh (c+d x) (b \text{csch}(c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};-\sinh ^2(c+d x)\right )}{d (1-n) \sqrt{\cosh ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0346077, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ \frac{b \cosh (c+d x) (b \text{csch}(c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};-\sinh ^2(c+d x)\right )}{d (1-n) \sqrt{\cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int (b \text{csch}(c+d x))^n \, dx &=(b \text{csch}(c+d x))^n \left (\frac{\sinh (c+d x)}{b}\right )^n \int \left (\frac{\sinh (c+d x)}{b}\right )^{-n} \, dx\\ &=\frac{\cosh (c+d x) (b \text{csch}(c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};-\sinh ^2(c+d x)\right ) \sinh (c+d x)}{d (1-n) \sqrt{\cosh ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0971341, size = 67, normalized size = 0.91 \[ -\frac{\sinh (c+d x) \cosh (c+d x) \left (-\sinh ^2(c+d x)\right )^{\frac{n-1}{2}} (b \text{csch}(c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cosh ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.214, size = 0, normalized size = 0. \begin{align*} \int \left ( b{\rm csch} \left (dx+c\right ) \right ) ^{n}\, 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*} \int \left (b \operatorname{csch}\left (d x + c\right )\right )^{n}\,{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 (\left (b \operatorname{csch}\left (d x + c\right )\right )^{n}, 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 \left (b \operatorname{csch}{\left (c + d x \right )}\right )^{n}\, 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 \left (b \operatorname{csch}\left (d x + c\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]