Optimal. Leaf size=94 \[ \frac{x^{-n} (e x)^n \cosh \left (c+d x^n\right ) \left (b \sinh \left (c+d x^n\right )\right )^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};-\sinh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt{\cosh ^2\left (c+d x^n\right )}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.103774, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5322, 5320, 2643} \[ \frac{x^{-n} (e x)^n \cosh \left (c+d x^n\right ) \left (b \sinh \left (c+d x^n\right )\right )^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};-\sinh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt{\cosh ^2\left (c+d x^n\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5322
Rule 5320
Rule 2643
Rubi steps
\begin{align*} \int (e x)^{-1+n} \left (b \sinh \left (c+d x^n\right )\right )^p \, dx &=\frac{\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (b \sinh \left (c+d x^n\right )\right )^p \, dx}{e}\\ &=\frac{\left (x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int (b \sinh (c+d x))^p \, dx,x,x^n\right )}{e n}\\ &=\frac{x^{-n} (e x)^n \cosh \left (c+d x^n\right ) \, _2F_1\left (\frac{1}{2},\frac{1+p}{2};\frac{3+p}{2};-\sinh ^2\left (c+d x^n\right )\right ) \left (b \sinh \left (c+d x^n\right )\right )^{1+p}}{b d e n (1+p) \sqrt{\cosh ^2\left (c+d x^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.138132, size = 93, normalized size = 0.99 \[ -\frac{x^{-n} (e x)^n \sinh \left (2 \left (c+d x^n\right )\right ) \left (-\sinh ^2\left (c+d x^n\right )\right )^{\frac{1}{2} (-p-1)} \left (b \sinh \left (c+d x^n\right )\right )^p \, _2F_1\left (\frac{1}{2},\frac{1-p}{2};\frac{3}{2};\cosh ^2\left (d x^n+c\right )\right )}{2 d e n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.838, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+n} \left ( b\sinh \left ( c+d{x}^{n} \right ) \right ) ^{p}\, 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 (e x\right )^{n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}\,{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 (e x\right )^{n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (e x\right )^{n - 1} \left (b \sinh \left (d x^{n} + c\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]