Optimal. Leaf size=95 \[ -\frac{x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (b \cosh \left (c+d x^n\right )\right )^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cosh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt{-\sinh ^2\left (c+d x^n\right )}} \]
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Rubi [A] time = 0.10428, antiderivative size = 95, 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 = {5323, 5321, 2643} \[ -\frac{x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (b \cosh \left (c+d x^n\right )\right )^{p+1} \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cosh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt{-\sinh ^2\left (c+d x^n\right )}} \]
Antiderivative was successfully verified.
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Rule 5323
Rule 5321
Rule 2643
Rubi steps
\begin{align*} \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx &=\frac{\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx}{e}\\ &=\frac{\left (x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int (b \cosh (c+d x))^p \, dx,x,x^n\right )}{e n}\\ &=-\frac{x^{-n} (e x)^n \left (b \cosh \left (c+d x^n\right )\right )^{1+p} \, _2F_1\left (\frac{1}{2},\frac{1+p}{2};\frac{3+p}{2};\cosh ^2\left (c+d x^n\right )\right ) \sinh \left (c+d x^n\right )}{b d e n (1+p) \sqrt{-\sinh ^2\left (c+d x^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.140312, size = 94, normalized size = 0.99 \[ -\frac{x^{-n} (e x)^n \sinh \left (2 \left (c+d x^n\right )\right ) \left (b \cosh \left (c+d x^n\right )\right )^p \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cosh ^2\left (d x^n+c\right )\right )}{2 d e n (p+1) \sqrt{-\sinh ^2\left (c+d x^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.802, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+n} \left ( b\cosh \left ( c+d{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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