Optimal. Leaf size=131 \[ \frac{\sqrt{2} x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac{a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} F_1\left (\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};\frac{1}{2} \left (1-\cosh \left (d x^n+c\right )\right ),\frac{b \left (1-\cosh \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt{\cosh \left (c+d x^n\right )+1}} \]
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Rubi [A] time = 0.186944, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {5323, 5321, 2665, 139, 138} \[ \frac{\sqrt{2} x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac{a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} F_1\left (\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};\frac{1}{2} \left (1-\cosh \left (d x^n+c\right )\right ),\frac{b \left (1-\cosh \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt{\cosh \left (c+d x^n\right )+1}} \]
Antiderivative was successfully verified.
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Rule 5323
Rule 5321
Rule 2665
Rule 139
Rule 138
Rubi steps
\begin{align*} \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx &=\frac{\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx}{e}\\ &=\frac{\left (x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int (a+b \cosh (c+d x))^p \, dx,x,x^n\right )}{e n}\\ &=-\frac{\left (x^{-n} (e x)^n \sinh \left (c+d x^n\right )\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^p}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cosh \left (c+d x^n\right )\right )}{d e n \sqrt{1-\cosh \left (c+d x^n\right )} \sqrt{1+\cosh \left (c+d x^n\right )}}\\ &=-\frac{\left (x^{-n} (e x)^n \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (-\frac{a+b \cosh \left (c+d x^n\right )}{-a-b}\right )^{-p} \sinh \left (c+d x^n\right )\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^p}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cosh \left (c+d x^n\right )\right )}{d e n \sqrt{1-\cosh \left (c+d x^n\right )} \sqrt{1+\cosh \left (c+d x^n\right )}}\\ &=\frac{\sqrt{2} x^{-n} (e x)^n F_1\left (\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};\frac{1}{2} \left (1-\cosh \left (c+d x^n\right )\right ),\frac{b \left (1-\cosh \left (c+d x^n\right )\right )}{a+b}\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac{a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} \sinh \left (c+d x^n\right )}{d e n \sqrt{1+\cosh \left (c+d x^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.358825, size = 148, normalized size = 1.13 \[ \frac{x^{-n} (e x)^n \text{csch}\left (c+d x^n\right ) \sqrt{-\frac{b \left (\cosh \left (c+d x^n\right )-1\right )}{a+b}} \sqrt{\frac{b \left (\cosh \left (c+d x^n\right )+1\right )}{b-a}} \left (a+b \cosh \left (c+d x^n\right )\right )^{p+1} F_1\left (p+1;\frac{1}{2},\frac{1}{2};p+2;\frac{a+b \cosh \left (d x^n+c\right )}{a+b},\frac{a+b \cosh \left (d x^n+c\right )}{a-b}\right )}{b d e n (p+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.759, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+n} \left ( a+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 ) + a\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 ) + a\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 ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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