Optimal. Leaf size=111 \[ \frac{2 \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac{6 i \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{5 b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]
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Rubi [A] time = 0.0652123, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2635, 2640, 2639} \[ \frac{2 \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac{6 i \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{5 b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sinh ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sinh ^{\frac{5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n}-\frac{3 \operatorname{Subst}\left (\int \sqrt{\sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n}-\frac{\left (3 \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname{Subst}\left (\int \sqrt{i \sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}}\\ &=\frac{6 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}}{5 b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}}+\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n}\\ \end{align*}
Mathematica [A] time = 0.0883684, size = 96, normalized size = 0.86 \[ \frac{\sinh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-6 \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right )\right |2\right )}{5 b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 227, normalized size = 2.1 \begin{align*}{\frac{1}{bn\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) } \left ( -{\frac{6\,\sqrt{2}}{5}\sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{1+i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) }+{\frac{3\,\sqrt{2}}{5}\sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{1+i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) }+{\frac{2\, \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}}{5}}-{\frac{2\, \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{5}} \right ){\frac{1}{\sqrt{\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}}} \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 \frac{\sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}\,{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 (\frac{\sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}, 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 \frac{\sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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