Optimal. Leaf size=67 \[ \frac{2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )}}{3 b n}-\frac{2 i \text{EllipticF}\left (\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{3 b n} \]
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Rubi [A] time = 0.0443683, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2635, 2641} \[ \frac{2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )}}{3 b n}-\frac{2 i F\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cosh ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cosh ^{\frac{3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\cosh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac{2 i F\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n}+\frac{2 \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end{align*}
Mathematica [C] time = 0.121836, size = 114, normalized size = 1.7 \[ \frac{2 \sqrt{\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.09, size = 237, normalized size = 3.5 \begin{align*}{\frac{2}{3\,nb}\sqrt{ \left ( 2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}} \left ( 4\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{5}-6\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}+\sqrt{- \left ( \sinh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cosh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) ,\sqrt{2} \right ) +2\,\cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ){\frac{1}{\sqrt{2\, \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}}}} \left ( \sinh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}}}} \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{\cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{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{\cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{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{\cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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