Optimal. Leaf size=43 \[ \frac{\cosh (2 a) \text{Chi}\left (2 b x^n\right )}{2 n}+\frac{\sinh (2 a) \text{Shi}\left (2 b x^n\right )}{2 n}+\frac{\log (x)}{2} \]
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Rubi [A] time = 0.0630822, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5363, 5319, 5317, 5316} \[ \frac{\cosh (2 a) \text{Chi}\left (2 b x^n\right )}{2 n}+\frac{\sinh (2 a) \text{Shi}\left (2 b x^n\right )}{2 n}+\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 5363
Rule 5319
Rule 5317
Rule 5316
Rubi steps
\begin{align*} \int \frac{\cosh ^2\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac{1}{2 x}+\frac{\cosh \left (2 a+2 b x^n\right )}{2 x}\right ) \, dx\\ &=\frac{\log (x)}{2}+\frac{1}{2} \int \frac{\cosh \left (2 a+2 b x^n\right )}{x} \, dx\\ &=\frac{\log (x)}{2}+\frac{1}{2} \cosh (2 a) \int \frac{\cosh \left (2 b x^n\right )}{x} \, dx+\frac{1}{2} \sinh (2 a) \int \frac{\sinh \left (2 b x^n\right )}{x} \, dx\\ &=\frac{\cosh (2 a) \text{Chi}\left (2 b x^n\right )}{2 n}+\frac{\log (x)}{2}+\frac{\sinh (2 a) \text{Shi}\left (2 b x^n\right )}{2 n}\\ \end{align*}
Mathematica [A] time = 0.0284064, size = 36, normalized size = 0.84 \[ \frac{\cosh (2 a) \text{Chi}\left (2 b x^n\right )+\sinh (2 a) \text{Shi}\left (2 b x^n\right )+n \log (x)}{2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 40, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{2}}-{\frac{{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,b{x}^{n} \right ) }{4\,n}}-{\frac{{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,b{x}^{n} \right ) }{4\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14735, size = 50, normalized size = 1.16 \begin{align*} \frac{{\rm Ei}\left (2 \, b x^{n}\right ) e^{\left (2 \, a\right )}}{4 \, n} + \frac{{\rm Ei}\left (-2 \, b x^{n}\right ) e^{\left (-2 \, a\right )}}{4 \, n} + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8367, size = 217, normalized size = 5.05 \begin{align*} \frac{{\left (\cosh \left (2 \, a\right ) + \sinh \left (2 \, a\right )\right )}{\rm Ei}\left (2 \, b \cosh \left (n \log \left (x\right )\right ) + 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) +{\left (\cosh \left (2 \, a\right ) - \sinh \left (2 \, a\right )\right )}{\rm Ei}\left (-2 \, b \cosh \left (n \log \left (x\right )\right ) - 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 2 \, n \log \left (x\right )}{4 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{2}{\left (a + b x^{n} \right )}}{x}\, dx \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 x^{n} + a\right )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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