Optimal. Leaf size=67 \[ \frac{3 \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{\cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{\sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n} \]
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Rubi [A] time = 0.100218, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5363, 5319, 5317, 5316} \[ \frac{3 \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{\cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{\sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n} \]
Antiderivative was successfully verified.
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Rule 5363
Rule 5319
Rule 5317
Rule 5316
Rubi steps
\begin{align*} \int \frac{\cosh ^3\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac{3 \cosh \left (a+b x^n\right )}{4 x}+\frac{\cosh \left (3 a+3 b x^n\right )}{4 x}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\cosh \left (3 a+3 b x^n\right )}{x} \, dx+\frac{3}{4} \int \frac{\cosh \left (a+b x^n\right )}{x} \, dx\\ &=\frac{1}{4} (3 \cosh (a)) \int \frac{\cosh \left (b x^n\right )}{x} \, dx+\frac{1}{4} \cosh (3 a) \int \frac{\cosh \left (3 b x^n\right )}{x} \, dx+\frac{1}{4} (3 \sinh (a)) \int \frac{\sinh \left (b x^n\right )}{x} \, dx+\frac{1}{4} \sinh (3 a) \int \frac{\sinh \left (3 b x^n\right )}{x} \, dx\\ &=\frac{3 \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{\cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{\sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n}\\ \end{align*}
Mathematica [A] time = 0.0480971, size = 52, normalized size = 0.78 \[ \frac{3 \cosh (a) \text{Chi}\left (b x^n\right )+\cosh (3 a) \text{Chi}\left (3 b x^n\right )+3 \sinh (a) \text{Shi}\left (b x^n\right )+\sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 67, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,b{x}^{n} \right ) }{8\,n}}-{\frac{3\,{{\rm e}^{-a}}{\it Ei} \left ( 1,b{x}^{n} \right ) }{8\,n}}-{\frac{{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,b{x}^{n} \right ) }{8\,n}}-{\frac{3\,{{\rm e}^{a}}{\it Ei} \left ( 1,-b{x}^{n} \right ) }{8\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27732, size = 84, normalized size = 1.25 \begin{align*} \frac{{\rm Ei}\left (3 \, b x^{n}\right ) e^{\left (3 \, a\right )}}{8 \, n} + \frac{3 \,{\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{8 \, n} + \frac{{\rm Ei}\left (-3 \, b x^{n}\right ) e^{\left (-3 \, a\right )}}{8 \, n} + \frac{3 \,{\rm Ei}\left (b x^{n}\right ) e^{a}}{8 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76263, size = 374, normalized size = 5.58 \begin{align*} \frac{{\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )}{\rm Ei}\left (3 \, b \cosh \left (n \log \left (x\right )\right ) + 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \,{\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )}{\rm Ei}\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \,{\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )}{\rm Ei}\left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right )\right ) +{\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )}{\rm Ei}\left (-3 \, b \cosh \left (n \log \left (x\right )\right ) - 3 \, b \sinh \left (n \log \left (x\right )\right )\right )}{8 \, 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 ^{3}{\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 )^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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