Optimal. Leaf size=25 \[ \frac{\cosh (a) \text{Chi}\left (b x^n\right )}{n}+\frac{\sinh (a) \text{Shi}\left (b x^n\right )}{n} \]
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Rubi [A] time = 0.0382602, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5319, 5317, 5316} \[ \frac{\cosh (a) \text{Chi}\left (b x^n\right )}{n}+\frac{\sinh (a) \text{Shi}\left (b x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 5319
Rule 5317
Rule 5316
Rubi steps
\begin{align*} \int \frac{\cosh \left (a+b x^n\right )}{x} \, dx &=\cosh (a) \int \frac{\cosh \left (b x^n\right )}{x} \, dx+\sinh (a) \int \frac{\sinh \left (b x^n\right )}{x} \, dx\\ &=\frac{\cosh (a) \text{Chi}\left (b x^n\right )}{n}+\frac{\sinh (a) \text{Shi}\left (b x^n\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0214314, size = 23, normalized size = 0.92 \[ \frac{\cosh (a) \text{Chi}\left (b x^n\right )+\sinh (a) \text{Shi}\left (b x^n\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 33, normalized size = 1.3 \begin{align*} -{\frac{{{\rm e}^{-a}}{\it Ei} \left ( 1,b{x}^{n} \right ) }{2\,n}}-{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-b{x}^{n} \right ) }{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15545, size = 41, normalized size = 1.64 \begin{align*} \frac{{\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{2 \, n} + \frac{{\rm Ei}\left (b x^{n}\right ) e^{a}}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8384, size = 178, normalized size = 7.12 \begin{align*} \frac{{\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 (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 )}{2 \, 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{\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 )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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