Optimal. Leaf size=37 \[ \frac{1}{4} \cosh (2 a) \text{Chi}\left (2 b x^2\right )+\frac{1}{4} \sinh (2 a) \text{Shi}\left (2 b x^2\right )+\frac{\log (x)}{2} \]
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Rubi [A] time = 0.0607286, antiderivative size = 37, 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 = {5341, 5319, 5317, 5316} \[ \frac{1}{4} \cosh (2 a) \text{Chi}\left (2 b x^2\right )+\frac{1}{4} \sinh (2 a) \text{Shi}\left (2 b x^2\right )+\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 5341
Rule 5319
Rule 5317
Rule 5316
Rubi steps
\begin{align*} \int \frac{\cosh ^2\left (a+b x^2\right )}{x} \, dx &=\int \left (\frac{1}{2 x}+\frac{\cosh \left (2 a+2 b x^2\right )}{2 x}\right ) \, dx\\ &=\frac{\log (x)}{2}+\frac{1}{2} \int \frac{\cosh \left (2 a+2 b x^2\right )}{x} \, dx\\ &=\frac{\log (x)}{2}+\frac{1}{2} \cosh (2 a) \int \frac{\cosh \left (2 b x^2\right )}{x} \, dx+\frac{1}{2} \sinh (2 a) \int \frac{\sinh \left (2 b x^2\right )}{x} \, dx\\ &=\frac{1}{4} \cosh (2 a) \text{Chi}\left (2 b x^2\right )+\frac{\log (x)}{2}+\frac{1}{4} \sinh (2 a) \text{Shi}\left (2 b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0196797, size = 33, normalized size = 0.89 \[ \frac{1}{4} \left (\cosh (2 a) \text{Chi}\left (2 b x^2\right )+\sinh (2 a) \text{Shi}\left (2 b x^2\right )+2 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 34, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{2}}-{\frac{{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,b{x}^{2} \right ) }{8}}-{\frac{{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,b{x}^{2} \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3177, size = 42, normalized size = 1.14 \begin{align*} \frac{1}{8} \,{\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} + \frac{1}{8} \,{\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} + \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 = 1.84838, size = 138, normalized size = 3.73 \begin{align*} \frac{1}{8} \,{\left ({\rm Ei}\left (2 \, b x^{2}\right ) +{\rm Ei}\left (-2 \, b x^{2}\right )\right )} \cosh \left (2 \, a\right ) + \frac{1}{8} \,{\left ({\rm Ei}\left (2 \, b x^{2}\right ) -{\rm Ei}\left (-2 \, b x^{2}\right )\right )} \sinh \left (2 \, a\right ) + \frac{1}{2} \, \log \left (x\right ) \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^{2} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30248, size = 47, normalized size = 1.27 \begin{align*} \frac{1}{8} \,{\rm Ei}\left (2 \, b x^{2}\right ) e^{\left (2 \, a\right )} + \frac{1}{8} \,{\rm Ei}\left (-2 \, b x^{2}\right ) e^{\left (-2 \, a\right )} + \frac{1}{4} \, \log \left (b x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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