Optimal. Leaf size=33 \[ \frac{1}{8} \cosh (4 a) \text{Chi}(4 b x)+\frac{1}{8} \sinh (4 a) \text{Shi}(4 b x)-\frac{\log (x)}{8} \]
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Rubi [A] time = 0.0848538, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5448, 3303, 3298, 3301} \[ \frac{1}{8} \cosh (4 a) \text{Chi}(4 b x)+\frac{1}{8} \sinh (4 a) \text{Shi}(4 b x)-\frac{\log (x)}{8} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh ^2(a+b x) \sinh ^2(a+b x)}{x} \, dx &=\int \left (-\frac{1}{8 x}+\frac{\cosh (4 a+4 b x)}{8 x}\right ) \, dx\\ &=-\frac{\log (x)}{8}+\frac{1}{8} \int \frac{\cosh (4 a+4 b x)}{x} \, dx\\ &=-\frac{\log (x)}{8}+\frac{1}{8} \cosh (4 a) \int \frac{\cosh (4 b x)}{x} \, dx+\frac{1}{8} \sinh (4 a) \int \frac{\sinh (4 b x)}{x} \, dx\\ &=\frac{1}{8} \cosh (4 a) \text{Chi}(4 b x)-\frac{\log (x)}{8}+\frac{1}{8} \sinh (4 a) \text{Shi}(4 b x)\\ \end{align*}
Mathematica [A] time = 0.0974682, size = 32, normalized size = 0.97 \[ \frac{1}{8} (\cosh (4 a) \text{Chi}(4 b x)+\sinh (4 a) \text{Shi}(4 b x)-\log (2 b x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 30, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( x \right ) }{8}}-{\frac{{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{16}}-{\frac{{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26553, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{16} \,{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac{1}{16} \,{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - \frac{1}{8} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83623, size = 130, normalized size = 3.94 \begin{align*} \frac{1}{16} \,{\left ({\rm Ei}\left (4 \, b x\right ) +{\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) + \frac{1}{16} \,{\left ({\rm Ei}\left (4 \, b x\right ) -{\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) - \frac{1}{8} \, \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{\sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15044, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{16} \,{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac{1}{16} \,{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - \frac{1}{8} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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