Optimal. Leaf size=52 \[ \frac{1}{2} b \sinh (4 a) \text{Chi}(4 b x)+\frac{1}{2} b \cosh (4 a) \text{Shi}(4 b x)-\frac{\cosh (4 a+4 b x)}{8 x}+\frac{1}{8 x} \]
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Rubi [A] time = 0.111748, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5448, 3297, 3303, 3298, 3301} \[ \frac{1}{2} b \sinh (4 a) \text{Chi}(4 b x)+\frac{1}{2} b \cosh (4 a) \text{Shi}(4 b x)-\frac{\cosh (4 a+4 b x)}{8 x}+\frac{1}{8 x} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh ^2(a+b x) \sinh ^2(a+b x)}{x^2} \, dx &=\int \left (-\frac{1}{8 x^2}+\frac{\cosh (4 a+4 b x)}{8 x^2}\right ) \, dx\\ &=\frac{1}{8 x}+\frac{1}{8} \int \frac{\cosh (4 a+4 b x)}{x^2} \, dx\\ &=\frac{1}{8 x}-\frac{\cosh (4 a+4 b x)}{8 x}+\frac{1}{2} b \int \frac{\sinh (4 a+4 b x)}{x} \, dx\\ &=\frac{1}{8 x}-\frac{\cosh (4 a+4 b x)}{8 x}+\frac{1}{2} (b \cosh (4 a)) \int \frac{\sinh (4 b x)}{x} \, dx+\frac{1}{2} (b \sinh (4 a)) \int \frac{\cosh (4 b x)}{x} \, dx\\ &=\frac{1}{8 x}-\frac{\cosh (4 a+4 b x)}{8 x}+\frac{1}{2} b \text{Chi}(4 b x) \sinh (4 a)+\frac{1}{2} b \cosh (4 a) \text{Shi}(4 b x)\\ \end{align*}
Mathematica [A] time = 0.0932268, size = 45, normalized size = 0.87 \[ \frac{4 b x \sinh (4 a) \text{Chi}(4 b x)+4 b x \cosh (4 a) \text{Shi}(4 b x)-\cosh (4 (a+b x))+1}{8 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 61, normalized size = 1.2 \begin{align*}{\frac{1}{8\,x}}-{\frac{{{\rm e}^{-4\,bx-4\,a}}}{16\,x}}+{\frac{b{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{4}}-{\frac{{{\rm e}^{4\,bx+4\,a}}}{16\,x}}-{\frac{b{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3941, size = 43, normalized size = 0.83 \begin{align*} -\frac{1}{4} \, b e^{\left (-4 \, a\right )} \Gamma \left (-1, 4 \, b x\right ) + \frac{1}{4} \, b e^{\left (4 \, a\right )} \Gamma \left (-1, -4 \, b x\right ) + \frac{1}{8 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87895, size = 240, normalized size = 4.62 \begin{align*} -\frac{\cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{4} - 2 \,{\left (b x{\rm Ei}\left (4 \, b x\right ) - b x{\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) - 2 \,{\left (b x{\rm Ei}\left (4 \, b x\right ) + b x{\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) - 1}{8 \, x} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1669, size = 74, normalized size = 1.42 \begin{align*} \frac{4 \, b x{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} - 4 \, b x{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + 2}{16 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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