Optimal. Leaf size=92 \[ \frac{4}{3} b^3 \sinh (4 a) \text{Chi}(4 b x)+\frac{4}{3} b^3 \cosh (4 a) \text{Shi}(4 b x)-\frac{b^2 \cosh (4 a+4 b x)}{3 x}-\frac{b \sinh (4 a+4 b x)}{12 x^2}-\frac{\cosh (4 a+4 b x)}{24 x^3}+\frac{1}{24 x^3} \]
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Rubi [A] time = 0.169529, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{4}{3} b^3 \sinh (4 a) \text{Chi}(4 b x)+\frac{4}{3} b^3 \cosh (4 a) \text{Shi}(4 b x)-\frac{b^2 \cosh (4 a+4 b x)}{3 x}-\frac{b \sinh (4 a+4 b x)}{12 x^2}-\frac{\cosh (4 a+4 b x)}{24 x^3}+\frac{1}{24 x^3} \]
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^4} \, dx &=\int \left (-\frac{1}{8 x^4}+\frac{\cosh (4 a+4 b x)}{8 x^4}\right ) \, dx\\ &=\frac{1}{24 x^3}+\frac{1}{8} \int \frac{\cosh (4 a+4 b x)}{x^4} \, dx\\ &=\frac{1}{24 x^3}-\frac{\cosh (4 a+4 b x)}{24 x^3}+\frac{1}{6} b \int \frac{\sinh (4 a+4 b x)}{x^3} \, dx\\ &=\frac{1}{24 x^3}-\frac{\cosh (4 a+4 b x)}{24 x^3}-\frac{b \sinh (4 a+4 b x)}{12 x^2}+\frac{1}{3} b^2 \int \frac{\cosh (4 a+4 b x)}{x^2} \, dx\\ &=\frac{1}{24 x^3}-\frac{\cosh (4 a+4 b x)}{24 x^3}-\frac{b^2 \cosh (4 a+4 b x)}{3 x}-\frac{b \sinh (4 a+4 b x)}{12 x^2}+\frac{1}{3} \left (4 b^3\right ) \int \frac{\sinh (4 a+4 b x)}{x} \, dx\\ &=\frac{1}{24 x^3}-\frac{\cosh (4 a+4 b x)}{24 x^3}-\frac{b^2 \cosh (4 a+4 b x)}{3 x}-\frac{b \sinh (4 a+4 b x)}{12 x^2}+\frac{1}{3} \left (4 b^3 \cosh (4 a)\right ) \int \frac{\sinh (4 b x)}{x} \, dx+\frac{1}{3} \left (4 b^3 \sinh (4 a)\right ) \int \frac{\cosh (4 b x)}{x} \, dx\\ &=\frac{1}{24 x^3}-\frac{\cosh (4 a+4 b x)}{24 x^3}-\frac{b^2 \cosh (4 a+4 b x)}{3 x}+\frac{4}{3} b^3 \text{Chi}(4 b x) \sinh (4 a)-\frac{b \sinh (4 a+4 b x)}{12 x^2}+\frac{4}{3} b^3 \cosh (4 a) \text{Shi}(4 b x)\\ \end{align*}
Mathematica [A] time = 0.191029, size = 79, normalized size = 0.86 \[ -\frac{-32 b^3 x^3 \sinh (4 a) \text{Chi}(4 b x)-32 b^3 x^3 \cosh (4 a) \text{Shi}(4 b x)+8 b^2 x^2 \cosh (4 (a+b x))+2 b x \sinh (4 (a+b x))+\cosh (4 (a+b x))-1}{24 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 129, normalized size = 1.4 \begin{align*}{\frac{1}{24\,{x}^{3}}}-{\frac{{b}^{2}{{\rm e}^{-4\,bx-4\,a}}}{6\,x}}+{\frac{b{{\rm e}^{-4\,bx-4\,a}}}{24\,{x}^{2}}}-{\frac{{{\rm e}^{-4\,bx-4\,a}}}{48\,{x}^{3}}}+{\frac{2\,{b}^{3}{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{3}}-{\frac{{{\rm e}^{4\,bx+4\,a}}}{48\,{x}^{3}}}-{\frac{b{{\rm e}^{4\,bx+4\,a}}}{24\,{x}^{2}}}-{\frac{{b}^{2}{{\rm e}^{4\,bx+4\,a}}}{6\,x}}-{\frac{2\,{b}^{3}{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33638, size = 49, normalized size = 0.53 \begin{align*} -4 \, b^{3} e^{\left (-4 \, a\right )} \Gamma \left (-3, 4 \, b x\right ) + 4 \, b^{3} e^{\left (4 \, a\right )} \Gamma \left (-3, -4 \, b x\right ) + \frac{1}{24 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8123, size = 436, normalized size = 4.74 \begin{align*} -\frac{8 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 8 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} +{\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} + 6 \,{\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4} - 16 \,{\left (b^{3} x^{3}{\rm Ei}\left (4 \, b x\right ) - b^{3} x^{3}{\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) - 16 \,{\left (b^{3} x^{3}{\rm Ei}\left (4 \, b x\right ) + b^{3} x^{3}{\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) - 1}{24 \, x^{3}} \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^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1609, size = 166, normalized size = 1.8 \begin{align*} \frac{32 \, b^{3} x^{3}{\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} - 32 \, b^{3} x^{3}{\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - 8 \, b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} - 8 \, b^{2} x^{2} e^{\left (-4 \, b x - 4 \, a\right )} - 2 \, b x e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b x e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + 2}{48 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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