Optimal. Leaf size=92 \[ \frac {4}{3} b^3 \sinh (4 a) \text {Chi}(4 b x)+\frac {4}{3} b^3 \cosh (4 a) \text {Shi}(4 b x)-\frac {b^2 \cosh (4 a+4 b x)}{3 x}-\frac {\cosh (4 a+4 b x)}{24 x^3}-\frac {b \sinh (4 a+4 b x)}{12 x^2}+\frac {1}{24 x^3} \]
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Rubi [A] time = 0.17, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5448
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*}
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Mathematica [A] time = 0.20, 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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fricas [B] time = 1.11, size = 172, normalized size = 1.87 \[ -\frac {8 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 8 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} + 6 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4} - 16 \, {\left (b^{3} x^{3} {\rm Ei}\left (4 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) - 16 \, {\left (b^{3} x^{3} {\rm Ei}\left (4 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) - 1}{24 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 123, normalized size = 1.34 \[ \frac {32 \, b^{3} x^{3} {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} - 32 \, b^{3} x^{3} {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - 8 \, b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} - 8 \, b^{2} x^{2} e^{\left (-4 \, b x - 4 \, a\right )} - 2 \, b x e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b x e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + 2}{48 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 129, normalized size = 1.40 \[ \frac {1}{24 x^{3}}-\frac {b^{2} {\mathrm e}^{-4 b x -4 a}}{6 x}+\frac {b \,{\mathrm e}^{-4 b x -4 a}}{24 x^{2}}-\frac {{\mathrm e}^{-4 b x -4 a}}{48 x^{3}}+\frac {2 b^{3} {\mathrm e}^{-4 a} \Ei \left (1, 4 b x \right )}{3}-\frac {{\mathrm e}^{4 b x +4 a}}{48 x^{3}}-\frac {b \,{\mathrm e}^{4 b x +4 a}}{24 x^{2}}-\frac {b^{2} {\mathrm e}^{4 b x +4 a}}{6 x}-\frac {2 b^{3} {\mathrm e}^{4 a} \Ei \left (1, -4 b x \right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 36, normalized size = 0.39 \[ -4 \, b^{3} e^{\left (-4 \, a\right )} \Gamma \left (-3, 4 \, b x\right ) + 4 \, b^{3} e^{\left (4 \, a\right )} \Gamma \left (-3, -4 \, b x\right ) + \frac {1}{24 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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