Optimal. Leaf size=89 \[ -\frac {3}{16} b \cosh (2 a) \text {Chi}(2 b x)+\frac {3}{16} b \cosh (6 a) \text {Chi}(6 b x)-\frac {3}{16} b \sinh (2 a) \text {Shi}(2 b x)+\frac {3}{16} b \sinh (6 a) \text {Shi}(6 b x)+\frac {3 \sinh (2 a+2 b x)}{32 x}-\frac {\sinh (6 a+6 b x)}{32 x} \]
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Rubi [A] time = 0.20, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {3}{16} b \cosh (2 a) \text {Chi}(2 b x)+\frac {3}{16} b \cosh (6 a) \text {Chi}(6 b x)-\frac {3}{16} b \sinh (2 a) \text {Shi}(2 b x)+\frac {3}{16} b \sinh (6 a) \text {Shi}(6 b x)+\frac {3 \sinh (2 a+2 b x)}{32 x}-\frac {\sinh (6 a+6 b x)}{32 x} \]
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 ^3(a+b x) \sinh ^3(a+b x)}{x^2} \, dx &=\int \left (-\frac {3 \sinh (2 a+2 b x)}{32 x^2}+\frac {\sinh (6 a+6 b x)}{32 x^2}\right ) \, dx\\ &=\frac {1}{32} \int \frac {\sinh (6 a+6 b x)}{x^2} \, dx-\frac {3}{32} \int \frac {\sinh (2 a+2 b x)}{x^2} \, dx\\ &=\frac {3 \sinh (2 a+2 b x)}{32 x}-\frac {\sinh (6 a+6 b x)}{32 x}-\frac {1}{16} (3 b) \int \frac {\cosh (2 a+2 b x)}{x} \, dx+\frac {1}{16} (3 b) \int \frac {\cosh (6 a+6 b x)}{x} \, dx\\ &=\frac {3 \sinh (2 a+2 b x)}{32 x}-\frac {\sinh (6 a+6 b x)}{32 x}-\frac {1}{16} (3 b \cosh (2 a)) \int \frac {\cosh (2 b x)}{x} \, dx+\frac {1}{16} (3 b \cosh (6 a)) \int \frac {\cosh (6 b x)}{x} \, dx-\frac {1}{16} (3 b \sinh (2 a)) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{16} (3 b \sinh (6 a)) \int \frac {\sinh (6 b x)}{x} \, dx\\ &=-\frac {3}{16} b \cosh (2 a) \text {Chi}(2 b x)+\frac {3}{16} b \cosh (6 a) \text {Chi}(6 b x)+\frac {3 \sinh (2 a+2 b x)}{32 x}-\frac {\sinh (6 a+6 b x)}{32 x}-\frac {3}{16} b \sinh (2 a) \text {Shi}(2 b x)+\frac {3}{16} b \sinh (6 a) \text {Shi}(6 b x)\\ \end {align*}
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Mathematica [A] time = 0.24, size = 78, normalized size = 0.88 \[ -\frac {6 b x \cosh (2 a) \text {Chi}(2 b x)-6 b x \cosh (6 a) \text {Chi}(6 b x)+6 b x \sinh (2 a) \text {Shi}(2 b x)-6 b x \sinh (6 a) \text {Shi}(6 b x)-3 \sinh (2 (a+b x))+\sinh (6 (a+b x))}{32 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 159, normalized size = 1.79 \[ -\frac {20 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - 3 \, {\left (b x {\rm Ei}\left (6 \, b x\right ) + b x {\rm Ei}\left (-6 \, b x\right )\right )} \cosh \left (6 \, a\right ) + 3 \, {\left (b x {\rm Ei}\left (2 \, b x\right ) + b x {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + 6 \, {\left (\cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 3 \, {\left (b x {\rm Ei}\left (6 \, b x\right ) - b x {\rm Ei}\left (-6 \, b x\right )\right )} \sinh \left (6 \, a\right ) + 3 \, {\left (b x {\rm Ei}\left (2 \, b x\right ) - b x {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{32 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 100, normalized size = 1.12 \[ \frac {6 \, b x {\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - 6 \, b x {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - 6 \, b x {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} + 6 \, b x {\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} - e^{\left (6 \, b x + 6 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 3 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{64 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 110, normalized size = 1.24 \[ \frac {{\mathrm e}^{-6 b x -6 a}}{64 x}-\frac {3 b \,{\mathrm e}^{-6 a} \Ei \left (1, 6 b x \right )}{32}-\frac {3 \,{\mathrm e}^{-2 b x -2 a}}{64 x}+\frac {3 b \,{\mathrm e}^{-2 a} \Ei \left (1, 2 b x \right )}{32}+\frac {3 \,{\mathrm e}^{2 b x +2 a}}{64 x}+\frac {3 b \,{\mathrm e}^{2 a} \Ei \left (1, -2 b x \right )}{32}-\frac {{\mathrm e}^{6 b x +6 a}}{64 x}-\frac {3 b \,{\mathrm e}^{6 a} \Ei \left (1, -6 b x \right )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 53, normalized size = 0.60 \[ \frac {3}{32} \, b e^{\left (-6 \, a\right )} \Gamma \left (-1, 6 \, b x\right ) - \frac {3}{32} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x\right ) - \frac {3}{32} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x\right ) + \frac {3}{32} \, b e^{\left (6 \, a\right )} \Gamma \left (-1, -6 \, b x\right ) \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{x^2} \,d x \]
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 \[ \int \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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