Optimal. Leaf size=169 \[ -\frac {1}{8} b^3 \cosh (2 a) \text {Chi}(2 b x)+\frac {9}{8} b^3 \cosh (6 a) \text {Chi}(6 b x)-\frac {1}{8} b^3 \sinh (2 a) \text {Shi}(2 b x)+\frac {9}{8} b^3 \sinh (6 a) \text {Shi}(6 b x)+\frac {b^2 \sinh (2 a+2 b x)}{16 x}-\frac {3 b^2 \sinh (6 a+6 b x)}{16 x}+\frac {\sinh (2 a+2 b x)}{32 x^3}-\frac {\sinh (6 a+6 b x)}{96 x^3}+\frac {b \cosh (2 a+2 b x)}{32 x^2}-\frac {b \cosh (6 a+6 b x)}{32 x^2} \]
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Rubi [A] time = 0.32, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ -\frac {1}{8} b^3 \cosh (2 a) \text {Chi}(2 b x)+\frac {9}{8} b^3 \cosh (6 a) \text {Chi}(6 b x)-\frac {1}{8} b^3 \sinh (2 a) \text {Shi}(2 b x)+\frac {9}{8} b^3 \sinh (6 a) \text {Shi}(6 b x)+\frac {b^2 \sinh (2 a+2 b x)}{16 x}-\frac {3 b^2 \sinh (6 a+6 b x)}{16 x}+\frac {\sinh (2 a+2 b x)}{32 x^3}-\frac {\sinh (6 a+6 b x)}{96 x^3}+\frac {b \cosh (2 a+2 b x)}{32 x^2}-\frac {b \cosh (6 a+6 b x)}{32 x^2} \]
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^4} \, dx &=\int \left (-\frac {3 \sinh (2 a+2 b x)}{32 x^4}+\frac {\sinh (6 a+6 b x)}{32 x^4}\right ) \, dx\\ &=\frac {1}{32} \int \frac {\sinh (6 a+6 b x)}{x^4} \, dx-\frac {3}{32} \int \frac {\sinh (2 a+2 b x)}{x^4} \, dx\\ &=\frac {\sinh (2 a+2 b x)}{32 x^3}-\frac {\sinh (6 a+6 b x)}{96 x^3}-\frac {1}{16} b \int \frac {\cosh (2 a+2 b x)}{x^3} \, dx+\frac {1}{16} b \int \frac {\cosh (6 a+6 b x)}{x^3} \, dx\\ &=\frac {b \cosh (2 a+2 b x)}{32 x^2}-\frac {b \cosh (6 a+6 b x)}{32 x^2}+\frac {\sinh (2 a+2 b x)}{32 x^3}-\frac {\sinh (6 a+6 b x)}{96 x^3}-\frac {1}{16} b^2 \int \frac {\sinh (2 a+2 b x)}{x^2} \, dx+\frac {1}{16} \left (3 b^2\right ) \int \frac {\sinh (6 a+6 b x)}{x^2} \, dx\\ &=\frac {b \cosh (2 a+2 b x)}{32 x^2}-\frac {b \cosh (6 a+6 b x)}{32 x^2}+\frac {\sinh (2 a+2 b x)}{32 x^3}+\frac {b^2 \sinh (2 a+2 b x)}{16 x}-\frac {\sinh (6 a+6 b x)}{96 x^3}-\frac {3 b^2 \sinh (6 a+6 b x)}{16 x}-\frac {1}{8} b^3 \int \frac {\cosh (2 a+2 b x)}{x} \, dx+\frac {1}{8} \left (9 b^3\right ) \int \frac {\cosh (6 a+6 b x)}{x} \, dx\\ &=\frac {b \cosh (2 a+2 b x)}{32 x^2}-\frac {b \cosh (6 a+6 b x)}{32 x^2}+\frac {\sinh (2 a+2 b x)}{32 x^3}+\frac {b^2 \sinh (2 a+2 b x)}{16 x}-\frac {\sinh (6 a+6 b x)}{96 x^3}-\frac {3 b^2 \sinh (6 a+6 b x)}{16 x}-\frac {1}{8} \left (b^3 \cosh (2 a)\right ) \int \frac {\cosh (2 b x)}{x} \, dx+\frac {1}{8} \left (9 b^3 \cosh (6 a)\right ) \int \frac {\cosh (6 b x)}{x} \, dx-\frac {1}{8} \left (b^3 \sinh (2 a)\right ) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{8} \left (9 b^3 \sinh (6 a)\right ) \int \frac {\sinh (6 b x)}{x} \, dx\\ &=\frac {b \cosh (2 a+2 b x)}{32 x^2}-\frac {b \cosh (6 a+6 b x)}{32 x^2}-\frac {1}{8} b^3 \cosh (2 a) \text {Chi}(2 b x)+\frac {9}{8} b^3 \cosh (6 a) \text {Chi}(6 b x)+\frac {\sinh (2 a+2 b x)}{32 x^3}+\frac {b^2 \sinh (2 a+2 b x)}{16 x}-\frac {\sinh (6 a+6 b x)}{96 x^3}-\frac {3 b^2 \sinh (6 a+6 b x)}{16 x}-\frac {1}{8} b^3 \sinh (2 a) \text {Shi}(2 b x)+\frac {9}{8} b^3 \sinh (6 a) \text {Shi}(6 b x)\\ \end {align*}
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Mathematica [A] time = 0.34, size = 150, normalized size = 0.89 \[ -\frac {12 b^3 x^3 \cosh (2 a) \text {Chi}(2 b x)-108 b^3 x^3 \cosh (6 a) \text {Chi}(6 b x)+12 b^3 x^3 \sinh (2 a) \text {Shi}(2 b x)-108 b^3 x^3 \sinh (6 a) \text {Shi}(6 b x)-6 b^2 x^2 \sinh (2 (a+b x))+18 b^2 x^2 \sinh (6 (a+b x))-3 \sinh (2 (a+b x))+\sinh (6 (a+b x))-3 b x \cosh (2 (a+b x))+3 b x \cosh (6 (a+b x))}{96 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 315, normalized size = 1.86 \[ -\frac {3 \, b x \cosh \left (b x + a\right )^{6} + 45 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + 3 \, b x \sinh \left (b x + a\right )^{6} + 20 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 6 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - 3 \, b x \cosh \left (b x + a\right )^{2} + 3 \, {\left (15 \, b x \cosh \left (b x + a\right )^{4} - b x\right )} \sinh \left (b x + a\right )^{2} - 54 \, {\left (b^{3} x^{3} {\rm Ei}\left (6 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-6 \, b x\right )\right )} \cosh \left (6 \, a\right ) + 6 \, {\left (b^{3} x^{3} {\rm Ei}\left (2 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + 6 \, {\left ({\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{5} - {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 54 \, {\left (b^{3} x^{3} {\rm Ei}\left (6 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-6 \, b x\right )\right )} \sinh \left (6 \, a\right ) + 6 \, {\left (b^{3} x^{3} {\rm Ei}\left (2 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right )}{96 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 236, normalized size = 1.40 \[ \frac {108 \, b^{3} x^{3} {\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - 12 \, b^{3} x^{3} {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - 12 \, b^{3} x^{3} {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} + 108 \, b^{3} x^{3} {\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} - 18 \, b^{2} x^{2} e^{\left (6 \, b x + 6 \, a\right )} + 6 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (-2 \, b x - 2 \, a\right )} + 18 \, b^{2} x^{2} e^{\left (-6 \, b x - 6 \, a\right )} - 3 \, b x e^{\left (6 \, b x + 6 \, a\right )} + 3 \, b x e^{\left (2 \, b x + 2 \, a\right )} + 3 \, b x e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, b x e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (6 \, b x + 6 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 3 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{192 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 246, normalized size = 1.46 \[ \frac {3 b^{2} {\mathrm e}^{-6 b x -6 a}}{32 x}-\frac {b \,{\mathrm e}^{-6 b x -6 a}}{64 x^{2}}+\frac {{\mathrm e}^{-6 b x -6 a}}{192 x^{3}}-\frac {9 b^{3} {\mathrm e}^{-6 a} \Ei \left (1, 6 b x \right )}{16}-\frac {b^{2} {\mathrm e}^{-2 b x -2 a}}{32 x}+\frac {b \,{\mathrm e}^{-2 b x -2 a}}{64 x^{2}}-\frac {{\mathrm e}^{-2 b x -2 a}}{64 x^{3}}+\frac {b^{3} {\mathrm e}^{-2 a} \Ei \left (1, 2 b x \right )}{16}+\frac {{\mathrm e}^{2 b x +2 a}}{64 x^{3}}+\frac {b \,{\mathrm e}^{2 b x +2 a}}{64 x^{2}}+\frac {b^{2} {\mathrm e}^{2 b x +2 a}}{32 x}+\frac {b^{3} {\mathrm e}^{2 a} \Ei \left (1, -2 b x \right )}{16}-\frac {{\mathrm e}^{6 b x +6 a}}{192 x^{3}}-\frac {b \,{\mathrm e}^{6 b x +6 a}}{64 x^{2}}-\frac {3 b^{2} {\mathrm e}^{6 b x +6 a}}{32 x}-\frac {9 b^{3} {\mathrm e}^{6 a} \Ei \left (1, -6 b x \right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 61, normalized size = 0.36 \[ \frac {27}{8} \, b^{3} e^{\left (-6 \, a\right )} \Gamma \left (-3, 6 \, b x\right ) - \frac {3}{8} \, b^{3} e^{\left (-2 \, a\right )} \Gamma \left (-3, 2 \, b x\right ) - \frac {3}{8} \, b^{3} e^{\left (2 \, a\right )} \Gamma \left (-3, -2 \, b x\right ) + \frac {27}{8} \, b^{3} e^{\left (6 \, a\right )} \Gamma \left (-3, -6 \, b x\right ) \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{x^4} \,d x \]
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 \[ \int \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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