Optimal. Leaf size=73 \[ -\frac {1}{8} \sinh (a) \text {Chi}(b x)-\frac {1}{16} \sinh (3 a) \text {Chi}(3 b x)+\frac {1}{16} \sinh (5 a) \text {Chi}(5 b x)-\frac {1}{8} \cosh (a) \text {Shi}(b x)-\frac {1}{16} \cosh (3 a) \text {Shi}(3 b x)+\frac {1}{16} \cosh (5 a) \text {Shi}(5 b x) \]
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Rubi [A] time = 0.18, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5448, 3303, 3298, 3301} \[ -\frac {1}{8} \sinh (a) \text {Chi}(b x)-\frac {1}{16} \sinh (3 a) \text {Chi}(3 b x)+\frac {1}{16} \sinh (5 a) \text {Chi}(5 b x)-\frac {1}{8} \cosh (a) \text {Shi}(b x)-\frac {1}{16} \cosh (3 a) \text {Shi}(3 b x)+\frac {1}{16} \cosh (5 a) \text {Shi}(5 b x) \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rubi steps
\begin {align*} \int \frac {\cosh ^2(a+b x) \sinh ^3(a+b x)}{x} \, dx &=\int \left (-\frac {\sinh (a+b x)}{8 x}-\frac {\sinh (3 a+3 b x)}{16 x}+\frac {\sinh (5 a+5 b x)}{16 x}\right ) \, dx\\ &=-\left (\frac {1}{16} \int \frac {\sinh (3 a+3 b x)}{x} \, dx\right )+\frac {1}{16} \int \frac {\sinh (5 a+5 b x)}{x} \, dx-\frac {1}{8} \int \frac {\sinh (a+b x)}{x} \, dx\\ &=-\left (\frac {1}{8} \cosh (a) \int \frac {\sinh (b x)}{x} \, dx\right )-\frac {1}{16} \cosh (3 a) \int \frac {\sinh (3 b x)}{x} \, dx+\frac {1}{16} \cosh (5 a) \int \frac {\sinh (5 b x)}{x} \, dx-\frac {1}{8} \sinh (a) \int \frac {\cosh (b x)}{x} \, dx-\frac {1}{16} \sinh (3 a) \int \frac {\cosh (3 b x)}{x} \, dx+\frac {1}{16} \sinh (5 a) \int \frac {\cosh (5 b x)}{x} \, dx\\ &=-\frac {1}{8} \text {Chi}(b x) \sinh (a)-\frac {1}{16} \text {Chi}(3 b x) \sinh (3 a)+\frac {1}{16} \text {Chi}(5 b x) \sinh (5 a)-\frac {1}{8} \cosh (a) \text {Shi}(b x)-\frac {1}{16} \cosh (3 a) \text {Shi}(3 b x)+\frac {1}{16} \cosh (5 a) \text {Shi}(5 b x)\\ \end {align*}
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Mathematica [A] time = 0.12, size = 63, normalized size = 0.86 \[ \frac {1}{16} (-2 \sinh (a) \text {Chi}(b x)-\sinh (3 a) \text {Chi}(3 b x)+\sinh (5 a) \text {Chi}(5 b x)-2 \cosh (a) \text {Shi}(b x)-\cosh (3 a) \text {Shi}(3 b x)+\cosh (5 a) \text {Shi}(5 b x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 103, normalized size = 1.41 \[ \frac {1}{32} \, {\left ({\rm Ei}\left (5 \, b x\right ) - {\rm Ei}\left (-5 \, b x\right )\right )} \cosh \left (5 \, a\right ) - \frac {1}{32} \, {\left ({\rm Ei}\left (3 \, b x\right ) - {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) - \frac {1}{16} \, {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \relax (a) + \frac {1}{32} \, {\left ({\rm Ei}\left (5 \, b x\right ) + {\rm Ei}\left (-5 \, b x\right )\right )} \sinh \left (5 \, a\right ) - \frac {1}{32} \, {\left ({\rm Ei}\left (3 \, b x\right ) + {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) - \frac {1}{16} \, {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \sinh \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 64, normalized size = 0.88 \[ \frac {1}{32} \, {\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} - \frac {1}{32} \, {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + \frac {1}{16} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac {1}{32} \, {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - \frac {1}{32} \, {\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - \frac {1}{16} \, {\rm Ei}\left (b x\right ) e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 71, normalized size = 0.97 \[ \frac {{\mathrm e}^{-5 a} \Ei \left (1, 5 b x \right )}{32}-\frac {{\mathrm e}^{-3 a} \Ei \left (1, 3 b x \right )}{32}-\frac {{\mathrm e}^{-a} \Ei \left (1, b x \right )}{16}+\frac {{\mathrm e}^{a} \Ei \left (1, -b x \right )}{16}+\frac {{\mathrm e}^{3 a} \Ei \left (1, -3 b x \right )}{32}-\frac {{\mathrm e}^{5 a} \Ei \left (1, -5 b x \right )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 64, normalized size = 0.88 \[ \frac {1}{32} \, {\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} - \frac {1}{32} \, {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + \frac {1}{16} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac {1}{32} \, {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - \frac {1}{32} \, {\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - \frac {1}{16} \, {\rm Ei}\left (b x\right ) e^{a} \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{x} \,d x \]
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 \[ \int \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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