Optimal. Leaf size=53 \[ -\frac {3}{32} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{32} \sinh (6 a) \text {Chi}(6 b x)-\frac {3}{32} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{32} \cosh (6 a) \text {Shi}(6 b x) \]
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Rubi [A] time = 0.16, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5448, 3303, 3298, 3301} \[ -\frac {3}{32} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{32} \sinh (6 a) \text {Chi}(6 b x)-\frac {3}{32} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{32} \cosh (6 a) \text {Shi}(6 b x) \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rubi steps
\begin {align*} \int \frac {\cosh ^3(a+b x) \sinh ^3(a+b x)}{x} \, dx &=\int \left (-\frac {3 \sinh (2 a+2 b x)}{32 x}+\frac {\sinh (6 a+6 b x)}{32 x}\right ) \, dx\\ &=\frac {1}{32} \int \frac {\sinh (6 a+6 b x)}{x} \, dx-\frac {3}{32} \int \frac {\sinh (2 a+2 b x)}{x} \, dx\\ &=-\left (\frac {1}{32} (3 \cosh (2 a)) \int \frac {\sinh (2 b x)}{x} \, dx\right )+\frac {1}{32} \cosh (6 a) \int \frac {\sinh (6 b x)}{x} \, dx-\frac {1}{32} (3 \sinh (2 a)) \int \frac {\cosh (2 b x)}{x} \, dx+\frac {1}{32} \sinh (6 a) \int \frac {\cosh (6 b x)}{x} \, dx\\ &=-\frac {3}{32} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{32} \text {Chi}(6 b x) \sinh (6 a)-\frac {3}{32} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{32} \cosh (6 a) \text {Shi}(6 b x)\\ \end {align*}
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Mathematica [A] time = 0.18, size = 47, normalized size = 0.89 \[ \frac {1}{32} (\sinh (6 a) \text {Chi}(6 b x)-6 \sinh (a) \cosh (a) \text {Chi}(2 b x)-3 \cosh (2 a) \text {Shi}(2 b x)+\cosh (6 a) \text {Shi}(6 b x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 73, normalized size = 1.38 \[ \frac {1}{64} \, {\left ({\rm Ei}\left (6 \, b x\right ) - {\rm Ei}\left (-6 \, b x\right )\right )} \cosh \left (6 \, a\right ) - \frac {3}{64} \, {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \frac {1}{64} \, {\left ({\rm Ei}\left (6 \, b x\right ) + {\rm Ei}\left (-6 \, b x\right )\right )} \sinh \left (6 \, a\right ) - \frac {3}{64} \, {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 45, normalized size = 0.85 \[ \frac {1}{64} \, {\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - \frac {3}{64} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + \frac {3}{64} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac {1}{64} \, {\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 50, normalized size = 0.94 \[ \frac {{\mathrm e}^{-6 a} \Ei \left (1, 6 b x \right )}{64}-\frac {3 \,{\mathrm e}^{-2 a} \Ei \left (1, 2 b x \right )}{64}+\frac {3 \,{\mathrm e}^{2 a} \Ei \left (1, -2 b x \right )}{64}-\frac {{\mathrm e}^{6 a} \Ei \left (1, -6 b x \right )}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 45, normalized size = 0.85 \[ \frac {1}{64} \, {\rm Ei}\left (6 \, b x\right ) e^{\left (6 \, a\right )} - \frac {3}{64} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + \frac {3}{64} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac {1}{64} \, {\rm Ei}\left (-6 \, b x\right ) e^{\left (-6 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{x} \,d x \]
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 \[ \int \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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