Optimal. Leaf size=53 \[ -\frac {1}{4} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{8} \text {Chi}(4 b x) \sinh (4 a)-\frac {1}{4} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{8} \cosh (4 a) \text {Shi}(4 b x) \]
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Rubi [A]
time = 0.10, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5556, 3384,
3379, 3382} \begin {gather*} -\frac {1}{4} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{8} \sinh (4 a) \text {Chi}(4 b x)-\frac {1}{4} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{8} \cosh (4 a) \text {Shi}(4 b x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rubi steps
\begin {align*} \int \frac {\cosh (a+b x) \sinh ^3(a+b x)}{x} \, dx &=\int \left (-\frac {\sinh (2 a+2 b x)}{4 x}+\frac {\sinh (4 a+4 b x)}{8 x}\right ) \, dx\\ &=\frac {1}{8} \int \frac {\sinh (4 a+4 b x)}{x} \, dx-\frac {1}{4} \int \frac {\sinh (2 a+2 b x)}{x} \, dx\\ &=-\left (\frac {1}{4} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx\right )+\frac {1}{8} \cosh (4 a) \int \frac {\sinh (4 b x)}{x} \, dx-\frac {1}{4} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx+\frac {1}{8} \sinh (4 a) \int \frac {\cosh (4 b x)}{x} \, dx\\ &=-\frac {1}{4} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{8} \text {Chi}(4 b x) \sinh (4 a)-\frac {1}{4} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{8} \cosh (4 a) \text {Shi}(4 b x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 47, normalized size = 0.89 \begin {gather*} \frac {1}{8} (-2 \text {Chi}(2 b x) \sinh (2 a)+\text {Chi}(4 b x) \sinh (4 a)-2 \cosh (2 a) \text {Shi}(2 b x)+\cosh (4 a) \text {Shi}(4 b x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.29, size = 50, normalized size = 0.94
method | result | size |
risch | \(\frac {{\mathrm e}^{-4 a} \expIntegral \left (1, 4 b x \right )}{16}-\frac {{\mathrm e}^{-2 a} \expIntegral \left (1, 2 b x \right )}{8}+\frac {{\mathrm e}^{2 a} \expIntegral \left (1, -2 b x \right )}{8}-\frac {{\mathrm e}^{4 a} \expIntegral \left (1, -4 b x \right )}{16}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 45, normalized size = 0.85 \begin {gather*} \frac {1}{16} \, {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} - \frac {1}{8} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac {1}{16} \, {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 73, normalized size = 1.38 \begin {gather*} \frac {1}{16} \, {\left ({\rm Ei}\left (4 \, b x\right ) - {\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) - \frac {1}{8} \, {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \frac {1}{16} \, {\left ({\rm Ei}\left (4 \, b x\right ) + {\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) - \frac {1}{8} \, {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right ) \end {gather*}
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 \begin {gather*} \int \frac {\sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 45, normalized size = 0.85 \begin {gather*} \frac {1}{16} \, {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} - \frac {1}{8} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac {1}{16} \, {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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