Optimal. Leaf size=47 \[ \frac {9}{4} \sinh (a) \text {Chi}(b x)+\frac {1}{4} \sinh (3 a) \text {Chi}(3 b x)+\frac {9}{4} \cosh (a) \text {Shi}(b x)+\frac {1}{4} \cosh (3 a) \text {Shi}(3 b x) \]
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Rubi [A] time = 0.46, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6742, 3303, 3298, 3301, 5448} \[ \frac {9}{4} \sinh (a) \text {Chi}(b x)+\frac {1}{4} \sinh (3 a) \text {Chi}(3 b x)+\frac {9}{4} \cosh (a) \text {Shi}(b x)+\frac {1}{4} \cosh (3 a) \text {Shi}(3 b x) \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (2+\cosh ^2(a+b x)\right ) \sinh (a+b x)}{x} \, dx &=\int \left (\frac {2 \sinh (a+b x)}{x}+\frac {\cosh ^2(a+b x) \sinh (a+b x)}{x}\right ) \, dx\\ &=2 \int \frac {\sinh (a+b x)}{x} \, dx+\int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x} \, dx\\ &=(2 \cosh (a)) \int \frac {\sinh (b x)}{x} \, dx+(2 \sinh (a)) \int \frac {\cosh (b x)}{x} \, dx+\int \left (\frac {\sinh (a+b x)}{4 x}+\frac {\sinh (3 a+3 b x)}{4 x}\right ) \, dx\\ &=2 \text {Chi}(b x) \sinh (a)+2 \cosh (a) \text {Shi}(b x)+\frac {1}{4} \int \frac {\sinh (a+b x)}{x} \, dx+\frac {1}{4} \int \frac {\sinh (3 a+3 b x)}{x} \, dx\\ &=2 \text {Chi}(b x) \sinh (a)+2 \cosh (a) \text {Shi}(b x)+\frac {1}{4} \cosh (a) \int \frac {\sinh (b x)}{x} \, dx+\frac {1}{4} \cosh (3 a) \int \frac {\sinh (3 b x)}{x} \, dx+\frac {1}{4} \sinh (a) \int \frac {\cosh (b x)}{x} \, dx+\frac {1}{4} \sinh (3 a) \int \frac {\cosh (3 b x)}{x} \, dx\\ &=\frac {9}{4} \text {Chi}(b x) \sinh (a)+\frac {1}{4} \text {Chi}(3 b x) \sinh (3 a)+\frac {9}{4} \cosh (a) \text {Shi}(b x)+\frac {1}{4} \cosh (3 a) \text {Shi}(3 b x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 41, normalized size = 0.87 \[ \frac {1}{4} (9 \sinh (a) \text {Chi}(b x)+\sinh (3 a) \text {Chi}(3 b x)+9 \cosh (a) \text {Shi}(b x)+\cosh (3 a) \text {Shi}(3 b x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 67, normalized size = 1.43 \[ \frac {1}{8} \, {\left ({\rm Ei}\left (3 \, b x\right ) - {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + \frac {9}{8} \, {\left ({\rm Ei}\left (b x\right ) - {\rm Ei}\left (-b x\right )\right )} \cosh \relax (a) + \frac {1}{8} \, {\left ({\rm Ei}\left (3 \, b x\right ) + {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) + \frac {9}{8} \, {\left ({\rm Ei}\left (b x\right ) + {\rm Ei}\left (-b x\right )\right )} \sinh \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 42, normalized size = 0.89 \[ \frac {1}{8} \, {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - \frac {9}{8} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - \frac {1}{8} \, {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + \frac {9}{8} \, {\rm Ei}\left (b x\right ) e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 47, normalized size = 1.00 \[ \frac {{\mathrm e}^{-3 a} \Ei \left (1, 3 b x \right )}{8}+\frac {9 \,{\mathrm e}^{-a} \Ei \left (1, b x \right )}{8}-\frac {9 \,{\mathrm e}^{a} \Ei \left (1, -b x \right )}{8}-\frac {{\mathrm e}^{3 a} \Ei \left (1, -3 b x \right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 42, normalized size = 0.89 \[ \frac {1}{8} \, {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} - \frac {9}{8} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - \frac {1}{8} \, {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + \frac {9}{8} \, {\rm Ei}\left (b x\right ) e^{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {sinh}\left (a+b\,x\right )\,\left ({\mathrm {cosh}\left (a+b\,x\right )}^2+2\right )}{x} \,d x \]
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 \[ \int \frac {\left (\cosh ^{2}{\left (a + b x \right )} + 2\right ) \sinh {\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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