Optimal. Leaf size=27 \[ \frac {1}{2} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x) \]
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Rubi [A] time = 0.07, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5448, 12, 3303, 3298, 3301} \[ \frac {1}{2} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rubi steps
\begin {align*} \int \frac {\cosh (a+b x) \sinh (a+b x)}{x} \, dx &=\int \frac {\sinh (2 a+2 b x)}{2 x} \, dx\\ &=\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x} \, dx\\ &=\frac {1}{2} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx\\ &=\frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 25, normalized size = 0.93 \[ \frac {1}{2} (\sinh (2 a) \text {Chi}(2 b x)+\cosh (2 a) \text {Shi}(2 b x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 37, normalized size = 1.37 \[ \frac {1}{4} \, {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \frac {1}{4} \, {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 23, normalized size = 0.85 \[ \frac {1}{4} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{4} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 26, normalized size = 0.96 \[ \frac {{\mathrm e}^{-2 a} \Ei \left (1, 2 b x \right )}{4}-\frac {{\mathrm e}^{2 a} \Ei \left (1, -2 b x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 23, normalized size = 0.85 \[ \frac {1}{4} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{4} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )}{x} \,d x \]
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 \[ \int \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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