Optimal. Leaf size=80 \[ \frac {1}{4} b \cosh (a) \text {Chi}(b x)+\frac {3}{4} b \cosh (3 a) \text {Chi}(3 b x)-\frac {\sinh (a+b x)}{4 x}-\frac {\sinh (3 a+3 b x)}{4 x}+\frac {1}{4} b \sinh (a) \text {Shi}(b x)+\frac {3}{4} b \sinh (3 a) \text {Shi}(3 b x) \]
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Rubi [A]
time = 0.14, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5556, 3378,
3384, 3379, 3382} \begin {gather*} \frac {1}{4} b \cosh (a) \text {Chi}(b x)+\frac {3}{4} b \cosh (3 a) \text {Chi}(3 b x)+\frac {1}{4} b \sinh (a) \text {Shi}(b x)+\frac {3}{4} b \sinh (3 a) \text {Shi}(3 b x)-\frac {\sinh (a+b x)}{4 x}-\frac {\sinh (3 a+3 b x)}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rubi steps
\begin {align*} \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^2} \, dx &=\int \left (\frac {\sinh (a+b x)}{4 x^2}+\frac {\sinh (3 a+3 b x)}{4 x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sinh (a+b x)}{x^2} \, dx+\frac {1}{4} \int \frac {\sinh (3 a+3 b x)}{x^2} \, dx\\ &=-\frac {\sinh (a+b x)}{4 x}-\frac {\sinh (3 a+3 b x)}{4 x}+\frac {1}{4} b \int \frac {\cosh (a+b x)}{x} \, dx+\frac {1}{4} (3 b) \int \frac {\cosh (3 a+3 b x)}{x} \, dx\\ &=-\frac {\sinh (a+b x)}{4 x}-\frac {\sinh (3 a+3 b x)}{4 x}+\frac {1}{4} (b \cosh (a)) \int \frac {\cosh (b x)}{x} \, dx+\frac {1}{4} (3 b \cosh (3 a)) \int \frac {\cosh (3 b x)}{x} \, dx+\frac {1}{4} (b \sinh (a)) \int \frac {\sinh (b x)}{x} \, dx+\frac {1}{4} (3 b \sinh (3 a)) \int \frac {\sinh (3 b x)}{x} \, dx\\ &=\frac {1}{4} b \cosh (a) \text {Chi}(b x)+\frac {3}{4} b \cosh (3 a) \text {Chi}(3 b x)-\frac {\sinh (a+b x)}{4 x}-\frac {\sinh (3 a+3 b x)}{4 x}+\frac {1}{4} b \sinh (a) \text {Shi}(b x)+\frac {3}{4} b \sinh (3 a) \text {Shi}(3 b x)\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 70, normalized size = 0.88 \begin {gather*} \frac {b x \cosh (a) \text {Chi}(b x)+3 b x \cosh (3 a) \text {Chi}(3 b x)-\sinh (a+b x)-\sinh (3 (a+b x))+b x \sinh (a) \text {Shi}(b x)+3 b x \sinh (3 a) \text {Shi}(3 b x)}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.86, size = 104, normalized size = 1.30
method | result | size |
risch | \(\frac {{\mathrm e}^{-3 b x -3 a}}{8 x}-\frac {3 b \,{\mathrm e}^{-3 a} \expIntegral \left (1, 3 b x \right )}{8}+\frac {{\mathrm e}^{-b x -a}}{8 x}-\frac {b \,{\mathrm e}^{-a} \expIntegral \left (1, b x \right )}{8}-\frac {{\mathrm e}^{b x +a}}{8 x}-\frac {b \,{\mathrm e}^{a} \expIntegral \left (1, -b x \right )}{8}-\frac {{\mathrm e}^{3 b x +3 a}}{8 x}-\frac {3 b \,{\mathrm e}^{3 a} \expIntegral \left (1, -3 b x \right )}{8}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 50, normalized size = 0.62 \begin {gather*} \frac {3}{8} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x\right ) + \frac {1}{8} \, b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + \frac {1}{8} \, b e^{a} \Gamma \left (-1, -b x\right ) + \frac {3}{8} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 124, normalized size = 1.55 \begin {gather*} -\frac {2 \, \sinh \left (b x + a\right )^{3} - 3 \, {\left (b x {\rm Ei}\left (3 \, b x\right ) + b x {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) - {\left (b x {\rm Ei}\left (b x\right ) + b x {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) - 3 \, {\left (b x {\rm Ei}\left (3 \, b x\right ) - b x {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) - {\left (b x {\rm Ei}\left (b x\right ) - b x {\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right )}{8 \, x} \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 {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 90, normalized size = 1.12 \begin {gather*} \frac {3 \, b x {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 3 \, b x {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + b x {\rm Ei}\left (b x\right ) e^{a} - e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{8 \, x} \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 = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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