Optimal. Leaf size=94 \[ \frac {\cosh (a+b x)}{8 b^2}-\frac {\cosh (3 a+3 b x)}{144 b^2}-\frac {\cosh (5 a+5 b x)}{400 b^2}-\frac {x \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{48 b}+\frac {x \sinh (5 a+5 b x)}{80 b} \]
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Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5448, 3296, 2638} \[ \frac {\cosh (a+b x)}{8 b^2}-\frac {\cosh (3 a+3 b x)}{144 b^2}-\frac {\cosh (5 a+5 b x)}{400 b^2}-\frac {x \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{48 b}+\frac {x \sinh (5 a+5 b x)}{80 b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5448
Rubi steps
\begin {align*} \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {1}{8} x \cosh (a+b x)+\frac {1}{16} x \cosh (3 a+3 b x)+\frac {1}{16} x \cosh (5 a+5 b x)\right ) \, dx\\ &=\frac {1}{16} \int x \cosh (3 a+3 b x) \, dx+\frac {1}{16} \int x \cosh (5 a+5 b x) \, dx-\frac {1}{8} \int x \cosh (a+b x) \, dx\\ &=-\frac {x \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{48 b}+\frac {x \sinh (5 a+5 b x)}{80 b}-\frac {\int \sinh (5 a+5 b x) \, dx}{80 b}-\frac {\int \sinh (3 a+3 b x) \, dx}{48 b}+\frac {\int \sinh (a+b x) \, dx}{8 b}\\ &=\frac {\cosh (a+b x)}{8 b^2}-\frac {\cosh (3 a+3 b x)}{144 b^2}-\frac {\cosh (5 a+5 b x)}{400 b^2}-\frac {x \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{48 b}+\frac {x \sinh (5 a+5 b x)}{80 b}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 70, normalized size = 0.74 \[ \frac {-450 b x \sinh (a+b x)+75 b x \sinh (3 (a+b x))+45 b x \sinh (5 (a+b x))+450 \cosh (a+b x)-25 \cosh (3 (a+b x))-9 \cosh (5 (a+b x))}{3600 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 152, normalized size = 1.62 \[ \frac {45 \, b x \sinh \left (b x + a\right )^{5} - 9 \, \cosh \left (b x + a\right )^{5} - 45 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 75 \, {\left (6 \, b x \cosh \left (b x + a\right )^{2} + b x\right )} \sinh \left (b x + a\right )^{3} - 25 \, \cosh \left (b x + a\right )^{3} - 15 \, {\left (6 \, \cosh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 225 \, {\left (b x \cosh \left (b x + a\right )^{4} + b x \cosh \left (b x + a\right )^{2} - 2 \, b x\right )} \sinh \left (b x + a\right ) + 450 \, \cosh \left (b x + a\right )}{3600 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 116, normalized size = 1.23 \[ \frac {{\left (5 \, b x - 1\right )} e^{\left (5 \, b x + 5 \, a\right )}}{800 \, b^{2}} + \frac {{\left (3 \, b x - 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{288 \, b^{2}} - \frac {{\left (b x - 1\right )} e^{\left (b x + a\right )}}{16 \, b^{2}} + \frac {{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{16 \, b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{288 \, b^{2}} - \frac {{\left (5 \, b x + 1\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{800 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 129, normalized size = 1.37 \[ \frac {\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{15}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{15}-\frac {\left (\cosh ^{5}\left (b x +a \right )\right )}{25}+\frac {2 \cosh \left (b x +a \right )}{15}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{45}-a \left (\frac {\sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{5}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 129, normalized size = 1.37 \[ \frac {{\left (5 \, b x e^{\left (5 \, a\right )} - e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )}}{800 \, b^{2}} + \frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{288 \, b^{2}} - \frac {{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{16 \, b^{2}} + \frac {{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{16 \, b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{288 \, b^{2}} - \frac {{\left (5 \, b x + 1\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{800 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 83, normalized size = 0.88 \[ -\frac {b\,\left (\frac {2\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^5}{15}-\frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3}\right )-\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^4}{15}-\frac {26\,{\mathrm {cosh}\left (a+b\,x\right )}^5}{225}+\frac {13\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{45}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.82, size = 112, normalized size = 1.19 \[ \begin {cases} - \frac {2 x \sinh ^{5}{\left (a + b x \right )}}{15 b} + \frac {x \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{15 b^{2}} - \frac {13 \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{45 b^{2}} + \frac {26 \cosh ^{5}{\left (a + b x \right )}}{225 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \sinh ^{2}{\relax (a )} \cosh ^{3}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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