Optimal. Leaf size=94 \[ \frac{\sinh (a+b x)}{8 b^2}+\frac{\sinh (3 a+3 b x)}{144 b^2}-\frac{\sinh (5 a+5 b x)}{400 b^2}-\frac{x \cosh (a+b x)}{8 b}-\frac{x \cosh (3 a+3 b x)}{48 b}+\frac{x \cosh (5 a+5 b x)}{80 b} \]
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Rubi [A] time = 0.0903493, antiderivative size = 94, normalized size of antiderivative = 1., 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, 2637} \[ \frac{\sinh (a+b x)}{8 b^2}+\frac{\sinh (3 a+3 b x)}{144 b^2}-\frac{\sinh (5 a+5 b x)}{400 b^2}-\frac{x \cosh (a+b x)}{8 b}-\frac{x \cosh (3 a+3 b x)}{48 b}+\frac{x \cosh (5 a+5 b x)}{80 b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{1}{8} x \sinh (a+b x)-\frac{1}{16} x \sinh (3 a+3 b x)+\frac{1}{16} x \sinh (5 a+5 b x)\right ) \, dx\\ &=-\left (\frac{1}{16} \int x \sinh (3 a+3 b x) \, dx\right )+\frac{1}{16} \int x \sinh (5 a+5 b x) \, dx-\frac{1}{8} \int x \sinh (a+b x) \, dx\\ &=-\frac{x \cosh (a+b x)}{8 b}-\frac{x \cosh (3 a+3 b x)}{48 b}+\frac{x \cosh (5 a+5 b x)}{80 b}-\frac{\int \cosh (5 a+5 b x) \, dx}{80 b}+\frac{\int \cosh (3 a+3 b x) \, dx}{48 b}+\frac{\int \cosh (a+b x) \, dx}{8 b}\\ &=-\frac{x \cosh (a+b x)}{8 b}-\frac{x \cosh (3 a+3 b x)}{48 b}+\frac{x \cosh (5 a+5 b x)}{80 b}+\frac{\sinh (a+b x)}{8 b^2}+\frac{\sinh (3 a+3 b x)}{144 b^2}-\frac{\sinh (5 a+5 b x)}{400 b^2}\\ \end{align*}
Mathematica [A] time = 0.188888, size = 70, normalized size = 0.74 \[ \frac{450 \sinh (a+b x)+25 \sinh (3 (a+b x))-9 \sinh (5 (a+b x))-450 b x \cosh (a+b x)-75 b x \cosh (3 (a+b x))+45 b x \cosh (5 (a+b x))}{3600 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 149, normalized size = 1.6 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{5}}-{\frac{ \left ( 2\,bx+2\,a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{15}}-{\frac{ \left ( 2\,bx+2\,a \right ) \cosh \left ( bx+a \right ) }{15}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}\sinh \left ( bx+a \right ) }{25}}+{\frac{26\,\sinh \left ( bx+a \right ) }{225}}+{\frac{13\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{225}}-a \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{5}}-{\frac{2\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{15}}-{\frac{2\,\cosh \left ( bx+a \right ) }{15}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20229, size = 174, normalized size = 1.85 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69211, size = 433, normalized size = 4.61 \begin{align*} \frac{45 \, b x \cosh \left (b x + a\right )^{5} + 225 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 75 \, b x \cosh \left (b x + a\right )^{3} - 9 \, \sinh \left (b x + a\right )^{5} - 5 \,{\left (18 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right )^{3} - 450 \, b x \cosh \left (b x + a\right ) + 225 \,{\left (2 \, b x \cosh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 15 \,{\left (3 \, \cosh \left (b x + a\right )^{4} - 5 \, \cosh \left (b x + a\right )^{2} - 30\right )} \sinh \left (b x + a\right )}{3600 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.18373, size = 112, normalized size = 1.19 \begin{align*} \begin{cases} \frac{x \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac{2 x \cosh ^{5}{\left (a + b x \right )}}{15 b} + \frac{26 \sinh ^{5}{\left (a + b x \right )}}{225 b^{2}} - \frac{13 \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{45 b^{2}} + \frac{2 \sinh{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{15 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh ^{3}{\left (a \right )} \cosh ^{2}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1768, size = 157, normalized size = 1.67 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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