Optimal. Leaf size=45 \[ -\frac{\sinh ^3(a+b x)}{9 b^2}-\frac{\sinh (a+b x)}{3 b^2}+\frac{x \cosh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0313528, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5373, 2633} \[ -\frac{\sinh ^3(a+b x)}{9 b^2}-\frac{\sinh (a+b x)}{3 b^2}+\frac{x \cosh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 5373
Rule 2633
Rubi steps
\begin{align*} \int x \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac{x \cosh ^3(a+b x)}{3 b}-\frac{\int \cosh ^3(a+b x) \, dx}{3 b}\\ &=\frac{x \cosh ^3(a+b x)}{3 b}-\frac{i \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{3 b^2}\\ &=\frac{x \cosh ^3(a+b x)}{3 b}-\frac{\sinh (a+b x)}{3 b^2}-\frac{\sinh ^3(a+b x)}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.136738, size = 46, normalized size = 1.02 \[ -\frac{9 \sinh (a+b x)+\sinh (3 (a+b x))-9 b x \cosh (a+b x)-3 b x \cosh (3 (a+b x))}{36 b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 92, normalized size = 2. \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}+{\frac{ \left ( bx+a \right ) \cosh \left ( bx+a \right ) }{3}}-{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{9}}-{\frac{2\,\sinh \left ( bx+a \right ) }{9}}-a \left ({\frac{\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}}+{\frac{\cosh \left ( bx+a \right ) }{3}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19734, size = 113, normalized size = 2.51 \begin{align*} \frac{{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{72 \, b^{2}} + \frac{{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{2}} + \frac{{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} + \frac{{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30944, size = 205, normalized size = 4.56 \begin{align*} \frac{3 \, b x \cosh \left (b x + a\right )^{3} + 9 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 9 \, b x \cosh \left (b x + a\right ) - \sinh \left (b x + a\right )^{3} - 3 \,{\left (\cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )}{36 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13427, size = 61, normalized size = 1.36 \begin{align*} \begin{cases} \frac{x \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac{2 \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac{\sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh{\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 = 1.16303, size = 103, normalized size = 2.29 \begin{align*} \frac{{\left (3 \, b x - 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{72 \, b^{2}} + \frac{{\left (b x - 1\right )} e^{\left (b x + a\right )}}{8 \, b^{2}} + \frac{{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} + \frac{{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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