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 = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 2633
Rule 5373
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*}
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Mathematica [A] time = 0.15, 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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fricas [A] time = 0.72, size = 74, normalized size = 1.64 \[ \frac {3 \, b x \cosh \left (b x + a\right )^{3} + 9 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 9 \, b x \cosh \left (b x + a\right ) - \sinh \left (b x + a\right )^{3} - 3 \, {\left (\cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )}{36 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 76, normalized size = 1.69 \[ \frac {{\left (3 \, b x - 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{72 \, b^{2}} + \frac {{\left (b x - 1\right )} e^{\left (b x + a\right )}}{8 \, b^{2}} + \frac {{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} + \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 56, normalized size = 1.24 \[ \frac {\frac {\left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{3}-\frac {2 \sinh \left (b x +a \right )}{9}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{9}-\frac {a \left (\cosh ^{3}\left (b x +a \right )\right )}{3}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 84, normalized size = 1.87 \[ \frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{72 \, b^{2}} + \frac {{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{2}} + \frac {{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} + \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 41, normalized size = 0.91 \[ -\frac {-3\,b\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3+\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (a+b\,x\right )}^2+2\,\mathrm {sinh}\left (a+b\,x\right )}{9\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 61, normalized size = 1.36 \[ \begin {cases} \frac {x \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac {\sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \sinh {\relax (a )} \cosh ^{2}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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