Optimal. Leaf size=83 \[ \frac {2 \cosh ^3(a+b x)}{27 b^3}+\frac {4 \cosh (a+b x)}{9 b^3}-\frac {4 x \sinh (a+b x)}{9 b^2}-\frac {2 x \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2}+\frac {x^2 \cosh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5373, 3310, 3296, 2638} \[ -\frac {4 x \sinh (a+b x)}{9 b^2}+\frac {2 \cosh ^3(a+b x)}{27 b^3}+\frac {4 \cosh (a+b x)}{9 b^3}-\frac {2 x \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2}+\frac {x^2 \cosh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3310
Rule 5373
Rubi steps
\begin {align*} \int x^2 \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac {x^2 \cosh ^3(a+b x)}{3 b}-\frac {2 \int x \cosh ^3(a+b x) \, dx}{3 b}\\ &=\frac {2 \cosh ^3(a+b x)}{27 b^3}+\frac {x^2 \cosh ^3(a+b x)}{3 b}-\frac {2 x \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2}-\frac {4 \int x \cosh (a+b x) \, dx}{9 b}\\ &=\frac {2 \cosh ^3(a+b x)}{27 b^3}+\frac {x^2 \cosh ^3(a+b x)}{3 b}-\frac {4 x \sinh (a+b x)}{9 b^2}-\frac {2 x \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2}+\frac {4 \int \sinh (a+b x) \, dx}{9 b^2}\\ &=\frac {4 \cosh (a+b x)}{9 b^3}+\frac {2 \cosh ^3(a+b x)}{27 b^3}+\frac {x^2 \cosh ^3(a+b x)}{3 b}-\frac {4 x \sinh (a+b x)}{9 b^2}-\frac {2 x \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 65, normalized size = 0.78 \[ \frac {27 \left (b^2 x^2+2\right ) \cosh (a+b x)+\left (9 b^2 x^2+2\right ) \cosh (3 (a+b x))-6 b x (9 \sinh (a+b x)+\sinh (3 (a+b x)))}{108 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 105, normalized size = 1.27 \[ -\frac {6 \, b x \sinh \left (b x + a\right )^{3} - {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 27 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 18 \, {\left (b x \cosh \left (b x + a\right )^{2} + 3 \, b x\right )} \sinh \left (b x + a\right )}{108 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 108, normalized size = 1.30 \[ \frac {{\left (9 \, b^{2} x^{2} - 6 \, b x + 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{3}} + \frac {{\left (b^{2} x^{2} - 2 \, b x + 2\right )} e^{\left (b x + a\right )}}{8 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{8 \, b^{3}} + \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 131, normalized size = 1.58 \[ \frac {\frac {\left (b x +a \right )^{2} \left (\cosh ^{3}\left (b x +a \right )\right )}{3}-\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{9}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{9}+\frac {4 \cosh \left (b x +a \right )}{9}+\frac {2 \left (\cosh ^{3}\left (b x +a \right )\right )}{27}-2 a \left (\frac {\left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{3}-\frac {2 \sinh \left (b x +a \right )}{9}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{9}\right )+\frac {a^{2} \left (\cosh ^{3}\left (b x +a \right )\right )}{3}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 122, normalized size = 1.47 \[ \frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{216 \, b^{3}} + \frac {{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{8 \, b^{3}} + \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 69, normalized size = 0.83 \[ \frac {\frac {4\,\mathrm {cosh}\left (a+b\,x\right )}{9}-b\,\left (\frac {2\,x\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{9}+\frac {4\,x\,\mathrm {sinh}\left (a+b\,x\right )}{9}\right )+\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{27}+\frac {b^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3}}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.71, size = 105, normalized size = 1.27 \[ \begin {cases} \frac {x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {4 x \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac {2 x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} - \frac {4 \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{3}} + \frac {14 \cosh ^{3}{\left (a + b x \right )}}{27 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh {\relax (a )} \cosh ^{2}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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