Optimal. Leaf size=64 \[ \frac {\sinh ^2(a+b x)}{4 b^3}-\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {x^2}{4 b} \]
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Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5372, 3310, 30} \[ \frac {\sinh ^2(a+b x)}{4 b^3}-\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {x^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3310
Rule 5372
Rubi steps
\begin {align*} \int x^2 \cosh (a+b x) \sinh (a+b x) \, dx &=\frac {x^2 \sinh ^2(a+b x)}{2 b}-\frac {\int x \sinh ^2(a+b x) \, dx}{b}\\ &=-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {\int x \, dx}{2 b}\\ &=\frac {x^2}{4 b}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 39, normalized size = 0.61 \[ \frac {\left (2 b^2 x^2+1\right ) \cosh (2 (a+b x))-2 b x \sinh (2 (a+b x))}{8 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 62, normalized size = 0.97 \[ -\frac {4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - {\left (2 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{2}}{8 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 57, normalized size = 0.89 \[ \frac {{\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 114, normalized size = 1.78 \[ \frac {\frac {\left (b x +a \right )^{2} \left (\cosh ^{2}\left (b x +a \right )\right )}{2}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{4}-2 a \left (\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{2}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}-\frac {b x}{4}-\frac {a}{4}\right )+\frac {\left (\cosh ^{2}\left (b x +a \right )\right ) a^{2}}{2}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 64, normalized size = 1.00 \[ \frac {{\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{16 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 46, normalized size = 0.72 \[ \frac {\frac {\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2}-b\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+b^2\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 75, normalized size = 1.17 \[ \begin {cases} \frac {x^{2} \sinh ^{2}{\left (a + b x \right )}}{4 b} + \frac {x^{2} \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac {x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b^{2}} + \frac {\sinh ^{2}{\left (a + b x \right )}}{4 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh {\relax (a )} \cosh {\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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