Optimal. Leaf size=101 \[ \frac {\cosh ^4(a+b x)}{32 b^3}+\frac {3 \cosh ^2(a+b x)}{32 b^3}-\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{8 b^2}-\frac {3 x \sinh (a+b x) \cosh (a+b x)}{16 b^2}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {3 x^2}{32 b} \]
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Rubi [A] time = 0.08, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5373, 3310, 30} \[ \frac {\cosh ^4(a+b x)}{32 b^3}+\frac {3 \cosh ^2(a+b x)}{32 b^3}-\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{8 b^2}-\frac {3 x \sinh (a+b x) \cosh (a+b x)}{16 b^2}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {3 x^2}{32 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3310
Rule 5373
Rubi steps
\begin {align*} \int x^2 \cosh ^3(a+b x) \sinh (a+b x) \, dx &=\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {\int x \cosh ^4(a+b x) \, dx}{2 b}\\ &=\frac {\cosh ^4(a+b x)}{32 b^3}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {x \cosh ^3(a+b x) \sinh (a+b x)}{8 b^2}-\frac {3 \int x \cosh ^2(a+b x) \, dx}{8 b}\\ &=\frac {3 \cosh ^2(a+b x)}{32 b^3}+\frac {\cosh ^4(a+b x)}{32 b^3}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {3 x \cosh (a+b x) \sinh (a+b x)}{16 b^2}-\frac {x \cosh ^3(a+b x) \sinh (a+b x)}{8 b^2}-\frac {3 \int x \, dx}{16 b}\\ &=-\frac {3 x^2}{32 b}+\frac {3 \cosh ^2(a+b x)}{32 b^3}+\frac {\cosh ^4(a+b x)}{32 b^3}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {3 x \cosh (a+b x) \sinh (a+b x)}{16 b^2}-\frac {x \cosh ^3(a+b x) \sinh (a+b x)}{8 b^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 70, normalized size = 0.69 \[ \frac {16 \left (2 b^2 x^2+1\right ) \cosh (2 (a+b x))+\left (8 b^2 x^2+1\right ) \cosh (4 (a+b x))-4 b x (8 \sinh (2 (a+b x))+\sinh (4 (a+b x)))}{256 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 154, normalized size = 1.52 \[ -\frac {16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - {\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4} - 16 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (16 \, b^{2} x^{2} + 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} + 8\right )} \sinh \left (b x + a\right )^{2} + 16 \, {\left (b x \cosh \left (b x + a\right )^{3} + 4 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{256 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 113, normalized size = 1.12 \[ \frac {{\left (8 \, b^{2} x^{2} - 4 \, b x + 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{512 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{3}} + \frac {{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 161, normalized size = 1.59 \[ \frac {\frac {\left (b x +a \right )^{2} \left (\cosh ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{8}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{16}-\frac {3 \left (b x +a \right )^{2}}{32}+\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{32}+\frac {3 \left (\cosh ^{2}\left (b x +a \right )\right )}{32}-2 a \left (\frac {\left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{16}-\frac {3 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{32}-\frac {3 b x}{32}-\frac {3 a}{32}\right )+\frac {a^{2} \left (\cosh ^{4}\left (b x +a \right )\right )}{4}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 127, normalized size = 1.26 \[ \frac {{\left (8 \, b^{2} x^{2} e^{\left (4 \, a\right )} - 4 \, b x e^{\left (4 \, a\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{512 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{32 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{3}} + \frac {{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 89, normalized size = 0.88 \[ \frac {3\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{32\,b^3}-\frac {\frac {3\,x^2}{32}-\frac {x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^4}{4}}{b}-\frac {\frac {x\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{8}+\frac {3\,x\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right )}{16}}{b^2}+\frac {{\mathrm {cosh}\left (a+b\,x\right )}^4}{32\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.92, size = 150, normalized size = 1.49 \[ \begin {cases} - \frac {3 x^{2} \sinh ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16 b} + \frac {5 x^{2} \cosh ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 x \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16 b^{2}} - \frac {5 x \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{16 b^{2}} - \frac {3 \sinh ^{4}{\left (a + b x \right )}}{64 b^{3}} + \frac {5 \cosh ^{4}{\left (a + b x \right )}}{64 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh {\relax (a )} \cosh ^{3}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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