Optimal. Leaf size=105 \[ -\frac {3 \cosh (2 a+2 b x)}{128 b^3}+\frac {\cosh (6 a+6 b x)}{3456 b^3}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b} \]
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Rubi [A] time = 0.14, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5448, 3296, 2638} \[ \frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}-\frac {3 \cosh (2 a+2 b x)}{128 b^3}+\frac {\cosh (6 a+6 b x)}{3456 b^3}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5448
Rubi steps
\begin {align*} \int x^2 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{32} x^2 \sinh (2 a+2 b x)+\frac {1}{32} x^2 \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac {1}{32} \int x^2 \sinh (6 a+6 b x) \, dx-\frac {3}{32} \int x^2 \sinh (2 a+2 b x) \, dx\\ &=-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}-\frac {\int x \cosh (6 a+6 b x) \, dx}{96 b}+\frac {3 \int x \cosh (2 a+2 b x) \, dx}{32 b}\\ &=-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}+\frac {\int \sinh (6 a+6 b x) \, dx}{576 b^2}-\frac {3 \int \sinh (2 a+2 b x) \, dx}{64 b^2}\\ &=-\frac {3 \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {\cosh (6 a+6 b x)}{3456 b^3}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 72, normalized size = 0.69 \[ \frac {-81 \left (2 b^2 x^2+1\right ) \cosh (2 (a+b x))+\left (18 b^2 x^2+1\right ) \cosh (6 (a+b x))+6 b x (27 \sinh (2 (a+b x))-\sinh (6 (a+b x)))}{3456 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 202, normalized size = 1.92 \[ -\frac {120 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 36 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{6} - 15 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - {\left (18 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 81 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 3 \, {\left (5 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - 54 \, b^{2} x^{2} - 27\right )} \sinh \left (b x + a\right )^{2} + 36 \, {\left (b x \cosh \left (b x + a\right )^{5} - 9 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{3456 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 113, normalized size = 1.08 \[ \frac {{\left (18 \, b^{2} x^{2} - 6 \, b x + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{6912 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{256 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac {{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 276, normalized size = 2.63 \[ \frac {\frac {\left (b x +a \right )^{2} \left (\sinh ^{2}\left (b x +a \right )\right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{6}-\frac {\left (b x +a \right )^{2} \left (\cosh ^{4}\left (b x +a \right )\right )}{12}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{18}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{18}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{12}+\frac {\left (b x +a \right )^{2}}{24}+\frac {\left (\cosh ^{6}\left (b x +a \right )\right )}{108}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{72}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{24}-2 a \left (\frac {\left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{6}-\frac {\left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{12}-\frac {\sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{36}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{36}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{24}+\frac {b x}{24}+\frac {a}{24}\right )+a^{2} \left (\frac {\left (\cosh ^{4}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{12}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 127, normalized size = 1.21 \[ \frac {{\left (18 \, b^{2} x^{2} e^{\left (6 \, a\right )} - 6 \, b x e^{\left (6 \, a\right )} + e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )}}{6912 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{256 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac {{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 89, normalized size = 0.85 \[ -\frac {\frac {3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{128}-\frac {\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{3456}+b^2\,\left (\frac {3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{64}-\frac {x^2\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{192}\right )-b\,\left (\frac {3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}-\frac {x\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{576}\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.37, size = 212, normalized size = 2.02 \[ \begin {cases} - \frac {x^{2} \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x^{2} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac {x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac {x^{2} \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{12 b^{2}} - \frac {2 x \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {x \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{12 b^{2}} - \frac {\sinh ^{6}{\left (a + b x \right )}}{72 b^{3}} + \frac {\sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{18 b^{3}} - \frac {7 \cosh ^{6}{\left (a + b x \right )}}{216 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh ^{3}{\relax (a )} \cosh ^{3}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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