Optimal. Leaf size=143 \[ \frac{9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac{x^2 \sinh (6 a+6 b x)}{384 b^2}+\frac{9 \sinh (2 a+2 b x)}{256 b^4}-\frac{\sinh (6 a+6 b x)}{6912 b^4}-\frac{9 x \cosh (2 a+2 b x)}{128 b^3}+\frac{x \cosh (6 a+6 b x)}{1152 b^3}-\frac{3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac{x^3 \cosh (6 a+6 b x)}{192 b} \]
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Rubi [A] time = 0.197204, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5448, 3296, 2637} \[ \frac{9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac{x^2 \sinh (6 a+6 b x)}{384 b^2}+\frac{9 \sinh (2 a+2 b x)}{256 b^4}-\frac{\sinh (6 a+6 b x)}{6912 b^4}-\frac{9 x \cosh (2 a+2 b x)}{128 b^3}+\frac{x \cosh (6 a+6 b x)}{1152 b^3}-\frac{3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac{x^3 \cosh (6 a+6 b x)}{192 b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{32} x^3 \sinh (2 a+2 b x)+\frac{1}{32} x^3 \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac{1}{32} \int x^3 \sinh (6 a+6 b x) \, dx-\frac{3}{32} \int x^3 \sinh (2 a+2 b x) \, dx\\ &=-\frac{3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac{x^3 \cosh (6 a+6 b x)}{192 b}-\frac{\int x^2 \cosh (6 a+6 b x) \, dx}{64 b}+\frac{9 \int x^2 \cosh (2 a+2 b x) \, dx}{64 b}\\ &=-\frac{3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac{x^3 \cosh (6 a+6 b x)}{192 b}+\frac{9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac{x^2 \sinh (6 a+6 b x)}{384 b^2}+\frac{\int x \sinh (6 a+6 b x) \, dx}{192 b^2}-\frac{9 \int x \sinh (2 a+2 b x) \, dx}{64 b^2}\\ &=-\frac{9 x \cosh (2 a+2 b x)}{128 b^3}-\frac{3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{1152 b^3}+\frac{x^3 \cosh (6 a+6 b x)}{192 b}+\frac{9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac{x^2 \sinh (6 a+6 b x)}{384 b^2}-\frac{\int \cosh (6 a+6 b x) \, dx}{1152 b^3}+\frac{9 \int \cosh (2 a+2 b x) \, dx}{128 b^3}\\ &=-\frac{9 x \cosh (2 a+2 b x)}{128 b^3}-\frac{3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{1152 b^3}+\frac{x^3 \cosh (6 a+6 b x)}{192 b}+\frac{9 \sinh (2 a+2 b x)}{256 b^4}+\frac{9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac{\sinh (6 a+6 b x)}{6912 b^4}-\frac{x^2 \sinh (6 a+6 b x)}{384 b^2}\\ \end{align*}
Mathematica [A] time = 0.88507, size = 90, normalized size = 0.63 \[ -\frac{81 b x \left (2 b^2 x^2+3\right ) \cosh (2 (a+b x))-3 \left (6 b^3 x^3+b x\right ) \cosh (6 (a+b x))+\sinh (2 (a+b x)) \left (\left (18 b^2 x^2+1\right ) \cosh (4 (a+b x))-234 b^2 x^2-121\right )}{3456 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 622, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08534, size = 231, normalized size = 1.62 \begin{align*} \frac{{\left (36 \, b^{3} x^{3} e^{\left (6 \, a\right )} - 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 )}}{13824 \, b^{4}} - \frac{3 \,{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{512 \, b^{4}} - \frac{3 \,{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{512 \, b^{4}} + \frac{{\left (36 \, b^{3} x^{3} + 18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{13824 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76797, size = 617, normalized size = 4.31 \begin{align*} \frac{3 \,{\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{6} - 10 \,{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 45 \,{\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - 3 \,{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \,{\left (6 \, b^{3} x^{3} + b x\right )} \sinh \left (b x + a\right )^{6} - 81 \,{\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{2} - 9 \,{\left (18 \, b^{3} x^{3} - 5 \,{\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{4} + 27 \, b x\right )} \sinh \left (b x + a\right )^{2} - 3 \,{\left ({\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{5} - 81 \,{\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{3456 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 23.1727, size = 314, normalized size = 2.2 \begin{align*} \begin{cases} - \frac{x^{3} \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac{x^{3} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac{x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac{x^{3} \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac{x^{2} \sinh ^{5}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b^{2}} - \frac{x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac{x^{2} \sinh{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{8 b^{2}} - \frac{5 x \sinh ^{6}{\left (a + b x \right )}}{72 b^{3}} + \frac{x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{12 b^{3}} + \frac{x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{12 b^{3}} - \frac{5 x \cosh ^{6}{\left (a + b x \right )}}{72 b^{3}} + \frac{5 \sinh ^{5}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{72 b^{4}} - \frac{31 \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{216 b^{4}} + \frac{5 \sinh{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{72 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30538, size = 196, normalized size = 1.37 \begin{align*} \frac{{\left (36 \, b^{3} x^{3} - 18 \, b^{2} x^{2} + 6 \, b x - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{13824 \, b^{4}} - \frac{3 \,{\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} e^{\left (2 \, b x + 2 \, a\right )}}{512 \, b^{4}} - \frac{3 \,{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{512 \, b^{4}} + \frac{{\left (36 \, b^{3} x^{3} + 18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{13824 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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