Optimal. Leaf size=202 \[ \frac{3 x^2 \sinh (a+b x)}{8 b^2}+\frac{x^2 \sinh (3 a+3 b x)}{48 b^2}-\frac{3 x^2 \sinh (5 a+5 b x)}{400 b^2}+\frac{3 \sinh (a+b x)}{4 b^4}+\frac{\sinh (3 a+3 b x)}{216 b^4}-\frac{3 \sinh (5 a+5 b x)}{5000 b^4}-\frac{3 x \cosh (a+b x)}{4 b^3}-\frac{x \cosh (3 a+3 b x)}{72 b^3}+\frac{3 x \cosh (5 a+5 b x)}{1000 b^3}-\frac{x^3 \cosh (a+b x)}{8 b}-\frac{x^3 \cosh (3 a+3 b x)}{48 b}+\frac{x^3 \cosh (5 a+5 b x)}{80 b} \]
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Rubi [A] time = 0.262065, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5448, 3296, 2637} \[ \frac{3 x^2 \sinh (a+b x)}{8 b^2}+\frac{x^2 \sinh (3 a+3 b x)}{48 b^2}-\frac{3 x^2 \sinh (5 a+5 b x)}{400 b^2}+\frac{3 \sinh (a+b x)}{4 b^4}+\frac{\sinh (3 a+3 b x)}{216 b^4}-\frac{3 \sinh (5 a+5 b x)}{5000 b^4}-\frac{3 x \cosh (a+b x)}{4 b^3}-\frac{x \cosh (3 a+3 b x)}{72 b^3}+\frac{3 x \cosh (5 a+5 b x)}{1000 b^3}-\frac{x^3 \cosh (a+b x)}{8 b}-\frac{x^3 \cosh (3 a+3 b x)}{48 b}+\frac{x^3 \cosh (5 a+5 b x)}{80 b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{1}{8} x^3 \sinh (a+b x)-\frac{1}{16} x^3 \sinh (3 a+3 b x)+\frac{1}{16} x^3 \sinh (5 a+5 b x)\right ) \, dx\\ &=-\left (\frac{1}{16} \int x^3 \sinh (3 a+3 b x) \, dx\right )+\frac{1}{16} \int x^3 \sinh (5 a+5 b x) \, dx-\frac{1}{8} \int x^3 \sinh (a+b x) \, dx\\ &=-\frac{x^3 \cosh (a+b x)}{8 b}-\frac{x^3 \cosh (3 a+3 b x)}{48 b}+\frac{x^3 \cosh (5 a+5 b x)}{80 b}-\frac{3 \int x^2 \cosh (5 a+5 b x) \, dx}{80 b}+\frac{\int x^2 \cosh (3 a+3 b x) \, dx}{16 b}+\frac{3 \int x^2 \cosh (a+b x) \, dx}{8 b}\\ &=-\frac{x^3 \cosh (a+b x)}{8 b}-\frac{x^3 \cosh (3 a+3 b x)}{48 b}+\frac{x^3 \cosh (5 a+5 b x)}{80 b}+\frac{3 x^2 \sinh (a+b x)}{8 b^2}+\frac{x^2 \sinh (3 a+3 b x)}{48 b^2}-\frac{3 x^2 \sinh (5 a+5 b x)}{400 b^2}+\frac{3 \int x \sinh (5 a+5 b x) \, dx}{200 b^2}-\frac{\int x \sinh (3 a+3 b x) \, dx}{24 b^2}-\frac{3 \int x \sinh (a+b x) \, dx}{4 b^2}\\ &=-\frac{3 x \cosh (a+b x)}{4 b^3}-\frac{x^3 \cosh (a+b x)}{8 b}-\frac{x \cosh (3 a+3 b x)}{72 b^3}-\frac{x^3 \cosh (3 a+3 b x)}{48 b}+\frac{3 x \cosh (5 a+5 b x)}{1000 b^3}+\frac{x^3 \cosh (5 a+5 b x)}{80 b}+\frac{3 x^2 \sinh (a+b x)}{8 b^2}+\frac{x^2 \sinh (3 a+3 b x)}{48 b^2}-\frac{3 x^2 \sinh (5 a+5 b x)}{400 b^2}-\frac{3 \int \cosh (5 a+5 b x) \, dx}{1000 b^3}+\frac{\int \cosh (3 a+3 b x) \, dx}{72 b^3}+\frac{3 \int \cosh (a+b x) \, dx}{4 b^3}\\ &=-\frac{3 x \cosh (a+b x)}{4 b^3}-\frac{x^3 \cosh (a+b x)}{8 b}-\frac{x \cosh (3 a+3 b x)}{72 b^3}-\frac{x^3 \cosh (3 a+3 b x)}{48 b}+\frac{3 x \cosh (5 a+5 b x)}{1000 b^3}+\frac{x^3 \cosh (5 a+5 b x)}{80 b}+\frac{3 \sinh (a+b x)}{4 b^4}+\frac{3 x^2 \sinh (a+b x)}{8 b^2}+\frac{\sinh (3 a+3 b x)}{216 b^4}+\frac{x^2 \sinh (3 a+3 b x)}{48 b^2}-\frac{3 \sinh (5 a+5 b x)}{5000 b^4}-\frac{3 x^2 \sinh (5 a+5 b x)}{400 b^2}\\ \end{align*}
Mathematica [A] time = 0.490951, size = 136, normalized size = 0.67 \[ \frac{-33750 \left (b x \left (b^2 x^2+6\right ) \cosh (a+b x)-3 \left (b^2 x^2+2\right ) \sinh (a+b x)\right )-625 \left (\left (9 b^3 x^3+6 b x\right ) \cosh (3 (a+b x))-\left (9 b^2 x^2+2\right ) \sinh (3 (a+b x))\right )+27 \left (5 b x \left (25 b^2 x^2+6\right ) \cosh (5 (a+b x))-3 \left (25 b^2 x^2+2\right ) \sinh (5 (a+b x))\right )}{270000 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 542, normalized size = 2.7 \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 = 1.09067, size = 331, normalized size = 1.64 \begin{align*} \frac{{\left (125 \, b^{3} x^{3} e^{\left (5 \, a\right )} - 75 \, b^{2} x^{2} e^{\left (5 \, a\right )} + 30 \, b x e^{\left (5 \, a\right )} - 6 \, e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )}}{20000 \, b^{4}} - \frac{{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 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 )}}{864 \, b^{4}} - \frac{{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{16 \, b^{4}} - \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{16 \, b^{4}} - \frac{{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{4}} + \frac{{\left (125 \, b^{3} x^{3} + 75 \, b^{2} x^{2} + 30 \, b x + 6\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{20000 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77431, size = 722, normalized size = 3.57 \begin{align*} \frac{135 \,{\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right )^{5} + 675 \,{\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 81 \,{\left (25 \, b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{5} - 1875 \,{\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{3} + 5 \,{\left (1125 \, b^{2} x^{2} - 162 \,{\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 250\right )} \sinh \left (b x + a\right )^{3} + 225 \,{\left (6 \,{\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right )^{3} - 25 \,{\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 33750 \,{\left (b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right ) - 15 \,{\left (27 \,{\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{4} - 6750 \, b^{2} x^{2} - 125 \,{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - 13500\right )} \sinh \left (b x + a\right )}{270000 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.8801, size = 253, normalized size = 1.25 \begin{align*} \begin{cases} \frac{x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac{2 x^{3} \cosh ^{5}{\left (a + b x \right )}}{15 b} + \frac{26 x^{2} \sinh ^{5}{\left (a + b x \right )}}{75 b^{2}} - \frac{13 x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{15 b^{2}} + \frac{2 x^{2} \sinh{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{5 b^{2}} - \frac{52 x \sinh ^{4}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{75 b^{3}} + \frac{338 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{225 b^{3}} - \frac{856 x \cosh ^{5}{\left (a + b x \right )}}{1125 b^{3}} + \frac{12568 \sinh ^{5}{\left (a + b x \right )}}{16875 b^{4}} - \frac{5114 \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3375 b^{4}} + \frac{856 \sinh{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{1125 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh ^{3}{\left (a \right )} \cosh ^{2}{\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 = 1.17057, size = 286, normalized size = 1.42 \begin{align*} \frac{{\left (125 \, b^{3} x^{3} - 75 \, b^{2} x^{2} + 30 \, b x - 6\right )} e^{\left (5 \, b x + 5 \, a\right )}}{20000 \, b^{4}} - \frac{{\left (9 \, b^{3} x^{3} - 9 \, b^{2} x^{2} + 6 \, b x - 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{864 \, b^{4}} - \frac{{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{16 \, b^{4}} - \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{16 \, b^{4}} - \frac{{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{4}} + \frac{{\left (125 \, b^{3} x^{3} + 75 \, b^{2} x^{2} + 30 \, b x + 6\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{20000 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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