Optimal. Leaf size=202 \[ \frac {3 \cosh (a+b x)}{4 b^4}-\frac {\cosh (3 a+3 b x)}{216 b^4}-\frac {3 \cosh (5 a+5 b x)}{5000 b^4}-\frac {3 x \sinh (a+b x)}{4 b^3}+\frac {x \sinh (3 a+3 b x)}{72 b^3}+\frac {3 x \sinh (5 a+5 b x)}{1000 b^3}+\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {x^3 \sinh (5 a+5 b x)}{80 b} \]
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Rubi [A] time = 0.27, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5448, 3296, 2638} \[ \frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {3 x \sinh (a+b x)}{4 b^3}+\frac {x \sinh (3 a+3 b x)}{72 b^3}+\frac {3 x \sinh (5 a+5 b x)}{1000 b^3}+\frac {3 \cosh (a+b x)}{4 b^4}-\frac {\cosh (3 a+3 b x)}{216 b^4}-\frac {3 \cosh (5 a+5 b x)}{5000 b^4}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {x^3 \sinh (5 a+5 b x)}{80 b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5448
Rubi steps
\begin {align*} \int x^3 \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {1}{8} x^3 \cosh (a+b x)+\frac {1}{16} x^3 \cosh (3 a+3 b x)+\frac {1}{16} x^3 \cosh (5 a+5 b x)\right ) \, dx\\ &=\frac {1}{16} \int x^3 \cosh (3 a+3 b x) \, dx+\frac {1}{16} \int x^3 \cosh (5 a+5 b x) \, dx-\frac {1}{8} \int x^3 \cosh (a+b x) \, dx\\ &=-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {x^3 \sinh (5 a+5 b x)}{80 b}-\frac {3 \int x^2 \sinh (5 a+5 b x) \, dx}{80 b}-\frac {\int x^2 \sinh (3 a+3 b x) \, dx}{16 b}+\frac {3 \int x^2 \sinh (a+b x) \, dx}{8 b}\\ &=\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {x^3 \sinh (5 a+5 b x)}{80 b}+\frac {3 \int x \cosh (5 a+5 b x) \, dx}{200 b^2}+\frac {\int x \cosh (3 a+3 b x) \, dx}{24 b^2}-\frac {3 \int x \cosh (a+b x) \, dx}{4 b^2}\\ &=\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {3 x \sinh (a+b x)}{4 b^3}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{72 b^3}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {3 x \sinh (5 a+5 b x)}{1000 b^3}+\frac {x^3 \sinh (5 a+5 b x)}{80 b}-\frac {3 \int \sinh (5 a+5 b x) \, dx}{1000 b^3}-\frac {\int \sinh (3 a+3 b x) \, dx}{72 b^3}+\frac {3 \int \sinh (a+b x) \, dx}{4 b^3}\\ &=\frac {3 \cosh (a+b x)}{4 b^4}+\frac {3 x^2 \cosh (a+b x)}{8 b^2}-\frac {\cosh (3 a+3 b x)}{216 b^4}-\frac {x^2 \cosh (3 a+3 b x)}{48 b^2}-\frac {3 \cosh (5 a+5 b x)}{5000 b^4}-\frac {3 x^2 \cosh (5 a+5 b x)}{400 b^2}-\frac {3 x \sinh (a+b x)}{4 b^3}-\frac {x^3 \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{72 b^3}+\frac {x^3 \sinh (3 a+3 b x)}{48 b}+\frac {3 x \sinh (5 a+5 b x)}{1000 b^3}+\frac {x^3 \sinh (5 a+5 b x)}{80 b}\\ \end {align*}
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Mathematica [A] time = 1.11, size = 125, normalized size = 0.62 \[ \frac {101250 \left (b^2 x^2+2\right ) \cosh (a+b x)-625 \left (9 b^2 x^2+2\right ) \cosh (3 (a+b x))-81 \left (25 b^2 x^2+2\right ) \cosh (5 (a+b x))+30 b x \sinh (a+b x) \left (8 \left (75 b^2 x^2+38\right ) \cosh (2 (a+b x))+9 \left (25 b^2 x^2+6\right ) \cosh (4 (a+b x))-825 b^2 x^2-6598\right )}{270000 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 274, normalized size = 1.36 \[ -\frac {81 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{5} + 405 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 135 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \sinh \left (b x + a\right )^{5} + 625 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - 75 \, {\left (75 \, b^{3} x^{3} + 18 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right )^{2} + 50 \, b x\right )} \sinh \left (b x + a\right )^{3} + 15 \, {\left (54 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 125 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 101250 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 225 \, {\left (150 \, b^{3} x^{3} - 3 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right )^{4} - 25 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} + 900 \, b x\right )} \sinh \left (b x + a\right )}{270000 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 212, normalized size = 1.05 \[ \frac {{\left (125 \, b^{3} x^{3} - 75 \, b^{2} x^{2} + 30 \, b x - 6\right )} e^{\left (5 \, b x + 5 \, a\right )}}{20000 \, b^{4}} + \frac {{\left (9 \, b^{3} x^{3} - 9 \, b^{2} x^{2} + 6 \, b x - 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{864 \, b^{4}} - \frac {{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{16 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{16 \, b^{4}} - \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{4}} - \frac {{\left (125 \, b^{3} x^{3} + 75 \, b^{2} x^{2} + 30 \, b x + 6\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{20000 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 478, normalized size = 2.37 \[ \frac {\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {2 \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{15}-\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{15}-\frac {3 \left (b x +a \right )^{2} \left (\cosh ^{5}\left (b x +a \right )\right )}{25}-\frac {856 \left (b x +a \right ) \sinh \left (b x +a \right )}{1125}+\frac {6 \left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{125}+\frac {22 \left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{1125}+\frac {856 \cosh \left (b x +a \right )}{1125}-\frac {6 \left (\cosh ^{5}\left (b x +a \right )\right )}{625}-\frac {22 \left (\cosh ^{3}\left (b x +a \right )\right )}{3375}+\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{5}+\frac {\left (b x +a \right )^{2} \left (\cosh ^{3}\left (b x +a \right )\right )}{15}-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{15}-\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{15}-\frac {2 \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{25}-\frac {856 \sinh \left (b x +a \right )}{3375}+\frac {2 \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{125}+\frac {22 \left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{3375}+\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{15}+\frac {2 \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{45}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{15}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{15}-\frac {\left (\cosh ^{5}\left (b x +a \right )\right )}{25}+\frac {2 \cosh \left (b x +a \right )}{15}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{45}\right )-a^{3} \left (\frac {\sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{5}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 245, normalized size = 1.21 \[ \frac {{\left (125 \, b^{3} x^{3} e^{\left (5 \, a\right )} - 75 \, b^{2} x^{2} e^{\left (5 \, a\right )} + 30 \, b x e^{\left (5 \, a\right )} - 6 \, e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )}}{20000 \, b^{4}} + \frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{864 \, b^{4}} - \frac {{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{16 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{16 \, b^{4}} - \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{4}} - \frac {{\left (125 \, b^{3} x^{3} + 75 \, b^{2} x^{2} + 30 \, b x + 6\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{20000 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 167, normalized size = 0.83 \[ \frac {\frac {x\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{72}-\frac {3\,x\,\mathrm {sinh}\left (a+b\,x\right )}{4}+\frac {3\,x\,\mathrm {sinh}\left (5\,a+5\,b\,x\right )}{1000}}{b^3}+\frac {\frac {x^3\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{48}+\frac {x^3\,\mathrm {sinh}\left (5\,a+5\,b\,x\right )}{80}-\frac {x^3\,\mathrm {sinh}\left (a+b\,x\right )}{8}}{b}+\frac {3\,\mathrm {cosh}\left (a+b\,x\right )}{4\,b^4}-\frac {\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{216\,b^4}-\frac {3\,\mathrm {cosh}\left (5\,a+5\,b\,x\right )}{5000\,b^4}-\frac {\frac {x^2\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{48}-\frac {3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{8}+\frac {3\,x^2\,\mathrm {cosh}\left (5\,a+5\,b\,x\right )}{400}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.50, size = 253, normalized size = 1.25 \[ \begin {cases} - \frac {2 x^{3} \sinh ^{5}{\left (a + b x \right )}}{15 b} + \frac {x^{3} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 x^{2} \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{5 b^{2}} - \frac {13 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{15 b^{2}} + \frac {26 x^{2} \cosh ^{5}{\left (a + b x \right )}}{75 b^{2}} - \frac {856 x \sinh ^{5}{\left (a + b x \right )}}{1125 b^{3}} + \frac {338 x \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{225 b^{3}} - \frac {52 x \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{75 b^{3}} + \frac {856 \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{1125 b^{4}} - \frac {5114 \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3375 b^{4}} + \frac {12568 \cosh ^{5}{\left (a + b x \right )}}{16875 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{2}{\relax (a )} \cosh ^{3}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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