Optimal. Leaf size=117 \[ -\frac {2 \sinh ^3(a+b x)}{27 b^4}-\frac {14 \sinh (a+b x)}{9 b^4}+\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {4 x \cosh (a+b x)}{3 b^3}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b^2}+\frac {x^3 \cosh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.12, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5373, 3311, 3296, 2637, 2633} \[ -\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b^2}-\frac {2 \sinh ^3(a+b x)}{27 b^4}-\frac {14 \sinh (a+b x)}{9 b^4}+\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {4 x \cosh (a+b x)}{3 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2637
Rule 3296
Rule 3311
Rule 5373
Rubi steps
\begin {align*} \int x^3 \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {\int x^2 \cosh ^3(a+b x) \, dx}{b}\\ &=\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 \int \cosh ^3(a+b x) \, dx}{9 b^3}-\frac {2 \int x^2 \cosh (a+b x) \, dx}{3 b}\\ &=\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {(2 i) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{9 b^4}+\frac {4 \int x \sinh (a+b x) \, dx}{3 b^2}\\ &=\frac {4 x \cosh (a+b x)}{3 b^3}+\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {2 \sinh (a+b x)}{9 b^4}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 \sinh ^3(a+b x)}{27 b^4}-\frac {4 \int \cosh (a+b x) \, dx}{3 b^3}\\ &=\frac {4 x \cosh (a+b x)}{3 b^3}+\frac {2 x \cosh ^3(a+b x)}{9 b^3}+\frac {x^3 \cosh ^3(a+b x)}{3 b}-\frac {14 \sinh (a+b x)}{9 b^4}-\frac {2 x^2 \sinh (a+b x)}{3 b^2}-\frac {x^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 \sinh ^3(a+b x)}{27 b^4}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 86, normalized size = 0.74 \[ \frac {\left (9 b^3 x^3+6 b x\right ) \cosh (3 (a+b x))+27 b x \left (b^2 x^2+6\right ) \cosh (a+b x)-2 \sinh (a+b x) \left (\left (9 b^2 x^2+2\right ) \cosh (2 (a+b x))+45 b^2 x^2+82\right )}{108 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 135, normalized size = 1.15 \[ \frac {3 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{3} + 9 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (9 \, b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{3} + 27 \, {\left (b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right ) - 3 \, {\left (27 \, b^{2} x^{2} + {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 54\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 140, normalized size = 1.20 \[ \frac {{\left (9 \, b^{3} x^{3} - 9 \, b^{2} x^{2} + 6 \, b x - 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} + \frac {{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} + \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 244, normalized size = 2.09 \[ \frac {\frac {\left (b x +a \right )^{3} \left (\cosh ^{3}\left (b x +a \right )\right )}{3}-\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}-\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{3}+\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}-\frac {40 \sinh \left (b x +a \right )}{27}+\frac {2 \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{9}-\frac {2 \left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{27}-3 a \left (\frac {\left (b x +a \right )^{2} \left (\cosh ^{3}\left (b x +a \right )\right )}{3}-\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{9}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{9}+\frac {4 \cosh \left (b x +a \right )}{9}+\frac {2 \left (\cosh ^{3}\left (b x +a \right )\right )}{27}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{3}-\frac {2 \sinh \left (b x +a \right )}{9}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{9}\right )-\frac {a^{3} \left (\cosh ^{3}\left (b x +a \right )\right )}{3}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 160, normalized size = 1.37 \[ \frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{216 \, b^{4}} + \frac {{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} + \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 108, normalized size = 0.92 \[ \frac {\frac {2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{9}+\frac {4\,x\,\mathrm {cosh}\left (a+b\,x\right )}{3}}{b^3}-\frac {\frac {2\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{3}+\frac {x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{3}}{b^2}-\frac {40\,\mathrm {sinh}\left (a+b\,x\right )}{27\,b^4}-\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{27\,b^4}+\frac {x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.90, size = 146, normalized size = 1.25 \[ \begin {cases} \frac {x^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 x^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} - \frac {x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {4 x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} + \frac {14 x \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {40 \sinh ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {14 \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh {\relax (a )} \cosh ^{2}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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