Optimal. Leaf size=117 \[ \frac {14 \cosh (a+b x)}{9 b^4}+\frac {2 x^2 \cosh (a+b x)}{3 b^2}-\frac {2 \cosh ^3(a+b x)}{27 b^4}-\frac {4 x \sinh (a+b x)}{3 b^3}-\frac {x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac {2 x \sinh ^3(a+b x)}{9 b^3}+\frac {x^3 \sinh ^3(a+b x)}{3 b} \]
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Rubi [A]
time = 0.10, 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 = {5480, 3392,
3377, 2718, 2713} \begin {gather*} -\frac {2 \cosh ^3(a+b x)}{27 b^4}+\frac {14 \cosh (a+b x)}{9 b^4}+\frac {2 x \sinh ^3(a+b x)}{9 b^3}-\frac {4 x \sinh (a+b x)}{3 b^3}+\frac {2 x^2 \cosh (a+b x)}{3 b^2}-\frac {x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b^2}+\frac {x^3 \sinh ^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 2718
Rule 3377
Rule 3392
Rule 5480
Rubi steps
\begin {align*} \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {\int x^2 \sinh ^3(a+b x) \, dx}{b}\\ &=-\frac {x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac {2 x \sinh ^3(a+b x)}{9 b^3}+\frac {x^3 \sinh ^3(a+b x)}{3 b}-\frac {2 \int \sinh ^3(a+b x) \, dx}{9 b^3}+\frac {2 \int x^2 \sinh (a+b x) \, dx}{3 b}\\ &=\frac {2 x^2 \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac {2 x \sinh ^3(a+b x)}{9 b^3}+\frac {x^3 \sinh ^3(a+b x)}{3 b}+\frac {2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{9 b^4}-\frac {4 \int x \cosh (a+b x) \, dx}{3 b^2}\\ &=\frac {2 \cosh (a+b x)}{9 b^4}+\frac {2 x^2 \cosh (a+b x)}{3 b^2}-\frac {2 \cosh ^3(a+b x)}{27 b^4}-\frac {4 x \sinh (a+b x)}{3 b^3}-\frac {x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac {2 x \sinh ^3(a+b x)}{9 b^3}+\frac {x^3 \sinh ^3(a+b x)}{3 b}+\frac {4 \int \sinh (a+b x) \, dx}{3 b^3}\\ &=\frac {14 \cosh (a+b x)}{9 b^4}+\frac {2 x^2 \cosh (a+b x)}{3 b^2}-\frac {2 \cosh ^3(a+b x)}{27 b^4}-\frac {4 x \sinh (a+b x)}{3 b^3}-\frac {x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac {2 x \sinh ^3(a+b x)}{9 b^3}+\frac {x^3 \sinh ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 84, normalized size = 0.72 \begin {gather*} \frac {81 \left (2+b^2 x^2\right ) \cosh (a+b x)-\left (2+9 b^2 x^2\right ) \cosh (3 (a+b x))+6 b x \left (-26-3 b^2 x^2+\left (2+3 b^2 x^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{108 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs.
\(2(103)=206\).
time = 1.63, size = 314, normalized size = 2.68
method | result | size |
risch | \(\frac {\left (9 b^{3} x^{3}-9 b^{2} x^{2}+6 b x -2\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{4}}-\frac {\left (b^{3} x^{3}-3 b^{2} x^{2}+6 b x -6\right ) {\mathrm e}^{b x +a}}{8 b^{4}}+\frac {\left (b^{3} x^{3}+3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x -a}}{8 b^{4}}-\frac {\left (9 b^{3} x^{3}+9 b^{2} x^{2}+6 b x +2\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{4}}\) | \(141\) |
default | \(-\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )-3 a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )+3 a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )-a^{3} \sinh \left (b x +a \right )}{4 b^{4}}+\frac {\left (3 b x +3 a \right )^{3} \sinh \left (3 b x +3 a \right )-3 \left (3 b x +3 a \right )^{2} \cosh \left (3 b x +3 a \right )+6 \left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-6 \cosh \left (3 b x +3 a \right )-9 a \left (\left (3 b x +3 a \right )^{2} \sinh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )+2 \sinh \left (3 b x +3 a \right )\right )+27 a^{2} \left (\left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-\cosh \left (3 b x +3 a \right )\right )-27 a^{3} \sinh \left (3 b x +3 a \right )}{324 b^{4}}\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 160, normalized size = 1.37 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 135, normalized size = 1.15 \begin {gather*} -\frac {{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \sinh \left (b x + a\right )^{3} - 81 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 9 \, {\left (3 \, b^{3} x^{3} - {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} + 18 \, b x\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.43, size = 146, normalized size = 1.25 \begin {gather*} \begin {cases} \frac {x^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} - \frac {x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} + \frac {2 x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {14 x \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {4 x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {14 \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{4}} + \frac {40 \cosh ^{3}{\left (a + b x \right )}}{27 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{2}{\left (a \right )} \cosh {\left (a \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 140, normalized size = 1.20 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 119, normalized size = 1.02 \begin {gather*} \frac {\frac {14\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{9}-\frac {4\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{3}}{b^3}+\frac {\frac {2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3}-x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2}+\frac {40\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{27\,b^4}-\frac {14\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{9\,b^4}+\frac {x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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