Optimal. Leaf size=95 \[ -\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {2 \sinh (a+b x)}{b^3}+\frac {x^2 \sinh (a+b x)}{b} \]
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Rubi [A]
time = 0.10, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5558, 3377,
2717, 5527, 4267, 2317, 2438} \begin {gather*} -\frac {2 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {2 \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {2 \sinh (a+b x)}{b^3}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {x^2 \sinh (a+b x)}{b}-\frac {x^2 \text {csch}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 2717
Rule 3377
Rule 4267
Rule 5527
Rule 5558
Rubi steps
\begin {align*} \int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx &=\int x^2 \cosh (a+b x) \, dx+\int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx\\ &=-\frac {x^2 \text {csch}(a+b x)}{b}+\frac {x^2 \sinh (a+b x)}{b}+\frac {2 \int x \text {csch}(a+b x) \, dx}{b}-\frac {2 \int x \sinh (a+b x) \, dx}{b}\\ &=-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}+\frac {x^2 \sinh (a+b x)}{b}+\frac {2 \int \cosh (a+b x) \, dx}{b^2}-\frac {2 \int \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac {2 \int \log \left (1+e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}+\frac {2 \sinh (a+b x)}{b^3}+\frac {x^2 \sinh (a+b x)}{b}-\frac {2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {2 \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {2 \sinh (a+b x)}{b^3}+\frac {x^2 \sinh (a+b x)}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(230\) vs. \(2(95)=190\).
time = 0.31, size = 230, normalized size = 2.42 \begin {gather*} \frac {\text {csch}\left (\frac {1}{2} (a+b x)\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \left (-2-3 b^2 x^2+2 \cosh (2 (a+b x))+b^2 x^2 \cosh (2 (a+b x))+4 a \log \left (1-e^{-a-b x}\right ) \sinh (a+b x)+4 b x \log \left (1-e^{-a-b x}\right ) \sinh (a+b x)-4 a \log \left (1+e^{-a-b x}\right ) \sinh (a+b x)-4 b x \log \left (1+e^{-a-b x}\right ) \sinh (a+b x)-4 a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right ) \sinh (a+b x)+4 \text {PolyLog}\left (2,-e^{-a-b x}\right ) \sinh (a+b x)-4 \text {PolyLog}\left (2,e^{-a-b x}\right ) \sinh (a+b x)-2 b x \sinh (2 (a+b x))\right )}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.30, size = 185, normalized size = 1.95
method | result | size |
risch | \(\frac {\left (b^{2} x^{2}-2 b x +2\right ) {\mathrm e}^{b x +a}}{2 b^{3}}-\frac {\left (b^{2} x^{2}+2 b x +2\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}-\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{3}}-\frac {2 \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 157, normalized size = 1.65 \begin {gather*} \frac {{\left (b^{2} x^{2} e^{\left (4 \, a\right )} - 2 \, b x e^{\left (4 \, a\right )} + 2 \, e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} - 2 \, {\left (3 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{3} e^{\left (2 \, b x + 3 \, a\right )} - b^{3} e^{a}\right )}} - \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 731 vs.
\(2 (90) = 180\).
time = 0.40, size = 731, normalized size = 7.69 \begin {gather*} \frac {{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} - 2 \, {\left (3 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, b^{2} x^{2} - 3 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x + 4 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 4 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 4 \, {\left (b x \cosh \left (b x + a\right )^{3} + 3 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right ) + {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 4 \, {\left (a \cosh \left (b x + a\right )^{3} + 3 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + a \sinh \left (b x + a\right )^{3} - a \cosh \left (b x + a\right ) + {\left (3 \, a \cosh \left (b x + a\right )^{2} - a\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + {\left (b x + a\right )} \sinh \left (b x + a\right )^{3} - {\left (b x + a\right )} \cosh \left (b x + a\right ) + {\left (3 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} - b x - a\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 4 \, {\left ({\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{3} - {\left (3 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 2}{2 \, {\left (b^{3} \cosh \left (b x + a\right )^{3} + 3 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{3} \sinh \left (b x + a\right )^{3} - b^{3} \cosh \left (b x + a\right ) + {\left (3 \, b^{3} \cosh \left (b x + a\right )^{2} - b^{3}\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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