Optimal. Leaf size=82 \[ \frac {x}{2 b}-\frac {x^2}{2}-\frac {\coth (a+b x)}{2 b^2}-\frac {x \coth ^2(a+b x)}{2 b}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3801, 3554, 8,
3797, 2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}-\frac {\coth (a+b x)}{2 b^2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x \coth ^2(a+b x)}{2 b}+\frac {x}{2 b}-\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3554
Rule 3797
Rule 3801
Rubi steps
\begin {align*} \int x \coth ^3(a+b x) \, dx &=-\frac {x \coth ^2(a+b x)}{2 b}+\frac {\int \coth ^2(a+b x) \, dx}{2 b}+\int x \coth (a+b x) \, dx\\ &=-\frac {x^2}{2}-\frac {\coth (a+b x)}{2 b^2}-\frac {x \coth ^2(a+b x)}{2 b}-2 \int \frac {e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx+\frac {\int 1 \, dx}{2 b}\\ &=\frac {x}{2 b}-\frac {x^2}{2}-\frac {\coth (a+b x)}{2 b^2}-\frac {x \coth ^2(a+b x)}{2 b}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {\int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {x}{2 b}-\frac {x^2}{2}-\frac {\coth (a+b x)}{2 b^2}-\frac {x \coth ^2(a+b x)}{2 b}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^2}\\ &=\frac {x}{2 b}-\frac {x^2}{2}-\frac {\coth (a+b x)}{2 b^2}-\frac {x \coth ^2(a+b x)}{2 b}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.56, size = 173, normalized size = 2.11 \begin {gather*} \frac {i b \pi x+b^2 x^2 \coth (a)-b x \text {csch}^2(a+b x)-i \pi \log \left (1+e^{2 b x}\right )+2 b x \log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )+i \pi \log (\cosh (b x))+2 \tanh ^{-1}(\tanh (a)) \left (b x+\log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )-\log \left (i \sinh \left (b x+\tanh ^{-1}(\tanh (a))\right )\right )\right )-\text {PolyLog}\left (2,e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )-b^2 e^{-\tanh ^{-1}(\tanh (a))} x^2 \coth (a) \sqrt {\text {sech}^2(a)}+\text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{2 b^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(163\) vs.
\(2(72)=144\).
time = 1.36, size = 164, normalized size = 2.00
method | result | size |
risch | \(-\frac {x^{2}}{2}-\frac {2 b x \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 b x +2 a}-1}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}+\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 149 vs.
\(2 (71) = 142\).
time = 0.31, size = 149, normalized size = 1.82 \begin {gather*} -x^{2} + \frac {b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{2} - 2 \, {\left (b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 2}{2 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} + \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \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. 975 vs.
\(2 (71) = 142\).
time = 0.46, size = 975, normalized size = 11.89 \begin {gather*} -\frac {{\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} - 2 \, {\left (b^{2} x^{2} - 2 \, a^{2} - 2 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - 3 \, {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, a^{2} - 2 \, b x - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, a^{2} - 2 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \, b x \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left (b x \cosh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (a \cosh \left (b x + a\right )^{4} + 4 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + a \sinh \left (b x + a\right )^{4} - 2 \, a \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, a \cosh \left (b x + a\right )^{2} - a\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (a \cosh \left (b x + a\right )^{3} - a \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 2 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b x + a\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} - b x - a\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{3} - {\left (b x + a\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + a\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 \, a^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (b^{2} x^{2} - 2 \, a^{2} - 2 \, b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 2}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{4} + 4 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{2} \sinh \left (b x + a\right )^{4} - 2 \, b^{2} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} - b^{2}\right )} \sinh \left (b x + a\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} - b^{2} \cosh \left (b x + a\right )\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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \coth ^{3}{\left (a + b x \right )}\, dx \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 x\,{\mathrm {coth}\left (a+b\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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