Optimal. Leaf size=65 \[ \frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}+\frac {2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {x^2 \coth (a+b x)}{b}-\frac {x^2}{b}+\frac {x^3}{3} \]
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Rubi [A] time = 0.12, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3720, 3716, 2190, 2279, 2391, 30} \[ \frac {\text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}+\frac {2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {x^2 \coth (a+b x)}{b}-\frac {x^2}{b}+\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 3720
Rubi steps
\begin {align*} \int x^2 \coth ^2(a+b x) \, dx &=-\frac {x^2 \coth (a+b x)}{b}+\frac {2 \int x \coth (a+b x) \, dx}{b}+\int x^2 \, dx\\ &=-\frac {x^2}{b}+\frac {x^3}{3}-\frac {x^2 \coth (a+b x)}{b}-\frac {4 \int \frac {e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx}{b}\\ &=-\frac {x^2}{b}+\frac {x^3}{3}-\frac {x^2 \coth (a+b x)}{b}+\frac {2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {2 \int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {x^2}{b}+\frac {x^3}{3}-\frac {x^2 \coth (a+b x)}{b}+\frac {2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{b^3}\\ &=-\frac {x^2}{b}+\frac {x^3}{3}-\frac {x^2 \coth (a+b x)}{b}+\frac {2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {\text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}\\ \end {align*}
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Mathematica [C] time = 5.41, size = 163, normalized size = 2.51 \[ \frac {-b^2 x^2 e^{-\tanh ^{-1}(\tanh (a))} \coth (a) \sqrt {\text {sech}^2(a)}-\text {Li}_2\left (e^{-2 \left (b x+\tanh ^{-1}(\tanh (a))\right )}\right )+2 b x \log \left (1-e^{-2 \left (\tanh ^{-1}(\tanh (a))+b x\right )}\right )+2 \tanh ^{-1}(\tanh (a)) \left (\log \left (1-e^{-2 \left (\tanh ^{-1}(\tanh (a))+b x\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}(\tanh (a))+b x\right )\right )+b x\right )+i \pi b x-i \pi \log \left (e^{2 b x}+1\right )+i \pi \log (\cosh (b x))}{b^3}+\frac {x^2 \text {csch}(a) \sinh (b x) \text {csch}(a+b x)}{b}+\frac {x^3}{3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.47, size = 453, normalized size = 6.97 \[ -\frac {b^{3} x^{3} - {\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \sinh \left (b x + a\right )^{2} + 6 \, a^{2} - 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 6 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2} - a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 6 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2} - b x - a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{3 \, {\left (b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} - b^{3}\right )}} \]
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 \[ \int x^{2} \coth \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 156, normalized size = 2.40 \[ \frac {x^{3}}{3}-\frac {2 x^{2}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {2 x^{2}}{b}-\frac {4 a x}{b^{2}}-\frac {2 a^{2}}{b^{3}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 108, normalized size = 1.66 \[ -\frac {2 \, x^{2}}{b} + \frac {b x^{3} e^{\left (2 \, b x + 2 \, a\right )} - b x^{3} - 6 \, x^{2}}{3 \, {\left (b e^{\left (2 \, b x + 2 \, a\right )} - b\right )}} + \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,{\mathrm {coth}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int x^{2} \coth ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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