Optimal. Leaf size=63 \[ -\frac {\text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x^3}{3} \]
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Rubi [A] time = 0.13, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3716, 2190, 2531, 2282, 6589} \[ \frac {x \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^2}-\frac {\text {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 6589
Rubi steps
\begin {align*} \int x^2 \coth (a+b x) \, dx &=-\frac {x^3}{3}-2 \int \frac {e^{2 (a+b x)} x^2}{1-e^{2 (a+b x)}} \, dx\\ &=-\frac {x^3}{3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {2 \int x \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac {x^3}{3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {\int \text {Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {x^3}{3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^3}\\ &=-\frac {x^3}{3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {\text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 66, normalized size = 1.05 \[ -\frac {\text {Li}_3\left (e^{2 a+2 b x}\right )}{2 b^3}+\frac {x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}+\frac {x^2 \log \left (1-e^{2 a+2 b x}\right )}{b}-\frac {x^3}{3} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.00, size = 168, normalized size = 2.67 \[ -\frac {b^{3} x^{3} - 3 \, b^{2} x^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 6 \, b x {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, b x {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 3 \, a^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{3 \, b^{3}} \]
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 \[ \int x^{2} \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 166, normalized size = 2.63 \[ -\frac {x^{3}}{3}+\frac {a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {2 a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 a^{2} x}{b^{2}}+\frac {4 a^{3}}{3 b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {2 \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{b}+\frac {2 \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 96, normalized size = 1.52 \[ -\frac {1}{3} \, x^{3} + \frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} + \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}} \]
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 \[ \int \frac {x^2\,\mathrm {cosh}\left (a+b\,x\right )}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x^{2} \cosh {\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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