Optimal. Leaf size=179 \[ \frac{3 x^2 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{3 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}+\frac{3 \text{PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x^3 \coth ^2(a+b x)}{2 b}-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{x^4}{4} \]
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Rubi [A] time = 0.336685, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {3720, 3716, 2190, 2279, 2391, 30, 2531, 6609, 2282, 6589} \[ \frac{3 x^2 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{3 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}+\frac{3 \text{PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x^3 \coth ^2(a+b x)}{2 b}-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Rule 3720
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rule 30
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \coth ^3(a+b x) \, dx &=-\frac{x^3 \coth ^2(a+b x)}{2 b}+\frac{3 \int x^2 \coth ^2(a+b x) \, dx}{2 b}+\int x^3 \coth (a+b x) \, dx\\ &=-\frac{x^4}{4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \coth ^2(a+b x)}{2 b}-2 \int \frac{e^{2 (a+b x)} x^3}{1-e^{2 (a+b x)}} \, dx+\frac{3 \int x \coth (a+b x) \, dx}{b^2}+\frac{3 \int x^2 \, dx}{2 b}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{x^4}{4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \coth ^2(a+b x)}{2 b}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{6 \int \frac{e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx}{b^2}-\frac{3 \int x^2 \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{x^4}{4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \coth ^2(a+b x)}{2 b}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{3 x^2 \text{Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 \int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^3}-\frac{3 \int x \text{Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{x^4}{4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \coth ^2(a+b x)}{2 b}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{3 x^2 \text{Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}+\frac{3 \int \text{Li}_3\left (e^{2 (a+b x)}\right ) \, dx}{2 b^3}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{x^4}{4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \coth ^2(a+b x)}{2 b}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{3 \text{Li}_2\left (e^{2 (a+b x)}\right )}{2 b^4}+\frac{3 x^2 \text{Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{4 b^4}\\ &=-\frac{3 x^2}{2 b^2}+\frac{x^3}{2 b}-\frac{x^4}{4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \coth ^2(a+b x)}{2 b}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{3 \text{Li}_2\left (e^{2 (a+b x)}\right )}{2 b^4}+\frac{3 x^2 \text{Li}_2\left (e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}+\frac{3 \text{Li}_4\left (e^{2 (a+b x)}\right )}{4 b^4}\\ \end{align*}
Mathematica [B] time = 3.33511, size = 390, normalized size = 2.18 \[ \frac{1}{4} \left (-\frac{2 e^{2 a} \left (6 \left (1-e^{-2 a}\right ) \left (b^2 x^2 \text{PolyLog}\left (2,-e^{-a-b x}\right )+2 \left (b x \text{PolyLog}\left (3,-e^{-a-b x}\right )+\text{PolyLog}\left (4,-e^{-a-b x}\right )\right )\right )+6 \left (1-e^{-2 a}\right ) \left (b^2 x^2 \text{PolyLog}\left (2,e^{-a-b x}\right )+2 \left (b x \text{PolyLog}\left (3,e^{-a-b x}\right )+\text{PolyLog}\left (4,e^{-a-b x}\right )\right )\right )+6 \left (1-e^{-2 a}\right ) \text{PolyLog}\left (2,-e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \text{PolyLog}\left (2,e^{-a-b x}\right )+e^{-2 a} b^4 x^4+6 e^{-2 a} b^2 x^2-2 e^{-2 a} \left (e^{2 a}-1\right ) b^3 x^3 \log \left (1-e^{-a-b x}\right )-2 e^{-2 a} \left (e^{2 a}-1\right ) b^3 x^3 \log \left (e^{-a-b x}+1\right )-6 \left (1-e^{-2 a}\right ) b x \log \left (1-e^{-a-b x}\right )-6 \left (1-e^{-2 a}\right ) b x \log \left (e^{-a-b x}+1\right )\right )}{\left (e^{2 a}-1\right ) b^4}+\frac{6 x^2 \text{csch}(a) \sinh (b x) \text{csch}(a+b x)}{b^2}-\frac{2 x^3 \text{csch}^2(a+b x)}{b}+x^4 \coth (a)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.091, size = 375, normalized size = 2.1 \begin{align*} 3\,{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+3\,{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-{\frac{{x}^{2} \left ( 2\,bx{{\rm e}^{2\,bx+2\,a}}+3\,{{\rm e}^{2\,bx+2\,a}}-3 \right ) }{{b}^{2} \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}+3\,{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) a}{{b}^{4}}}-3\,{\frac{a\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{4}}}-{\frac{3\,{a}^{4}}{2\,{b}^{4}}}+6\,{\frac{{\it polylog} \left ( 4,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+6\,{\frac{{\it polylog} \left ( 4,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-2\,{\frac{{a}^{3}x}{{b}^{3}}}-{\frac{{a}^{3}\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{4}}}+2\,{\frac{{a}^{3}\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{3}}{{b}^{4}}}+3\,{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}-6\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{3}}{b}}+3\,{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}-6\,{\frac{{\it polylog} \left ( 3,{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{3}}{b}}-6\,{\frac{ax}{{b}^{3}}}+6\,{\frac{a\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-3\,{\frac{{x}^{2}}{{b}^{2}}}-3\,{\frac{{a}^{2}}{{b}^{4}}}-{\frac{{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42995, size = 408, normalized size = 2.28 \begin{align*} \frac{b^{2} x^{4} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{4} + 12 \, x^{2} - 2 \,{\left (b^{2} x^{4} e^{\left (2 \, a\right )} + 4 \, b x^{3} e^{\left (2 \, a\right )} + 6 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{4 \,{\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^{4} x^{4} + 6 \, b^{2} x^{2}}{2 \, b^{4}} + \frac{b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2}{\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x{\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \,{\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac{b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2}{\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x{\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \,{\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} + \frac{3 \,{\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^{4}} + \frac{3 \,{\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^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.70712, size = 5023, normalized size = 28.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \cosh \left (b x + a\right )^{3} \operatorname{csch}\left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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