Optimal. Leaf size=180 \[ \frac{3 x^2 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{3 \text{PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac{3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 x}{8 b^3}+\frac{x^3}{4 b}-\frac{x^4}{4} \]
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Rubi [A] time = 0.236801, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5450, 5372, 3311, 30, 2635, 8, 3716, 2190, 2531, 6609, 2282, 6589} \[ \frac{3 x^2 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{3 \text{PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac{3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 x}{8 b^3}+\frac{x^3}{4 b}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Rule 5450
Rule 5372
Rule 3311
Rule 30
Rule 2635
Rule 8
Rule 3716
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx &=\int x^3 \coth (a+b x) \, dx+\int x^3 \cosh (a+b x) \sinh (a+b x) \, dx\\ &=-\frac{x^4}{4}+\frac{x^3 \sinh ^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^2 \sinh ^2(a+b x) \, dx}{2 b}\\ &=-\frac{x^4}{4}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}+\frac{x^3 \sinh ^2(a+b x)}{2 b}-\frac{3 \int \sinh ^2(a+b x) \, dx}{4 b^3}+\frac{3 \int x^2 \, dx}{4 b}-\frac{3 \int x^2 \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac{x^3}{4 b}-\frac{x^4}{4}+\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 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac{3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 \int 1 \, dx}{8 b^3}-\frac{3 \int x \text{Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=\frac{3 x}{8 b^3}+\frac{x^3}{4 b}-\frac{x^4}{4}+\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 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac{3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 \int \text{Li}_3\left (e^{2 (a+b x)}\right ) \, dx}{2 b^3}\\ &=\frac{3 x}{8 b^3}+\frac{x^3}{4 b}-\frac{x^4}{4}+\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 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac{3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\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}{8 b^3}+\frac{x^3}{4 b}-\frac{x^4}{4}+\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 \text{Li}_4\left (e^{2 (a+b x)}\right )}{4 b^4}-\frac{3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac{3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}+\frac{x^3 \sinh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 2.52679, size = 236, normalized size = 1.31 \[ \frac{\sinh (a) (\sinh (a)+\cosh (a)) \left (-48 b^2 x^2 \text{PolyLog}\left (2,-e^{-a-b x}\right )-48 b^2 x^2 \text{PolyLog}\left (2,e^{-a-b x}\right )-96 b x \text{PolyLog}\left (3,-e^{-a-b x}\right )-96 b x \text{PolyLog}\left (3,e^{-a-b x}\right )-96 \text{PolyLog}\left (4,-e^{-a-b x}\right )-96 \text{PolyLog}\left (4,e^{-a-b x}\right )+16 b^3 x^3 \log \left (1-e^{-a-b x}\right )+16 b^3 x^3 \log \left (e^{-a-b x}+1\right )-6 b^2 x^2 \sinh (2 (a+b x))+4 b^3 x^3 \cosh (2 (a+b x))-3 \sinh (2 (a+b x))+6 b x \cosh (2 (a+b x))+4 b^4 x^4\right )}{8 \left (e^{2 a}-1\right ) b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 272, normalized size = 1.5 \begin{align*} -{\frac{{x}^{4}}{4}}+{\frac{ \left ( 4\,{x}^{3}{b}^{3}-6\,{x}^{2}{b}^{2}+6\,bx-3 \right ){{\rm e}^{2\,bx+2\,a}}}{32\,{b}^{4}}}+{\frac{ \left ( 4\,{x}^{3}{b}^{3}+6\,{x}^{2}{b}^{2}+6\,bx+3 \right ){{\rm e}^{-2\,bx-2\,a}}}{32\,{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}}}+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}}}-{\frac{3\,{a}^{4}}{2\,{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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45046, size = 304, normalized size = 1.69 \begin{align*} -\frac{1}{2} \, x^{4} + \frac{{\left (8 \, b^{4} x^{4} e^{\left (2 \, a\right )} +{\left (4 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 6 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} +{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{32 \, 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.1776, size = 2325, normalized size = 12.92 \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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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