Optimal. Leaf size=143 \[ -\frac{6 x \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac{6 x \text{PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac{6 \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac{6 \text{PolyLog}\left (3,e^{a+b x}\right )}{b^4}-\frac{3 x^2 \cosh (a+b x)}{b^2}-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac{6 x \sinh (a+b x)}{b^3}-\frac{6 \cosh (a+b x)}{b^4}+\frac{x^3 \sinh (a+b x)}{b}-\frac{x^3 \text{csch}(a+b x)}{b} \]
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Rubi [A] time = 0.184215, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {5450, 3296, 2638, 5419, 4182, 2531, 2282, 6589} \[ -\frac{6 x \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac{6 x \text{PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac{6 \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac{6 \text{PolyLog}\left (3,e^{a+b x}\right )}{b^4}-\frac{3 x^2 \cosh (a+b x)}{b^2}-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac{6 x \sinh (a+b x)}{b^3}-\frac{6 \cosh (a+b x)}{b^4}+\frac{x^3 \sinh (a+b x)}{b}-\frac{x^3 \text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5450
Rule 3296
Rule 2638
Rule 5419
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \cosh (a+b x) \coth ^2(a+b x) \, dx &=\int x^3 \cosh (a+b x) \, dx+\int x^3 \coth (a+b x) \text{csch}(a+b x) \, dx\\ &=-\frac{x^3 \text{csch}(a+b x)}{b}+\frac{x^3 \sinh (a+b x)}{b}+\frac{3 \int x^2 \text{csch}(a+b x) \, dx}{b}-\frac{3 \int x^2 \sinh (a+b x) \, dx}{b}\\ &=-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \cosh (a+b x)}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b}+\frac{x^3 \sinh (a+b x)}{b}+\frac{6 \int x \cosh (a+b x) \, dx}{b^2}-\frac{6 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac{6 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \cosh (a+b x)}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b}-\frac{6 x \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{6 x \text{Li}_2\left (e^{a+b x}\right )}{b^3}+\frac{6 x \sinh (a+b x)}{b^3}+\frac{x^3 \sinh (a+b x)}{b}+\frac{6 \int \text{Li}_2\left (-e^{a+b x}\right ) \, dx}{b^3}-\frac{6 \int \text{Li}_2\left (e^{a+b x}\right ) \, dx}{b^3}-\frac{6 \int \sinh (a+b x) \, dx}{b^3}\\ &=-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{6 \cosh (a+b x)}{b^4}-\frac{3 x^2 \cosh (a+b x)}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b}-\frac{6 x \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{6 x \text{Li}_2\left (e^{a+b x}\right )}{b^3}+\frac{6 x \sinh (a+b x)}{b^3}+\frac{x^3 \sinh (a+b x)}{b}+\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{6 \cosh (a+b x)}{b^4}-\frac{3 x^2 \cosh (a+b x)}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b}-\frac{6 x \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{6 x \text{Li}_2\left (e^{a+b x}\right )}{b^3}+\frac{6 \text{Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac{6 \text{Li}_3\left (e^{a+b x}\right )}{b^4}+\frac{6 x \sinh (a+b x)}{b^3}+\frac{x^3 \sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.325346, size = 225, normalized size = 1.57 \[ \frac{\text{csch}\left (\frac{1}{2} (a+b x)\right ) \text{sech}\left (\frac{1}{2} (a+b x)\right ) \left (-12 b x \sinh (a+b x) \text{PolyLog}(2,-\sinh (a+b x)-\cosh (a+b x))+12 b x \sinh (a+b x) \text{PolyLog}(2,\sinh (a+b x)+\cosh (a+b x))+12 \sinh (a+b x) \text{PolyLog}(3,-\sinh (a+b x)-\cosh (a+b x))-12 \sinh (a+b x) \text{PolyLog}(3,\sinh (a+b x)+\cosh (a+b x))-3 b^2 x^2 \sinh (2 (a+b x))+b^3 x^3 \cosh (2 (a+b x))-12 b^2 x^2 \sinh (a+b x) \tanh ^{-1}(\sinh (a+b x)+\cosh (a+b x))-6 \sinh (2 (a+b x))+6 b x \cosh (2 (a+b x))-3 b^3 x^3-6 b x\right )}{4 b^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.079, size = 241, normalized size = 1.7 \begin{align*}{\frac{ \left ({x}^{3}{b}^{3}-3\,{x}^{2}{b}^{2}+6\,bx-6 \right ){{\rm e}^{bx+a}}}{2\,{b}^{4}}}-{\frac{ \left ({x}^{3}{b}^{3}+3\,{x}^{2}{b}^{2}+6\,bx+6 \right ){{\rm e}^{-bx-a}}}{2\,{b}^{4}}}-2\,{\frac{{{\rm e}^{bx+a}}{x}^{3}}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}-6\,{\frac{{a}^{2}{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-3\,{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}+3\,{\frac{{a}^{2}\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-6\,{\frac{x{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+6\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}-3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{2}}{{b}^{4}}}+6\,{\frac{x{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}-6\,{\frac{{\it polylog} \left ( 3,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46294, size = 292, normalized size = 2.04 \begin{align*} \frac{{\left (b^{3} x^{3} e^{\left (4 \, a\right )} - 3 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 6 \, b x e^{\left (4 \, a\right )} - 6 \, e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} - 6 \,{\left (b^{3} x^{3} e^{\left (2 \, a\right )} + 2 \, b x e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} +{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x\right )}}{2 \,{\left (b^{4} e^{\left (2 \, b x + 3 \, a\right )} - b^{4} e^{a}\right )}} - \frac{3 \,{\left (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 )})\right )}}{b^{4}} + \frac{3 \,{\left (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 )})\right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.57744, size = 2685, normalized size = 18.78 \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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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