Optimal. Leaf size=129 \[ \frac{3 x^2 \text{sech}^2(x) \text{PolyLog}\left (2,e^{2 x}\right )}{2 \sqrt{a \text{sech}^4(x)}}-\frac{3 x \text{sech}^2(x) \text{PolyLog}\left (3,e^{2 x}\right )}{2 \sqrt{a \text{sech}^4(x)}}+\frac{3 \text{sech}^2(x) \text{PolyLog}\left (4,e^{2 x}\right )}{4 \sqrt{a \text{sech}^4(x)}}-\frac{x^4 \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}+\frac{x^3 \log \left (1-e^{2 x}\right ) \text{sech}^2(x)}{\sqrt{a \text{sech}^4(x)}} \]
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Rubi [A] time = 0.553772, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {6720, 3716, 2190, 2531, 6609, 2282, 6589} \[ \frac{3 x^2 \text{sech}^2(x) \text{PolyLog}\left (2,e^{2 x}\right )}{2 \sqrt{a \text{sech}^4(x)}}-\frac{3 x \text{sech}^2(x) \text{PolyLog}\left (3,e^{2 x}\right )}{2 \sqrt{a \text{sech}^4(x)}}+\frac{3 \text{sech}^2(x) \text{PolyLog}\left (4,e^{2 x}\right )}{4 \sqrt{a \text{sech}^4(x)}}-\frac{x^4 \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}+\frac{x^3 \log \left (1-e^{2 x}\right ) \text{sech}^2(x)}{\sqrt{a \text{sech}^4(x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3716
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^3 \text{csch}(x) \text{sech}(x)}{\sqrt{a \text{sech}^4(x)}} \, dx &=\frac{\text{sech}^2(x) \int x^3 \coth (x) \, dx}{\sqrt{a \text{sech}^4(x)}}\\ &=-\frac{x^4 \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}-\frac{\left (2 \text{sech}^2(x)\right ) \int \frac{e^{2 x} x^3}{1-e^{2 x}} \, dx}{\sqrt{a \text{sech}^4(x)}}\\ &=-\frac{x^4 \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}+\frac{x^3 \log \left (1-e^{2 x}\right ) \text{sech}^2(x)}{\sqrt{a \text{sech}^4(x)}}-\frac{\left (3 \text{sech}^2(x)\right ) \int x^2 \log \left (1-e^{2 x}\right ) \, dx}{\sqrt{a \text{sech}^4(x)}}\\ &=-\frac{x^4 \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}+\frac{x^3 \log \left (1-e^{2 x}\right ) \text{sech}^2(x)}{\sqrt{a \text{sech}^4(x)}}+\frac{3 x^2 \text{Li}_2\left (e^{2 x}\right ) \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}-\frac{\left (3 \text{sech}^2(x)\right ) \int x \text{Li}_2\left (e^{2 x}\right ) \, dx}{\sqrt{a \text{sech}^4(x)}}\\ &=-\frac{x^4 \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}+\frac{x^3 \log \left (1-e^{2 x}\right ) \text{sech}^2(x)}{\sqrt{a \text{sech}^4(x)}}+\frac{3 x^2 \text{Li}_2\left (e^{2 x}\right ) \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}-\frac{3 x \text{Li}_3\left (e^{2 x}\right ) \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}+\frac{\left (3 \text{sech}^2(x)\right ) \int \text{Li}_3\left (e^{2 x}\right ) \, dx}{2 \sqrt{a \text{sech}^4(x)}}\\ &=-\frac{x^4 \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}+\frac{x^3 \log \left (1-e^{2 x}\right ) \text{sech}^2(x)}{\sqrt{a \text{sech}^4(x)}}+\frac{3 x^2 \text{Li}_2\left (e^{2 x}\right ) \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}-\frac{3 x \text{Li}_3\left (e^{2 x}\right ) \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}+\frac{\left (3 \text{sech}^2(x)\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 x}\right )}{4 \sqrt{a \text{sech}^4(x)}}\\ &=-\frac{x^4 \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}+\frac{x^3 \log \left (1-e^{2 x}\right ) \text{sech}^2(x)}{\sqrt{a \text{sech}^4(x)}}+\frac{3 x^2 \text{Li}_2\left (e^{2 x}\right ) \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}-\frac{3 x \text{Li}_3\left (e^{2 x}\right ) \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}+\frac{3 \text{Li}_4\left (e^{2 x}\right ) \text{sech}^2(x)}{4 \sqrt{a \text{sech}^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0542898, size = 68, normalized size = 0.53 \[ -\frac{\text{sech}^2(x) \left (-6 x^2 \text{PolyLog}\left (2,e^{2 x}\right )+6 x \text{PolyLog}\left (3,e^{2 x}\right )-3 \text{PolyLog}\left (4,e^{2 x}\right )+x^4-4 x^3 \log \left (1-e^{2 x}\right )\right )}{4 \sqrt{a \text{sech}^4(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.072, size = 329, normalized size = 2.6 \begin{align*} -{\frac{{{\rm e}^{2\,x}}{x}^{4}}{4\, \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{{{\rm e}^{2\,x}}{x}^{3}\ln \left ({{\rm e}^{x}}+1 \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+3\,{\frac{{{\rm e}^{2\,x}}{x}^{2}{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-6\,{\frac{x{{\rm e}^{2\,x}}{\it polylog} \left ( 3,-{{\rm e}^{x}} \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+6\,{\frac{{{\rm e}^{2\,x}}{\it polylog} \left ( 4,-{{\rm e}^{x}} \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{{{\rm e}^{2\,x}}{x}^{3}\ln \left ( 1-{{\rm e}^{x}} \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+3\,{\frac{{{\rm e}^{2\,x}}{x}^{2}{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-6\,{\frac{x{{\rm e}^{2\,x}}{\it polylog} \left ( 3,{{\rm e}^{x}} \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+6\,{\frac{{{\rm e}^{2\,x}}{\it polylog} \left ( 4,{{\rm e}^{x}} \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.91817, size = 117, normalized size = 0.91 \begin{align*} -\frac{x^{4}}{4 \, \sqrt{a}} + \frac{x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2}{\rm Li}_2\left (-e^{x}\right ) - 6 \, x{\rm Li}_{3}(-e^{x}) + 6 \,{\rm Li}_{4}(-e^{x})}{\sqrt{a}} + \frac{x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2}{\rm Li}_2\left (e^{x}\right ) - 6 \, x{\rm Li}_{3}(e^{x}) + 6 \,{\rm Li}_{4}(e^{x})}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.17801, size = 1223, normalized size = 9.48 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{csch}{\left (x \right )} \operatorname{sech}{\left (x \right )}}{\sqrt{a \operatorname{sech}^{4}{\left (x \right )}}}\, dx \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 \frac{x^{3} \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )}{\sqrt{a \operatorname{sech}\left (x\right )^{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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