Optimal. Leaf size=129 \[ \frac {3 x^2 \text {Li}_2\left (e^{2 x}\right ) \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}-\frac {3 x \text {Li}_3\left (e^{2 x}\right ) \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}+\frac {3 \text {Li}_4\left (e^{2 x}\right ) \text {sech}^2(x)}{4 \sqrt {a \text {sech}^4(x)}}-\frac {x^4 \text {sech}^2(x)}{4 \sqrt {a \text {sech}^4(x)}}+\frac {x^3 \log \left (1-e^{2 x}\right ) \text {sech}^2(x)}{\sqrt {a \text {sech}^4(x)}} \]
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Rubi [A] time = 0.55, antiderivative size = 129, normalized size of antiderivative = 1.00, 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 2190
Rule 2282
Rule 2531
Rule 3716
Rule 6589
Rule 6609
Rule 6720
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*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 0.53 \[ -\frac {\text {sech}^2(x) \left (-6 x^2 \text {Li}_2\left (e^{2 x}\right )+6 x \text {Li}_3\left (e^{2 x}\right )-3 \text {Li}_4\left (e^{2 x}\right )+x^4-4 x^3 \log \left (1-e^{2 x}\right )\right )}{4 \sqrt {a \text {sech}^4(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.48, size = 427, normalized size = 3.31 \[ \frac {{\left (24 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )} {\rm polylog}\left (4, \cosh \relax (x) + \sinh \relax (x)\right ) + 24 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )} {\rm polylog}\left (4, -\cosh \relax (x) - \sinh \relax (x)\right ) - 24 \, {\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )} {\rm polylog}\left (3, \cosh \relax (x) + \sinh \relax (x)\right ) - 24 \, {\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )} {\rm polylog}\left (3, -\cosh \relax (x) - \sinh \relax (x)\right ) - {\left (x^{4} e^{\left (4 \, x\right )} + 2 \, x^{4} e^{\left (2 \, x\right )} + x^{4} - 12 \, {\left (x^{2} e^{\left (4 \, x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - 12 \, {\left (x^{2} e^{\left (4 \, x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) - 4 \, {\left (x^{3} e^{\left (4 \, x\right )} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - 4 \, {\left (x^{3} e^{\left (4 \, x\right )} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right )\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{4 \, a} \]
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 \[ \int \frac {x^{3} \operatorname {csch}\relax (x) \operatorname {sech}\relax (x)}{\sqrt {a \operatorname {sech}\relax (x)^{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 329, normalized size = 2.55 \[ -\frac {{\mathrm e}^{2 x} x^{4}}{4 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {{\mathrm e}^{2 x} x^{3} \ln \left ({\mathrm e}^{x}+1\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {3 \,{\mathrm e}^{2 x} x^{2} \polylog \left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {6 \,{\mathrm e}^{2 x} x \polylog \left (3, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {6 \,{\mathrm e}^{2 x} \polylog \left (4, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {{\mathrm e}^{2 x} x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {3 \,{\mathrm e}^{2 x} x^{2} \polylog \left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {6 \,{\mathrm e}^{2 x} x \polylog \left (3, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {6 \,{\mathrm e}^{2 x} \polylog \left (4, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 87, normalized size = 0.67 \[ -\frac {x^{4}}{4 \, \sqrt {a}} + \frac {x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (-e^{x}\right ) - 6 \, x {\rm Li}_{3}(-e^{x}) + 6 \, {\rm Li}_{4}(-e^{x})}{\sqrt {a}} + \frac {x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 6 \, x {\rm Li}_{3}(e^{x}) + 6 \, {\rm Li}_{4}(e^{x})}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)\,\sqrt {\frac {a}{{\mathrm {cosh}\relax (x)}^4}}} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {csch}{\relax (x )} \operatorname {sech}{\relax (x )}}{\sqrt {a \operatorname {sech}^{4}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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