Optimal. Leaf size=129 \[ -\frac {x^4 \text {sech}^2(x)}{4 \sqrt {a \text {sech}^4(x)}}+\frac {x^3 \log \left (1-e^{2 x}\right ) \text {sech}^2(x)}{\sqrt {a \text {sech}^4(x)}}+\frac {3 x^2 \text {PolyLog}\left (2,e^{2 x}\right ) \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}-\frac {3 x \text {PolyLog}\left (3,e^{2 x}\right ) \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}+\frac {3 \text {PolyLog}\left (4,e^{2 x}\right ) \text {sech}^2(x)}{4 \sqrt {a \text {sech}^4(x)}} \]
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Rubi [A]
time = 0.38, 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 = {6852, 3797,
2221, 2611, 6744, 2320, 6724} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 6724
Rule 6744
Rule 6852
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 ) \text {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 = 0.04, size = 68, normalized size = 0.53 \begin {gather*} -\frac {\left (x^4-4 x^3 \log \left (1-e^{2 x}\right )-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 )\right ) \text {sech}^2(x)}{4 \sqrt {a \text {sech}^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(328\) vs.
\(2(107)=214\).
time = 1.55, size = 329, normalized size = 2.55
method | result | size |
risch | \(-\frac {{\mathrm e}^{2 x} x^{4}}{4 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {{\mathrm e}^{2 x} x^{3} \ln \left ({\mathrm e}^{x}+1\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {3 \,{\mathrm e}^{2 x} x^{2} \polylog \left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {6 \,{\mathrm e}^{2 x} x \polylog \left (3, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {6 \,{\mathrm e}^{2 x} \polylog \left (4, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {{\mathrm e}^{2 x} x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {3 \,{\mathrm e}^{2 x} x^{2} \polylog \left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {6 \,{\mathrm e}^{2 x} x \polylog \left (3, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {6 \,{\mathrm e}^{2 x} \polylog \left (4, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 87, normalized size = 0.67 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 427 vs.
\(2 (106) = 212\).
time = 0.38, size = 427, normalized size = 3.31 \begin {gather*} \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 \left (x\right ) + \sinh \left (x\right )\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 \left (x\right ) - \sinh \left (x\right )\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 \left (x\right ) + \sinh \left (x\right )\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 \left (x\right ) - \sinh \left (x\right )\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 \left (x\right ) + \sinh \left (x\right )\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 \left (x\right ) - \sinh \left (x\right )\right ) - 4 \, {\left (x^{3} e^{\left (4 \, x\right )} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 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 \left (x\right ) - \sinh \left (x\right ) + 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} \end {gather*}
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 \begin {gather*} \int \frac {x^{3} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{\sqrt {a \operatorname {sech}^{4}{\left (x \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^3}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^4}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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