Optimal. Leaf size=104 \[ -\frac {2 x^2 \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 x \text {PolyLog}\left (2,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {2 x \text {PolyLog}\left (2,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {2 \text {PolyLog}\left (3,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 \text {PolyLog}\left (3,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \]
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Rubi [A]
time = 0.45, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6852, 4267,
2611, 2320, 6724} \begin {gather*} -\frac {2 x \text {Li}_2\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {2 x \text {Li}_2\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {2 \text {Li}_3\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 \text {Li}_3\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 x^2 \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4267
Rule 6724
Rule 6852
Rubi steps
\begin {align*} \int \frac {x^2 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx &=\frac {\text {sech}(x) \int x^2 \text {csch}(x) \, dx}{\sqrt {a \text {sech}^2(x)}}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {(2 \text {sech}(x)) \int x \log \left (1-e^x\right ) \, dx}{\sqrt {a \text {sech}^2(x)}}+\frac {(2 \text {sech}(x)) \int x \log \left (1+e^x\right ) \, dx}{\sqrt {a \text {sech}^2(x)}}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 x \text {Li}_2\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {2 x \text {Li}_2\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {(2 \text {sech}(x)) \int \text {Li}_2\left (-e^x\right ) \, dx}{\sqrt {a \text {sech}^2(x)}}-\frac {(2 \text {sech}(x)) \int \text {Li}_2\left (e^x\right ) \, dx}{\sqrt {a \text {sech}^2(x)}}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 x \text {Li}_2\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {2 x \text {Li}_2\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {(2 \text {sech}(x)) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^x\right )}{\sqrt {a \text {sech}^2(x)}}-\frac {(2 \text {sech}(x)) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^x\right )}{\sqrt {a \text {sech}^2(x)}}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 x \text {Li}_2\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {2 x \text {Li}_2\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {2 \text {Li}_3\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 \text {Li}_3\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 83, normalized size = 0.80 \begin {gather*} \frac {\left (x^2 \log \left (1-e^{-x}\right )-x^2 \log \left (1+e^{-x}\right )+2 x \text {PolyLog}\left (2,-e^{-x}\right )-2 x \text {PolyLog}\left (2,e^{-x}\right )+2 \text {PolyLog}\left (3,-e^{-x}\right )-2 \text {PolyLog}\left (3,e^{-x}\right )\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(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. \(208\) vs.
\(2(89)=178\).
time = 1.46, size = 209, normalized size = 2.01
method | result | size |
risch | \(-\frac {{\mathrm e}^{x} x^{2} \ln \left ({\mathrm e}^{x}+1\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {2 \,{\mathrm e}^{x} x \polylog \left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {2 \,{\mathrm e}^{x} \polylog \left (3, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{x} x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {2 \,{\mathrm e}^{x} x \polylog \left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {2 \,{\mathrm e}^{x} \polylog \left (3, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 60, normalized size = 0.58 \begin {gather*} -\frac {x^{2} \log \left (e^{x} + 1\right ) + 2 \, x {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\rm Li}_{3}(-e^{x})}{\sqrt {a}} + \frac {x^{2} \log \left (-e^{x} + 1\right ) + 2 \, x {\rm Li}_2\left (e^{x}\right ) - 2 \, {\rm Li}_{3}(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. 188 vs.
\(2 (87) = 174\).
time = 0.35, size = 188, normalized size = 1.81 \begin {gather*} -\frac {{\left (2 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} e^{x} {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} e^{x} {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (2 \, {\left (x e^{\left (2 \, x\right )} + x\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\left (x e^{\left (2 \, x\right )} + x\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}\right )} e^{\left (-x\right )}}{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^{2} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{\sqrt {a \operatorname {sech}^{2}{\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^2}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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