Optimal. Leaf size=146 \[ -\frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 x \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 i \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}+\frac {2 x \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {2 \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}+\frac {x^2 \text {sech}(a+b x)}{b} \]
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Rubi [A]
time = 0.18, antiderivative size = 146, normalized size of antiderivative = 1.00, number
of steps used = 17, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules
used = {2702, 327, 213, 5570, 14, 6408, 12, 4267, 2611, 2320, 6724, 4265, 2317, 2438}
\begin {gather*} -\frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}+\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x^2 \text {sech}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 213
Rule 327
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 2702
Rule 4265
Rule 4267
Rule 5570
Rule 6408
Rule 6724
Rubi steps
\begin {align*} \int x^2 \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx &=-\frac {x^2 \tanh ^{-1}(\cosh (a+b x))}{b}+\frac {x^2 \text {sech}(a+b x)}{b}-2 \int x \left (-\frac {\tanh ^{-1}(\cosh (a+b x))}{b}+\frac {\text {sech}(a+b x)}{b}\right ) \, dx\\ &=-\frac {x^2 \tanh ^{-1}(\cosh (a+b x))}{b}+\frac {x^2 \text {sech}(a+b x)}{b}-2 \int \left (-\frac {x \tanh ^{-1}(\cosh (a+b x))}{b}+\frac {x \text {sech}(a+b x)}{b}\right ) \, dx\\ &=-\frac {x^2 \tanh ^{-1}(\cosh (a+b x))}{b}+\frac {x^2 \text {sech}(a+b x)}{b}+\frac {2 \int x \tanh ^{-1}(\cosh (a+b x)) \, dx}{b}-\frac {2 \int x \text {sech}(a+b x) \, dx}{b}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {x^2 \text {sech}(a+b x)}{b}+\frac {(2 i) \int \log \left (1-i e^{a+b x}\right ) \, dx}{b^2}-\frac {(2 i) \int \log \left (1+i e^{a+b x}\right ) \, dx}{b^2}+\frac {\int b x^2 \text {csch}(a+b x) \, dx}{b}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {x^2 \text {sech}(a+b x)}{b}+\frac {(2 i) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {(2 i) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\int x^2 \text {csch}(a+b x) \, dx\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {2 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {x^2 \text {sech}(a+b x)}{b}+\frac {2 \int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac {2 \int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {x^2 \text {sech}(a+b x)}{b}+\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}+\frac {x^2 \text {sech}(a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 225, normalized size = 1.54 \begin {gather*} \frac {(-2 i a+\pi -2 i b x) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )-(-2 i a+\pi ) \log \left (\cot \left (\frac {1}{4} (2 i a+\pi +2 i b x)\right )\right )+2 i \left (\text {PolyLog}\left (2,-i e^{a+b x}\right )-\text {PolyLog}\left (2,i e^{a+b x}\right )\right )-2 \left (b^2 x^2 \tanh ^{-1}(\cosh (a+b x)+\sinh (a+b x))+b x \text {PolyLog}(2,-\cosh (a+b x)-\sinh (a+b x))-b x \text {PolyLog}(2,\cosh (a+b x)+\sinh (a+b x))-\text {PolyLog}(3,-\cosh (a+b x)-\sinh (a+b x))+\text {PolyLog}(3,\cosh (a+b x)+\sinh (a+b x))\right )+b^2 x^2 \text {sech}(a+b x)}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.24, size = 0, normalized size = 0.00 \[\int x^{2} \mathrm {csch}\left (b x +a \right ) \mathrm {sech}\left (b x +a \right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 937 vs. \(2 (126) = 252\).
time = 0.45, size = 937, normalized size = 6.42 \begin {gather*} \frac {2 \, b^{2} x^{2} \cosh \left (b x + a\right ) + 2 \, b^{2} x^{2} \sinh \left (b x + a\right ) + 2 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} + b x\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, {\left (i \, \cosh \left (b x + a\right )^{2} + 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )^{2} + i\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 2 \, {\left (-i \, \cosh \left (b x + a\right )^{2} - 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )^{2} - i\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - 2 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} + b x\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )^{2} + b^{2} x^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left (-i \, a \cosh \left (b x + a\right )^{2} - 2 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, a \sinh \left (b x + a\right )^{2} - i \, a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - 2 \, {\left (i \, a \cosh \left (b x + a\right )^{2} + 2 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, a \sinh \left (b x + a\right )^{2} + i \, a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + {\left (a^{2} \cosh \left (b x + a\right )^{2} + 2 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )^{2} + a^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 2 \, {\left ({\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (-i \, b x - i \, a\right )} \sinh \left (b x + a\right )^{2} - i \, b x - i \, a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left ({\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (i \, b x + i \, a\right )} \sinh \left (b x + a\right )^{2} + i \, b x + i \, a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) + {\left (b^{2} x^{2} + {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} + b^{3}} \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 x^{2} \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b 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 (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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