Optimal. Leaf size=97 \[ \frac {\text {Li}_3\left (-e^{2 a+2 b x}\right )}{2 b^3}-\frac {\text {Li}_3\left (e^{2 a+2 b x}\right )}{2 b^3}-\frac {x \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {2 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.11, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5461, 4182, 2531, 2282, 6589} \[ -\frac {x \text {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{b^2}+\frac {x \text {PolyLog}\left (2,e^{2 a+2 b x}\right )}{b^2}+\frac {\text {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{2 b^3}-\frac {\text {PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b^3}-\frac {2 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4182
Rule 5461
Rule 6589
Rubi steps
\begin {align*} \int x^2 \text {csch}(a+b x) \text {sech}(a+b x) \, dx &=2 \int x^2 \text {csch}(2 a+2 b x) \, dx\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {2 \int x \log \left (1-e^{2 a+2 b x}\right ) \, dx}{b}+\frac {2 \int x \log \left (1+e^{2 a+2 b x}\right ) \, dx}{b}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {x \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}+\frac {\int \text {Li}_2\left (-e^{2 a+2 b x}\right ) \, dx}{b^2}-\frac {\int \text {Li}_2\left (e^{2 a+2 b x}\right ) \, dx}{b^2}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {x \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}+\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^3}-\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^3}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {x \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}+\frac {\text {Li}_3\left (-e^{2 a+2 b x}\right )}{2 b^3}-\frac {\text {Li}_3\left (e^{2 a+2 b x}\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 4.31, size = 108, normalized size = 1.11 \[ \frac {2 b^2 x^2 \log \left (1-e^{2 (a+b x)}\right )-2 b^2 x^2 \log \left (e^{2 (a+b x)}+1\right )-2 b x \text {Li}_2\left (-e^{2 (a+b x)}\right )+2 b x \text {Li}_2\left (e^{2 (a+b x)}\right )+\text {Li}_3\left (-e^{2 (a+b x)}\right )-\text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.44, size = 351, normalized size = 3.62 \[ \frac {b^{2} x^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, b x {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, b x {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 2 \, b x {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 2 \, b x {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - a^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - a^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + a^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 2 \, {\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - 2 \, {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{3}} \]
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 \[ \int x^{2} \operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 186, normalized size = 1.92 \[ \frac {a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {2 \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\polylog \left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{3}}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}-\frac {x^{2} \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}-\frac {x \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}+\frac {2 \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{b}+\frac {2 \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 148, normalized size = 1.53 \[ -\frac {2 \, b^{2} x^{2} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )})}{2 \, b^{3}} + \frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} + \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}} \]
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 \[ \int \frac {x^2}{\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x^{2} \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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