Optimal. Leaf size=123 \[ -\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {x \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {x \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {\text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {\text {PolyLog}\left (3,e^{a+b x}\right )}{b^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5565, 4267,
2611, 2320, 6724, 4271, 3855} \begin {gather*} \frac {\text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {\text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 3855
Rule 4267
Rule 4271
Rule 5565
Rule 6724
Rubi steps
\begin {align*} \int x^2 \coth ^2(a+b x) \text {csch}(a+b x) \, dx &=\int x^2 \text {csch}(a+b x) \, dx+\int x^2 \text {csch}^3(a+b x) \, dx\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \int x^2 \text {csch}(a+b x) \, dx+\frac {\int \text {csch}(a+b x) \, dx}{b^2}-\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 {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\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 \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 {\int x \log \left (1-e^{a+b x}\right ) \, dx}{b}-\frac {\int x \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\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 {\int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}+\frac {\int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {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 {\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {\text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {\text {Li}_3\left (e^{a+b x}\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 2.07, size = 224, normalized size = 1.82 \begin {gather*} -\frac {16 \tanh ^{-1}(\cosh (a+b x)+\sinh (a+b x))+8 b^2 x^2 \tanh ^{-1}(\cosh (a+b x)+\sinh (a+b x))+8 b x \text {csch}(a)+b^2 x^2 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )+8 b x \text {PolyLog}(2,-\cosh (a+b x)-\sinh (a+b x))-8 b x \text {PolyLog}(2,\cosh (a+b x)+\sinh (a+b x))-8 \text {PolyLog}(3,-\cosh (a+b x)-\sinh (a+b x))+8 \text {PolyLog}(3,\cosh (a+b x)+\sinh (a+b x))+b^2 x^2 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )-4 b x \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )-4 b x \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.88, size = 210, normalized size = 1.71
method | result | size |
risch | \(-\frac {x \,{\mathrm e}^{b x +a} \left (b x \,{\mathrm e}^{2 b x +2 a}+b x +2 \,{\mathrm e}^{2 b x +2 a}-2\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{2 b}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}+\frac {x \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{2 b^{3}}-\frac {x \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(210\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 197, normalized size = 1.60 \begin {gather*} -\frac {{\left (b x^{2} e^{\left (3 \, a\right )} + 2 \, x e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b x^{2} e^{a} - 2 \, x e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \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 )})}{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 )})}{2 \, b^{3}} - \frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} \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. 1311 vs.
\(2 (114) = 228\).
time = 0.49, size = 1311, normalized size = 10.66 \begin {gather*} -\frac {2 \, {\left (b^{2} x^{2} + 2 \, b x\right )} \cosh \left (b x + a\right )^{3} + 6 \, {\left (b^{2} x^{2} + 2 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, b x\right )} \sinh \left (b x + a\right )^{3} + 2 \, {\left (b^{2} x^{2} - 2 \, b x\right )} \cosh \left (b x + a\right ) - 2 \, {\left (b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \, b x \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left (b x \cosh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\left (b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \, b x \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left (b x \cosh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left ({\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} - 2 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - 3 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left ({\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 2\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left ({\left (a^{2} + 2\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (a^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (a^{2} + 2\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (a^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - a^{2} - 2\right )} \sinh \left (b x + a\right )^{2} + a^{2} + 4 \, {\left ({\left (a^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - {\left (a^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 2\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} - 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2} - a^{2} + 4 \, {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\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 )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 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 )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 2 \, {\left (b^{2} x^{2} + 3 \, {\left (b^{2} x^{2} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} - 2 \, b x\right )} \sinh \left (b x + a\right )}{2 \, {\left (b^{3} \cosh \left (b x + a\right )^{4} + 4 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{3} \sinh \left (b x + a\right )^{4} - 2 \, b^{3} \cosh \left (b x + a\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (b x + a\right )^{2} - b^{3}\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b^{3} \cosh \left (b x + a\right )^{3} - b^{3} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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