Integrand size = 12, antiderivative size = 65 \[ \int x^2 \tanh ^2(a+b x) \, dx=-\frac {x^2}{b}+\frac {x^3}{3}+\frac {2 x \log \left (1+e^{2 (a+b x)}\right )}{b^2}+\frac {\operatorname {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}-\frac {x^2 \tanh (a+b x)}{b} \]
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Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3801, 3799, 2221, 2317, 2438, 30} \[ \int x^2 \tanh ^2(a+b x) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}+\frac {2 x \log \left (e^{2 (a+b x)}+1\right )}{b^2}-\frac {x^2 \tanh (a+b x)}{b}-\frac {x^2}{b}+\frac {x^3}{3} \]
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Rule 30
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 3801
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \tanh (a+b x)}{b}+\frac {2 \int x \tanh (a+b x) \, dx}{b}+\int x^2 \, dx \\ & = -\frac {x^2}{b}+\frac {x^3}{3}-\frac {x^2 \tanh (a+b x)}{b}+\frac {4 \int \frac {e^{2 (a+b x)} x}{1+e^{2 (a+b x)}} \, dx}{b} \\ & = -\frac {x^2}{b}+\frac {x^3}{3}+\frac {2 x \log \left (1+e^{2 (a+b x)}\right )}{b^2}-\frac {x^2 \tanh (a+b x)}{b}-\frac {2 \int \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {x^2}{b}+\frac {x^3}{3}+\frac {2 x \log \left (1+e^{2 (a+b x)}\right )}{b^2}-\frac {x^2 \tanh (a+b x)}{b}-\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{b^3} \\ & = -\frac {x^2}{b}+\frac {x^3}{3}+\frac {2 x \log \left (1+e^{2 (a+b x)}\right )}{b^2}+\frac {\operatorname {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}-\frac {x^2 \tanh (a+b x)}{b} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.18 \[ \int x^2 \tanh ^2(a+b x) \, dx=-\frac {\operatorname {PolyLog}\left (2,-e^{-2 (a+b x)}\right )}{b^3}+\frac {1}{3} x \left (\frac {6 x}{b+b e^{2 a}}+x^2+\frac {6 \log \left (1+e^{-2 (a+b x)}\right )}{b^2}-\frac {3 x \text {sech}(a) \text {sech}(a+b x) \sinh (b x)}{b}\right ) \]
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Time = 0.89 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.52
method | result | size |
risch | \(\frac {x^{3}}{3}+\frac {2 x^{2}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )}-\frac {2 x^{2}}{b}-\frac {4 a x}{b^{2}}-\frac {2 a^{2}}{b^{3}}+\frac {2 x \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}+\frac {4 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(99\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 515, normalized size of antiderivative = 7.92 \[ \int x^2 \tanh ^2(a+b x) \, dx=\frac {b^{3} x^{3} + {\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \sinh \left (b x + a\right )^{2} + 6 \, a^{2} + 6 \, {\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 Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 6 \, {\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 Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - 6 \, {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2} + a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - 6 \, {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2} + a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + 6 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2} + b x + a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2} + b x + a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right )}{3 \, {\left (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}\right )}} \]
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\[ \int x^2 \tanh ^2(a+b x) \, dx=\int x^{2} \sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.29 \[ \int x^2 \tanh ^2(a+b x) \, dx=-\frac {2 \, x^{2}}{b} + \frac {b x^{3} e^{\left (2 \, b x + 2 \, a\right )} + b x^{3} + 6 \, x^{2}}{3 \, {\left (b e^{\left (2 \, b x + 2 \, a\right )} + b\right )}} + \frac {2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )}{b^{3}} \]
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\[ \int x^2 \tanh ^2(a+b x) \, dx=\int { x^{2} \operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \tanh ^2(a+b x) \, dx=\int \frac {x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \]
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