Optimal. Leaf size=91 \[ -\frac{i \text{PolyLog}\left (2,-i e^{a+b x}\right )}{2 b^2}+\frac{i \text{PolyLog}\left (2,i e^{a+b x}\right )}{2 b^2}-\frac{\text{sech}(a+b x)}{2 b^2}+\frac{x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x \tanh (a+b x) \text{sech}(a+b x)}{2 b} \]
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Rubi [A] time = 0.101526, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5455, 4180, 2279, 2391, 4185} \[ -\frac{i \text{PolyLog}\left (2,-i e^{a+b x}\right )}{2 b^2}+\frac{i \text{PolyLog}\left (2,i e^{a+b x}\right )}{2 b^2}-\frac{\text{sech}(a+b x)}{2 b^2}+\frac{x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x \tanh (a+b x) \text{sech}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 5455
Rule 4180
Rule 2279
Rule 2391
Rule 4185
Rubi steps
\begin{align*} \int x \text{sech}(a+b x) \tanh ^2(a+b x) \, dx &=\int x \text{sech}(a+b x) \, dx-\int x \text{sech}^3(a+b x) \, dx\\ &=\frac{2 x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\text{sech}(a+b x)}{2 b^2}-\frac{x \text{sech}(a+b x) \tanh (a+b x)}{2 b}-\frac{1}{2} \int x \text{sech}(a+b x) \, dx-\frac{i \int \log \left (1-i e^{a+b x}\right ) \, dx}{b}+\frac{i \int \log \left (1+i e^{a+b x}\right ) \, dx}{b}\\ &=\frac{x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\text{sech}(a+b x)}{2 b^2}-\frac{x \text{sech}(a+b x) \tanh (a+b x)}{2 b}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac{i \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac{i \int \log \left (1-i e^{a+b x}\right ) \, dx}{2 b}-\frac{i \int \log \left (1+i e^{a+b x}\right ) \, dx}{2 b}\\ &=\frac{x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{i \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}+\frac{i \text{Li}_2\left (i e^{a+b x}\right )}{b^2}-\frac{\text{sech}(a+b x)}{2 b^2}-\frac{x \text{sech}(a+b x) \tanh (a+b x)}{2 b}+\frac{i \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}\\ &=\frac{x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{i \text{Li}_2\left (-i e^{a+b x}\right )}{2 b^2}+\frac{i \text{Li}_2\left (i e^{a+b x}\right )}{2 b^2}-\frac{\text{sech}(a+b x)}{2 b^2}-\frac{x \text{sech}(a+b x) \tanh (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.752659, size = 93, normalized size = 1.02 \[ -\frac{i \text{PolyLog}(2,-i (\sinh (a+b x)+\cosh (a+b x)))-i \text{PolyLog}(2,i (\sinh (a+b x)+\cosh (a+b x)))+\text{sech}(a+b x)+b x \tanh (a+b x) \text{sech}(a+b x)-2 b x \tan ^{-1}(\sinh (a+b x)+\cosh (a+b x))}{2 b^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.068, size = 178, normalized size = 2. \begin{align*} -{\frac{{{\rm e}^{bx+a}} \left ( bx{{\rm e}^{2\,bx+2\,a}}-bx+{{\rm e}^{2\,bx+2\,a}}+1 \right ) }{{b}^{2} \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) ^{2}}}-{\frac{{\frac{i}{2}}\ln \left ( 1+i{{\rm e}^{bx+a}} \right ) x}{b}}-{\frac{{\frac{i}{2}}\ln \left ( 1+i{{\rm e}^{bx+a}} \right ) a}{{b}^{2}}}+{\frac{{\frac{i}{2}}\ln \left ( 1-i{{\rm e}^{bx+a}} \right ) x}{b}}+{\frac{{\frac{i}{2}}\ln \left ( 1-i{{\rm e}^{bx+a}} \right ) a}{{b}^{2}}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+i{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( 1-i{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}-{\frac{a\arctan \left ({{\rm e}^{bx+a}} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b x e^{\left (3 \, a\right )} + e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} -{\left (b x e^{a} - 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}} + 2 \, \int \frac{x e^{\left (b x + a\right )}}{2 \,{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35389, size = 2963, normalized size = 32.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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