Optimal. Leaf size=103 \[ -\frac{3 d^2 (c+d x) \text{PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}+\frac{3 d^3 \text{PolyLog}\left (3,-e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 d (c+d x)^2 \log \left (e^{2 (a+b x)}+1\right )}{b^2}+\frac{(c+d x)^3 \tanh (a+b x)}{b}+\frac{(c+d x)^3}{b} \]
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Rubi [A] time = 0.205963, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4184, 3718, 2190, 2531, 2282, 6589} \[ -\frac{3 d^2 (c+d x) \text{PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}+\frac{3 d^3 \text{PolyLog}\left (3,-e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 d (c+d x)^2 \log \left (e^{2 (a+b x)}+1\right )}{b^2}+\frac{(c+d x)^3 \tanh (a+b x)}{b}+\frac{(c+d x)^3}{b} \]
Antiderivative was successfully verified.
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Rule 4184
Rule 3718
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (c+d x)^3 \text{sech}^2(a+b x) \, dx &=\frac{(c+d x)^3 \tanh (a+b x)}{b}-\frac{(3 d) \int (c+d x)^2 \tanh (a+b x) \, dx}{b}\\ &=\frac{(c+d x)^3}{b}+\frac{(c+d x)^3 \tanh (a+b x)}{b}-\frac{(6 d) \int \frac{e^{2 (a+b x)} (c+d x)^2}{1+e^{2 (a+b x)}} \, dx}{b}\\ &=\frac{(c+d x)^3}{b}-\frac{3 d (c+d x)^2 \log \left (1+e^{2 (a+b x)}\right )}{b^2}+\frac{(c+d x)^3 \tanh (a+b x)}{b}+\frac{\left (6 d^2\right ) \int (c+d x) \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=\frac{(c+d x)^3}{b}-\frac{3 d (c+d x)^2 \log \left (1+e^{2 (a+b x)}\right )}{b^2}-\frac{3 d^2 (c+d x) \text{Li}_2\left (-e^{2 (a+b x)}\right )}{b^3}+\frac{(c+d x)^3 \tanh (a+b x)}{b}+\frac{\left (3 d^3\right ) \int \text{Li}_2\left (-e^{2 (a+b x)}\right ) \, dx}{b^3}\\ &=\frac{(c+d x)^3}{b}-\frac{3 d (c+d x)^2 \log \left (1+e^{2 (a+b x)}\right )}{b^2}-\frac{3 d^2 (c+d x) \text{Li}_2\left (-e^{2 (a+b x)}\right )}{b^3}+\frac{(c+d x)^3 \tanh (a+b x)}{b}+\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}\\ &=\frac{(c+d x)^3}{b}-\frac{3 d (c+d x)^2 \log \left (1+e^{2 (a+b x)}\right )}{b^2}-\frac{3 d^2 (c+d x) \text{Li}_2\left (-e^{2 (a+b x)}\right )}{b^3}+\frac{3 d^3 \text{Li}_3\left (-e^{2 (a+b x)}\right )}{2 b^4}+\frac{(c+d x)^3 \tanh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 2.03888, size = 145, normalized size = 1.41 \[ \frac{2 \text{sech}(a) \sinh (b x) (c+d x)^3 \text{sech}(a+b x)-\frac{e^{2 a} d \left (-\frac{3 \left (e^{-2 a}+1\right ) d \left (2 b (c+d x) \text{PolyLog}\left (2,-e^{-2 (a+b x)}\right )+d \text{PolyLog}\left (3,-e^{-2 (a+b x)}\right )\right )}{b^3}+\frac{6 \left (e^{-2 a}+1\right ) (c+d x)^2 \log \left (e^{-2 (a+b x)}+1\right )}{b}+\frac{4 e^{-2 a} (c+d x)^3}{d}\right )}{e^{2 a}+1}}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.067, size = 298, normalized size = 2.9 \begin{align*} -2\,{\frac{{d}^{3}{x}^{3}+3\,c{d}^{2}{x}^{2}+3\,{c}^{2}dx+{c}^{3}}{b \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }}-3\,{\frac{{c}^{2}d\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }{{b}^{2}}}+6\,{\frac{{c}^{2}d\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+6\,{\frac{{d}^{3}{a}^{2}\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+2\,{\frac{{d}^{3}{x}^{3}}{b}}-6\,{\frac{{d}^{3}{a}^{2}x}{{b}^{3}}}-4\,{\frac{{a}^{3}{d}^{3}}{{b}^{4}}}-3\,{\frac{{d}^{3}\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ){x}^{2}}{{b}^{2}}}-3\,{\frac{{d}^{3}{\it polylog} \left ( 2,-{{\rm e}^{2\,bx+2\,a}} \right ) x}{{b}^{3}}}+{\frac{3\,{d}^{3}{\it polylog} \left ( 3,-{{\rm e}^{2\,bx+2\,a}} \right ) }{2\,{b}^{4}}}-12\,{\frac{c{d}^{2}a\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+6\,{\frac{c{d}^{2}{x}^{2}}{b}}+12\,{\frac{c{d}^{2}ax}{{b}^{2}}}+6\,{\frac{c{d}^{2}{a}^{2}}{{b}^{3}}}-6\,{\frac{c{d}^{2}\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) x}{{b}^{2}}}-3\,{\frac{c{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,bx+2\,a}} \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.694, size = 321, normalized size = 3.12 \begin{align*} 3 \, c^{2} d{\left (\frac{2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b} - \frac{\log \left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}\right )}{b^{2}}\right )} - \frac{3 \,{\left (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 )\right )} c d^{2}}{b^{3}} + \frac{2 \, c^{3}}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} - \frac{2 \,{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2}\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b} - \frac{3 \,{\left (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 )})\right )} d^{3}}{2 \, b^{4}} + \frac{2 \,{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2}\right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.42635, size = 3131, normalized size = 30.4 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{3} \operatorname{sech}^{2}{\left (a + b x \right )}\, dx \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{\left (d x + c\right )}^{3} \operatorname{sech}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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