Optimal. Leaf size=84 \[ \frac{d (c+d x) \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac{d^2 \text{PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac{(c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac{(c+d x)^3}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155263, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3718, 2190, 2531, 2282, 6589} \[ \frac{d (c+d x) \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac{d^2 \text{PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac{(c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac{(c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3718
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (c+d x)^2 \tanh (e+f x) \, dx &=-\frac{(c+d x)^3}{3 d}+2 \int \frac{e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx\\ &=-\frac{(c+d x)^3}{3 d}+\frac{(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac{(2 d) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=-\frac{(c+d x)^3}{3 d}+\frac{(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac{d (c+d x) \text{Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac{d^2 \int \text{Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac{(c+d x)^3}{3 d}+\frac{(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac{d (c+d x) \text{Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3}\\ &=-\frac{(c+d x)^3}{3 d}+\frac{(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac{d (c+d x) \text{Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac{d^2 \text{Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}\\ \end{align*}
Mathematica [A] time = 1.4149, size = 143, normalized size = 1.7 \[ \frac{e^{2 e} \left (-\frac{3 d \left (e^{-2 e}+1\right ) \left (2 f (c+d x) \text{PolyLog}\left (2,-e^{-2 (e+f x)}\right )+d \text{PolyLog}\left (3,-e^{-2 (e+f x)}\right )\right )}{f^3}+\frac{6 \left (e^{-2 e}+1\right ) (c+d x)^2 \log \left (e^{-2 (e+f x)}+1\right )}{f}+\frac{4 e^{-2 e} (c+d x)^3}{d}\right )}{6 \left (e^{2 e}+1\right )}+\frac{1}{3} x \tanh (e) \left (3 c^2+3 c d x+d^2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.037, size = 234, normalized size = 2.8 \begin{align*} -{\frac{{d}^{2}{x}^{3}}{3}}-cd{x}^{2}+{c}^{2}x+{\frac{{c}^{2}\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) }{f}}-2\,{\frac{{c}^{2}\ln \left ({{\rm e}^{fx+e}} \right ) }{f}}-2\,{\frac{{d}^{2}{e}^{2}\ln \left ({{\rm e}^{fx+e}} \right ) }{{f}^{3}}}+2\,{\frac{{d}^{2}{e}^{2}x}{{f}^{2}}}+{\frac{4\,{d}^{2}{e}^{3}}{3\,{f}^{3}}}+{\frac{{d}^{2}\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ){x}^{2}}{f}}+{\frac{{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,fx+2\,e}} \right ) x}{{f}^{2}}}-{\frac{{d}^{2}{\it polylog} \left ( 3,-{{\rm e}^{2\,fx+2\,e}} \right ) }{2\,{f}^{3}}}+4\,{\frac{cde\ln \left ({{\rm e}^{fx+e}} \right ) }{{f}^{2}}}-4\,{\frac{cdex}{f}}-2\,{\frac{cd{e}^{2}}{{f}^{2}}}+2\,{\frac{cd\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) x}{f}}+{\frac{cd{\it polylog} \left ( 2,-{{\rm e}^{2\,fx+2\,e}} \right ) }{{f}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.75544, size = 238, normalized size = 2.83 \begin{align*} \frac{1}{3} \, d^{2} x^{3} + c d x^{2} + \frac{c^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{2 \, f} + \frac{c^{2} \log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{2 \, f} + \frac{{\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )} c d}{f^{2}} + \frac{{\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x{\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) -{\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} d^{2}}{2 \, f^{3}} - \frac{2 \,{\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2}\right )}}{3 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 1.84925, size = 857, normalized size = 10.2 \begin{align*} -\frac{d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} + 3 \, c^{2} f^{3} x + 6 \, d^{2}{\rm polylog}\left (3, i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right )\right ) + 6 \, d^{2}{\rm polylog}\left (3, -i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right )\right ) - 6 \,{\left (d^{2} f x + c d f\right )}{\rm Li}_2\left (i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right )\right ) - 6 \,{\left (d^{2} f x + c d f\right )}{\rm Li}_2\left (-i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right )\right ) - 3 \,{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + i\right ) - 3 \,{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - i\right ) - 3 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} \log \left (i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right ) + 1\right ) - 3 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} \log \left (-i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right ) + 1\right )}{3 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{2} \tanh{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2} \tanh \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]