Optimal. Leaf size=211 \[ \frac{2 a b d (c+d x) \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac{a b d^2 \text{PolyLog}\left (3,-e^{2 (e+f x)}\right )}{f^3}+\frac{b^2 d^2 \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac{a^2 (c+d x)^3}{3 d}+\frac{2 a b (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac{2 a b (c+d x)^3}{3 d}+\frac{2 b^2 d (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac{b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac{b^2 (c+d x)^2}{f}+\frac{b^2 (c+d x)^3}{3 d} \]
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Rubi [A] time = 0.396205, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3722, 3718, 2190, 2531, 2282, 6589, 3720, 2279, 2391, 32} \[ \frac{2 a b d (c+d x) \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac{a b d^2 \text{PolyLog}\left (3,-e^{2 (e+f x)}\right )}{f^3}+\frac{b^2 d^2 \text{PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac{a^2 (c+d x)^3}{3 d}+\frac{2 a b (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac{2 a b (c+d x)^3}{3 d}+\frac{2 b^2 d (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac{b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac{b^2 (c+d x)^2}{f}+\frac{b^2 (c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3722
Rule 3718
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 3720
Rule 2279
Rule 2391
Rule 32
Rubi steps
\begin{align*} \int (c+d x)^2 (a+b \tanh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \tanh (e+f x)+b^2 (c+d x)^2 \tanh ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}+(2 a b) \int (c+d x)^2 \tanh (e+f x) \, dx+b^2 \int (c+d x)^2 \tanh ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}-\frac{2 a b (c+d x)^3}{3 d}-\frac{b^2 (c+d x)^2 \tanh (e+f x)}{f}+(4 a b) \int \frac{e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx+b^2 \int (c+d x)^2 \, dx+\frac{\left (2 b^2 d\right ) \int (c+d x) \tanh (e+f x) \, dx}{f}\\ &=-\frac{b^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}-\frac{2 a b (c+d x)^3}{3 d}+\frac{b^2 (c+d x)^3}{3 d}+\frac{2 a b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac{b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac{(4 a b d) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}+\frac{\left (4 b^2 d\right ) \int \frac{e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx}{f}\\ &=-\frac{b^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}-\frac{2 a b (c+d x)^3}{3 d}+\frac{b^2 (c+d x)^3}{3 d}+\frac{2 b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac{2 a b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac{2 a b d (c+d x) \text{Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac{b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac{\left (2 a b d^2\right ) \int \text{Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac{\left (2 b^2 d^2\right ) \int \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac{b^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}-\frac{2 a b (c+d x)^3}{3 d}+\frac{b^2 (c+d x)^3}{3 d}+\frac{2 b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac{2 a b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac{2 a b d (c+d x) \text{Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac{b^2 (c+d x)^2 \tanh (e+f x)}{f}-\frac{\left (a b d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^3}-\frac{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^3}\\ &=-\frac{b^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}-\frac{2 a b (c+d x)^3}{3 d}+\frac{b^2 (c+d x)^3}{3 d}+\frac{2 b^2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac{2 a b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac{b^2 d^2 \text{Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac{2 a b d (c+d x) \text{Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac{a b d^2 \text{Li}_3\left (-e^{2 (e+f x)}\right )}{f^3}-\frac{b^2 (c+d x)^2 \tanh (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 5.51275, size = 232, normalized size = 1.1 \[ \frac{1}{3} \left (\frac{b \left (-3 d \text{PolyLog}\left (2,-e^{-2 (e+f x)}\right ) (2 a f (c+d x)+b d)-3 a d^2 \text{PolyLog}\left (3,-e^{-2 (e+f x)}\right )+2 f \left (\frac{f x \left (2 a f \left (-3 c^2 e^{2 e}+3 c d x+d^2 x^2\right )+3 b d \left (d x-2 c e^{2 e}\right )\right )}{e^{2 e}+1}+3 d x \log \left (e^{-2 (e+f x)}+1\right ) (a f (2 c+d x)+b d)+3 c \log \left (e^{2 (e+f x)}+1\right ) (a c f+b d)\right )\right )}{f^3}+x \left (3 c^2+3 c d x+d^2 x^2\right ) \left (a^2+2 a b \tanh (e)+b^2\right )-\frac{3 b^2 \text{sech}(e) (c+d x)^2 \sinh (f x) \text{sech}(e+f x)}{f}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.087, size = 510, normalized size = 2.4 \begin{align*}{\frac{{b}^{2}{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,fx+2\,e}} \right ) }{{f}^{3}}}+{\frac{{a}^{2}{d}^{2}{x}^{3}}{3}}+{\frac{{b}^{2}{d}^{2}{x}^{3}}{3}}+{c}^{2}{a}^{2}x+{c}^{2}{b}^{2}x-2\,{\frac{{b}^{2}{d}^{2}{x}^{2}}{f}}-2\,{\frac{{b}^{2}{d}^{2}{e}^{2}}{{f}^{3}}}+4\,{\frac{b\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) acdx}{f}}+8\,{\frac{cdbae\ln \left ({{\rm e}^{fx+e}} \right ) }{{f}^{2}}}-8\,{\frac{cdbaex}{f}}+{\frac{8\,ab{d}^{2}{e}^{3}}{3\,{f}^{3}}}-4\,{\frac{{b}^{2}{d}^{2}ex}{{f}^{2}}}+2\,{\frac{{b}^{2}{d}^{2}\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) x}{{f}^{2}}}-4\,{\frac{ab{c}^{2}\ln \left ({{\rm e}^{fx+e}} \right ) }{f}}+2\,{\frac{ab{c}^{2}\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) }{f}}-4\,{\frac{{b}^{2}cd\ln \left ({{\rm e}^{fx+e}} \right ) }{{f}^{2}}}+2\,{\frac{{b}^{2}cd\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) }{{f}^{2}}}+4\,{\frac{{b}^{2}{d}^{2}e\ln \left ({{\rm e}^{fx+e}} \right ) }{{f}^{3}}}+2\,{\frac{{b}^{2} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2} \right ) }{f \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ) }}-{\frac{2\,ab{d}^{2}{x}^{3}}{3}}+2\,ab{c}^{2}x-2\,abcd{x}^{2}-{\frac{ab{d}^{2}{\it polylog} \left ( 3,-{{\rm e}^{2\,fx+2\,e}} \right ) }{{f}^{3}}}+{a}^{2}cd{x}^{2}+{b}^{2}cd{x}^{2}-4\,{\frac{cdba{e}^{2}}{{f}^{2}}}+4\,{\frac{ab{d}^{2}{e}^{2}x}{{f}^{2}}}+2\,{\frac{ab{d}^{2}\ln \left ({{\rm e}^{2\,fx+2\,e}}+1 \right ){x}^{2}}{f}}+2\,{\frac{ab{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,fx+2\,e}} \right ) x}{{f}^{2}}}+2\,{\frac{cdba{\it polylog} \left ( 2,-{{\rm e}^{2\,fx+2\,e}} \right ) }{{f}^{2}}}-4\,{\frac{ab{d}^{2}{e}^{2}\ln \left ({{\rm e}^{fx+e}} \right ) }{{f}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46856, size = 555, normalized size = 2.63 \begin{align*} \frac{1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + b^{2} c^{2}{\left (x + \frac{e}{f} - \frac{2}{f{\left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}}\right )} + a^{2} c^{2} x + b^{2} c d{\left (\frac{f x^{2} +{\left (f x^{2} e^{\left (2 \, e\right )} - 4 \, x e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f e^{\left (2 \, f x + 2 \, e\right )} + f} + \frac{2 \, \log \left ({\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )} e^{\left (-2 \, e\right )}\right )}{f^{2}}\right )} + \frac{2 \, a b c^{2} \log \left (\cosh \left (f x + e\right )\right )}{f} + \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 )} a b d^{2}}{f^{3}} + \frac{{\left (2 \, a b d^{2} f + b^{2} d^{2} f\right )} x^{3} + 6 \,{\left (a b c d f + b^{2} d^{2}\right )} x^{2} +{\left (6 \, a b c d f x^{2} e^{\left (2 \, e\right )} +{\left (2 \, a b d^{2} f e^{\left (2 \, e\right )} + b^{2} d^{2} f e^{\left (2 \, e\right )}\right )} x^{3}\right )} e^{\left (2 \, f x\right )}}{3 \,{\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )}} + \frac{{\left (2 \, a b c d f + b^{2} d^{2}\right )}{\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 )}}{f^{3}} - \frac{2 \,{\left (2 \, a b d^{2} f^{3} x^{3} + 3 \,{\left (2 \, a b c d f + b^{2} d^{2}\right )} f^{2} x^{2}\right )}}{3 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.90322, size = 4886, normalized size = 23.16 \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 (a + b \tanh{\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}\, 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 )}^{2}{\left (b \tanh \left (f x + e\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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