Optimal. Leaf size=229 \[ \frac{2 i a b d (c+d x) \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac{a b d^2 \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}-\frac{i b^2 d^2 \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac{a^2 (c+d x)^3}{3 d}-\frac{2 a b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{2 i a b (c+d x)^3}{3 d}+\frac{2 b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac{b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac{i b^2 (c+d x)^2}{f}-\frac{b^2 (c+d x)^3}{3 d} \]
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Rubi [A] time = 0.412394, antiderivative size = 229, 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, 3719, 2190, 2531, 2282, 6589, 3720, 2279, 2391, 32} \[ \frac{a^2 (c+d x)^3}{3 d}+\frac{2 i a b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{2 a b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{2 i a b (c+d x)^3}{3 d}-\frac{a b d^2 \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac{2 b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac{b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac{i b^2 (c+d x)^2}{f}-\frac{b^2 (c+d x)^3}{3 d}-\frac{i b^2 d^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3} \]
Antiderivative was successfully verified.
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Rule 3722
Rule 3719
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 \tan (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \tan (e+f x)+b^2 (c+d x)^2 \tan ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}+(2 a b) \int (c+d x)^2 \tan (e+f x) \, dx+b^2 \int (c+d x)^2 \tan ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}+\frac{2 i a b (c+d x)^3}{3 d}+\frac{b^2 (c+d x)^2 \tan (e+f x)}{f}-(4 i a b) \int \frac{e^{2 i (e+f x)} (c+d x)^2}{1+e^{2 i (e+f x)}} \, dx-b^2 \int (c+d x)^2 \, dx-\frac{\left (2 b^2 d\right ) \int (c+d x) \tan (e+f x) \, dx}{f}\\ &=-\frac{i b^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}+\frac{2 i 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 i (e+f x)}\right )}{f}+\frac{b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac{(4 a b d) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}+\frac{\left (4 i b^2 d\right ) \int \frac{e^{2 i (e+f x)} (c+d x)}{1+e^{2 i (e+f x)}} \, dx}{f}\\ &=-\frac{i b^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}+\frac{2 i 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 i (e+f x)}\right )}{f^2}-\frac{2 a b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{2 i a b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}+\frac{b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac{\left (2 i a b d^2\right ) \int \text{Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac{\left (2 b^2 d^2\right ) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac{i b^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}+\frac{2 i 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 i (e+f x)}\right )}{f^2}-\frac{2 a b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{2 i a b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}+\frac{b^2 (c+d x)^2 \tan (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 i (e+f x)}\right )}{f^3}+\frac{\left (i b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{f^3}\\ &=-\frac{i b^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}+\frac{2 i 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 i (e+f x)}\right )}{f^2}-\frac{2 a b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac{i b^2 d^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac{2 i a b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{a b d^2 \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac{b^2 (c+d x)^2 \tan (e+f x)}{f}\\ \end{align*}
Mathematica [B] time = 7.09602, size = 649, normalized size = 2.83 \[ -\frac{2 a b c d \csc (e) \sec (e) \left (f^2 x^2 e^{-i \tan ^{-1}(\cot (e))}-\frac{\cot (e) \left (i \text{PolyLog}\left (2,e^{2 i \left (f x-\tan ^{-1}(\cot (e))\right )}\right )+i f x \left (-2 \tan ^{-1}(\cot (e))-\pi \right )-2 \left (f x-\tan ^{-1}(\cot (e))\right ) \log \left (1-e^{2 i \left (f x-\tan ^{-1}(\cot (e))\right )}\right )-2 \tan ^{-1}(\cot (e)) \log \left (\sin \left (f x-\tan ^{-1}(\cot (e))\right )\right )-\pi \log \left (1+e^{-2 i f x}\right )+\pi \log (\cos (f x))\right )}{\sqrt{\cot ^2(e)+1}}\right )}{f^2 \sqrt{\csc ^2(e) \left (\sin ^2(e)+\cos ^2(e)\right )}}-\frac{i a b d^2 e^{-i e} \sec (e) \left (6 \left (1+e^{2 i e}\right ) f x \text{PolyLog}\left (2,-e^{-2 i (e+f x)}\right )-3 i \left (1+e^{2 i e}\right ) \text{PolyLog}\left (3,-e^{-2 i (e+f x)}\right )+2 f^2 x^2 \left (2 f x-3 i \left (1+e^{2 i e}\right ) \log \left (1+e^{-2 i (e+f x)}\right )\right )\right )}{6 f^3}+\frac{b^2 d^2 \csc (e) \sec (e) \left (f^2 x^2 e^{-i \tan ^{-1}(\cot (e))}-\frac{\cot (e) \left (i \text{PolyLog}\left (2,e^{2 i \left (f x-\tan ^{-1}(\cot (e))\right )}\right )+i f x \left (-2 \tan ^{-1}(\cot (e))-\pi \right )-2 \left (f x-\tan ^{-1}(\cot (e))\right ) \log \left (1-e^{2 i \left (f x-\tan ^{-1}(\cot (e))\right )}\right )-2 \tan ^{-1}(\cot (e)) \log \left (\sin \left (f x-\tan ^{-1}(\cot (e))\right )\right )-\pi \log \left (1+e^{-2 i f x}\right )+\pi \log (\cos (f x))\right )}{\sqrt{\cot ^2(e)+1}}\right )}{f^3 \sqrt{\csc ^2(e) \left (\sin ^2(e)+\cos ^2(e)\right )}}+\frac{1}{3} x \sec (e) \left (3 c^2+3 c d x+d^2 x^2\right ) \left (a^2 \cos (e)+2 a b \sin (e)-b^2 \cos (e)\right )-\frac{2 a b c^2 \sec (e) (f x \sin (e)+\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x)))}{f \left (\sin ^2(e)+\cos ^2(e)\right )}+\frac{\sec (e) \sec (e+f x) \left (b^2 c^2 \sin (f x)+2 b^2 c d x \sin (f x)+b^2 d^2 x^2 \sin (f x)\right )}{f}+\frac{2 b^2 c d \sec (e) (f x \sin (e)+\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x)))}{f^2 \left (\sin ^2(e)+\cos ^2(e)\right )} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.109, size = 542, normalized size = 2.4 \begin{align*}{\frac{-i{b}^{2}{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{3}}}-2\,iab{c}^{2}x-{\frac{ab{d}^{2}{\it polylog} \left ( 3,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{3}}}+{a}^{2}cd{x}^{2}+{\frac{8\,ibacdex}{f}}-{\frac{2\,i{b}^{2}{d}^{2}{e}^{2}}{{f}^{3}}}-{\frac{2\,i{b}^{2}{d}^{2}{x}^{2}}{f}}+{\frac{2\,i}{3}}ab{d}^{2}{x}^{3}+2\,{\frac{{b}^{2}{d}^{2}\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) x}{{f}^{2}}}-2\,{\frac{ab{c}^{2}\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{f}}+4\,{\frac{ab{c}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{f}}+2\,{\frac{{b}^{2}cd\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{{f}^{2}}}-4\,{\frac{{b}^{2}cd\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+4\,{\frac{{b}^{2}{d}^{2}e\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}}}+4\,{\frac{ab{d}^{2}{e}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}}}-2\,{\frac{ab{d}^{2}\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ){x}^{2}}{f}}-{\frac{{\frac{8\,i}{3}}ba{d}^{2}{e}^{3}}{{f}^{3}}}-{\frac{4\,i{b}^{2}{d}^{2}ex}{{f}^{2}}}+{\frac{{a}^{2}{d}^{2}{x}^{3}}{3}}-{\frac{{b}^{2}{d}^{2}{x}^{3}}{3}}+{a}^{2}{c}^{2}x-{b}^{2}{c}^{2}x-{b}^{2}cd{x}^{2}+{\frac{2\,i{b}^{2} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2} \right ) }{f \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }}+2\,iabcd{x}^{2}-4\,{\frac{b\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) acdx}{f}}-8\,{\frac{abcde\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+{\frac{2\,iab{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) x}{{f}^{2}}}-{\frac{4\,iba{d}^{2}{e}^{2}x}{{f}^{2}}}+{\frac{4\,ibacd{e}^{2}}{{f}^{2}}}+{\frac{2\,iabcd{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.39943, size = 1713, normalized size = 7.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.77835, size = 1081, normalized size = 4.72 \begin{align*} \frac{2 \,{\left (a^{2} - b^{2}\right )} d^{2} f^{3} x^{3} + 6 \,{\left (a^{2} - b^{2}\right )} c d f^{3} x^{2} + 6 \,{\left (a^{2} - b^{2}\right )} c^{2} f^{3} x - 3 \, a b d^{2}{\rm polylog}\left (3, \frac{\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, a b d^{2}{\rm polylog}\left (3, \frac{\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) +{\left (-6 i \, a b d^{2} f x - 6 i \, a b c d f + 3 i \, b^{2} d^{2}\right )}{\rm Li}_2\left (\frac{2 \,{\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) +{\left (6 i \, a b d^{2} f x + 6 i \, a b c d f - 3 i \, b^{2} d^{2}\right )}{\rm Li}_2\left (\frac{2 \,{\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 6 \,{\left (a b d^{2} f^{2} x^{2} + a b c^{2} f^{2} - b^{2} c d f +{\left (2 \, a b c d f^{2} - b^{2} d^{2} f\right )} x\right )} \log \left (-\frac{2 \,{\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \,{\left (a b d^{2} f^{2} x^{2} + a b c^{2} f^{2} - b^{2} c d f +{\left (2 \, a b c d f^{2} - b^{2} d^{2} f\right )} x\right )} \log \left (-\frac{2 \,{\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \,{\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} \tan \left (f x + e\right )}{6 \, f^{3}} \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 \tan{\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 \tan \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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