Optimal. Leaf size=141 \[ -\frac{a^2 (c+d \tan (e+f x))^4}{4 d f}+\frac{2 i a^2 (c+d \tan (e+f x))^3}{3 f}+\frac{a^2 (d+i c) (c+d \tan (e+f x))^2}{f}+\frac{2 i a^2 d (c-i d)^2 \tan (e+f x)}{f}+\frac{2 a^2 (d+i c)^3 \log (\cos (e+f x))}{f}+2 a^2 x (c-i d)^3 \]
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Rubi [A] time = 0.199663, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3543, 3528, 3525, 3475} \[ -\frac{a^2 (c+d \tan (e+f x))^4}{4 d f}+\frac{2 i a^2 (c+d \tan (e+f x))^3}{3 f}+\frac{a^2 (d+i c) (c+d \tan (e+f x))^2}{f}+\frac{2 i a^2 d (c-i d)^2 \tan (e+f x)}{f}+\frac{2 a^2 (d+i c)^3 \log (\cos (e+f x))}{f}+2 a^2 x (c-i d)^3 \]
Antiderivative was successfully verified.
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Rule 3543
Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx &=-\frac{a^2 (c+d \tan (e+f x))^4}{4 d f}+\int \left (2 a^2+2 i a^2 \tan (e+f x)\right ) (c+d \tan (e+f x))^3 \, dx\\ &=\frac{2 i a^2 (c+d \tan (e+f x))^3}{3 f}-\frac{a^2 (c+d \tan (e+f x))^4}{4 d f}+\int (c+d \tan (e+f x))^2 \left (2 a^2 (c-i d)+2 a^2 (i c+d) \tan (e+f x)\right ) \, dx\\ &=\frac{a^2 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac{2 i a^2 (c+d \tan (e+f x))^3}{3 f}-\frac{a^2 (c+d \tan (e+f x))^4}{4 d f}+\int \left (2 a^2 (c-i d)^2+2 i a^2 (c-i d)^2 \tan (e+f x)\right ) (c+d \tan (e+f x)) \, dx\\ &=2 a^2 (c-i d)^3 x+\frac{2 i a^2 (c-i d)^2 d \tan (e+f x)}{f}+\frac{a^2 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac{2 i a^2 (c+d \tan (e+f x))^3}{3 f}-\frac{a^2 (c+d \tan (e+f x))^4}{4 d f}-\left (2 a^2 (i c+d)^3\right ) \int \tan (e+f x) \, dx\\ &=2 a^2 (c-i d)^3 x+\frac{2 a^2 (i c+d)^3 \log (\cos (e+f x))}{f}+\frac{2 i a^2 (c-i d)^2 d \tan (e+f x)}{f}+\frac{a^2 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac{2 i a^2 (c+d \tan (e+f x))^3}{3 f}-\frac{a^2 (c+d \tan (e+f x))^4}{4 d f}\\ \end{align*}
Mathematica [B] time = 7.85022, size = 733, normalized size = 5.2 \[ \frac{\sec ^2(e+f x) (a+i a \tan (e+f x))^2 \left (\frac{1}{24} \sec (e) (\cos (2 e)-i \sin (2 e)) \left (6 \cos (e) \left (c^2 d (-3-9 i f x)+3 c^3 f x+3 c d^2 (-3 f x+2 i)+d^3 (2+3 i f x)\right )+54 i c^2 d \sin (e+2 f x)-18 i c^2 d \sin (3 e+2 f x)+18 i c^2 d \sin (3 e+4 f x)-9 c^2 d \cos (3 e+2 f x)-36 i c^2 d f x \cos (3 e+2 f x)-9 i c^2 d f x \cos (3 e+4 f x)-9 i c^2 d f x \cos (5 e+4 f x)-54 i c^2 d \sin (e)-9 c^3 \sin (e+2 f x)+3 c^3 \sin (3 e+2 f x)-3 c^3 \sin (3 e+4 f x)+12 c^3 f x \cos (3 e+2 f x)+3 c^3 f x \cos (3 e+4 f x)+3 c^3 f x \cos (5 e+4 f x)+9 c^3 \sin (e)+57 c d^2 \sin (e+2 f x)-27 c d^2 \sin (3 e+2 f x)+21 c d^2 \sin (3 e+4 f x)+18 i c d^2 \cos (3 e+2 f x)-36 c d^2 f x \cos (3 e+2 f x)-9 c d^2 f x \cos (3 e+4 f x)-9 c d^2 f x \cos (5 e+4 f x)-63 c d^2 \sin (e)+3 (c-i d)^2 (4 c f x-4 i d f x-3 d) \cos (e+2 f x)-20 i d^3 \sin (e+2 f x)+12 i d^3 \sin (3 e+2 f x)-8 i d^3 \sin (3 e+4 f x)+9 d^3 \cos (3 e+2 f x)+12 i d^3 f x \cos (3 e+2 f x)+3 i d^3 f x \cos (3 e+4 f x)+3 i d^3 f x \cos (5 e+4 f x)+24 i d^3 \sin (e)\right )+2 f x (c-i d)^3 (\cos (2 e)-i \sin (2 e)) \cos ^4(e+f x)+(c-i d)^3 (-\sin (2 e)-i \cos (2 e)) \cos ^4(e+f x) \log \left (\cos ^2(e+f x)\right )-2 (c-i d)^3 (\cos (2 e)-i \sin (2 e)) \cos ^4(e+f x) \tan ^{-1}(\tan (3 e+f x))\right )}{f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 360, normalized size = 2.6 \begin{align*}{\frac{2\,i{a}^{2}\arctan \left ( \tan \left ( fx+e \right ) \right ){d}^{3}}{f}}-{\frac{{a}^{2}{d}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{4}}{4\,f}}+{\frac{6\,i{a}^{2}{c}^{2}d\tan \left ( fx+e \right ) }{f}}-{\frac{{a}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}c{d}^{2}}{f}}+{\frac{{\frac{2\,i}{3}}{a}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}{d}^{3}}{f}}+{\frac{i{a}^{2}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){c}^{3}}{f}}-{\frac{3\,{a}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{2}{c}^{2}d}{2\,f}}+{\frac{{a}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{2}{d}^{3}}{f}}-{\frac{{a}^{2}{c}^{3}\tan \left ( fx+e \right ) }{f}}+6\,{\frac{{a}^{2}\tan \left ( fx+e \right ) c{d}^{2}}{f}}-{\frac{6\,i{a}^{2}\arctan \left ( \tan \left ( fx+e \right ) \right ){c}^{2}d}{f}}-{\frac{3\,i{a}^{2}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) c{d}^{2}}{f}}+3\,{\frac{{a}^{2}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){c}^{2}d}{f}}-{\frac{{a}^{2}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){d}^{3}}{f}}+{\frac{3\,i{a}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{2}c{d}^{2}}{f}}-{\frac{2\,i{a}^{2}{d}^{3}\tan \left ( fx+e \right ) }{f}}+2\,{\frac{{a}^{2}\arctan \left ( \tan \left ( fx+e \right ) \right ){c}^{3}}{f}}-6\,{\frac{{a}^{2}\arctan \left ( \tan \left ( fx+e \right ) \right ) c{d}^{2}}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51292, size = 294, normalized size = 2.09 \begin{align*} -\frac{3 \, a^{2} d^{3} \tan \left (f x + e\right )^{4} +{\left (12 \, a^{2} c d^{2} - 8 i \, a^{2} d^{3}\right )} \tan \left (f x + e\right )^{3} + 6 \,{\left (3 \, a^{2} c^{2} d - 6 i \, a^{2} c d^{2} - 2 \, a^{2} d^{3}\right )} \tan \left (f x + e\right )^{2} - 12 \,{\left (2 \, a^{2} c^{3} - 6 i \, a^{2} c^{2} d - 6 \, a^{2} c d^{2} + 2 i \, a^{2} d^{3}\right )}{\left (f x + e\right )} - 12 \,{\left (i \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 3 i \, a^{2} c d^{2} - a^{2} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) +{\left (12 \, a^{2} c^{3} - 72 i \, a^{2} c^{2} d - 72 \, a^{2} c d^{2} + 24 i \, a^{2} d^{3}\right )} \tan \left (f x + e\right )}{12 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63543, size = 1141, normalized size = 8.09 \begin{align*} \frac{-6 i \, a^{2} c^{3} - 36 \, a^{2} c^{2} d + 42 i \, a^{2} c d^{2} + 16 \, a^{2} d^{3} +{\left (-6 i \, a^{2} c^{3} - 54 \, a^{2} c^{2} d + 90 i \, a^{2} c d^{2} + 42 \, a^{2} d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-18 i \, a^{2} c^{3} - 144 \, a^{2} c^{2} d + 198 i \, a^{2} c d^{2} + 72 \, a^{2} d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-18 i \, a^{2} c^{3} - 126 \, a^{2} c^{2} d + 150 i \, a^{2} c d^{2} + 58 \, a^{2} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-6 i \, a^{2} c^{3} - 18 \, a^{2} c^{2} d + 18 i \, a^{2} c d^{2} + 6 \, a^{2} d^{3} +{\left (-6 i \, a^{2} c^{3} - 18 \, a^{2} c^{2} d + 18 i \, a^{2} c d^{2} + 6 \, a^{2} d^{3}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-24 i \, a^{2} c^{3} - 72 \, a^{2} c^{2} d + 72 i \, a^{2} c d^{2} + 24 \, a^{2} d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-36 i \, a^{2} c^{3} - 108 \, a^{2} c^{2} d + 108 i \, a^{2} c d^{2} + 36 \, a^{2} d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-24 i \, a^{2} c^{3} - 72 \, a^{2} c^{2} d + 72 i \, a^{2} c d^{2} + 24 \, a^{2} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{3 \,{\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 89.5474, size = 343, normalized size = 2.43 \begin{align*} \frac{2 a^{2} \left (- i c^{3} - 3 c^{2} d + 3 i c d^{2} + d^{3}\right ) \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} + \frac{- \frac{\left (2 i a^{2} c^{3} + 18 a^{2} c^{2} d - 30 i a^{2} c d^{2} - 14 a^{2} d^{3}\right ) e^{- 2 i e} e^{6 i f x}}{f} - \frac{\left (6 i a^{2} c^{3} + 36 a^{2} c^{2} d - 42 i a^{2} c d^{2} - 16 a^{2} d^{3}\right ) e^{- 8 i e}}{3 f} - \frac{\left (6 i a^{2} c^{3} + 48 a^{2} c^{2} d - 66 i a^{2} c d^{2} - 24 a^{2} d^{3}\right ) e^{- 4 i e} e^{4 i f x}}{f} - \frac{\left (18 i a^{2} c^{3} + 126 a^{2} c^{2} d - 150 i a^{2} c d^{2} - 58 a^{2} d^{3}\right ) e^{- 6 i e} e^{2 i f x}}{3 f}}{e^{8 i f x} + 4 e^{- 2 i e} e^{6 i f x} + 6 e^{- 4 i e} e^{4 i f x} + 4 e^{- 6 i e} e^{2 i f x} + e^{- 8 i e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.14647, size = 1220, normalized size = 8.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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