Optimal. Leaf size=126 \[ -\frac{\tan ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac{11 a^3 \tan ^4(c+d x)}{20 d}+\frac{4 i a^3 \tan ^3(c+d x)}{3 d}+\frac{2 a^3 \tan ^2(c+d x)}{d}-\frac{4 i a^3 \tan (c+d x)}{d}+\frac{4 a^3 \log (\cos (c+d x))}{d}+4 i a^3 x \]
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Rubi [A] time = 0.171435, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3556, 3592, 3528, 3525, 3475} \[ -\frac{\tan ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac{11 a^3 \tan ^4(c+d x)}{20 d}+\frac{4 i a^3 \tan ^3(c+d x)}{3 d}+\frac{2 a^3 \tan ^2(c+d x)}{d}-\frac{4 i a^3 \tan (c+d x)}{d}+\frac{4 a^3 \log (\cos (c+d x))}{d}+4 i a^3 x \]
Antiderivative was successfully verified.
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Rule 3556
Rule 3592
Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \tan ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{\tan ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\frac{1}{5} a \int \tan ^3(c+d x) (a+i a \tan (c+d x)) (9 a+11 i a \tan (c+d x)) \, dx\\ &=-\frac{11 a^3 \tan ^4(c+d x)}{20 d}-\frac{\tan ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\frac{1}{5} a \int \tan ^3(c+d x) \left (20 a^2+20 i a^2 \tan (c+d x)\right ) \, dx\\ &=\frac{4 i a^3 \tan ^3(c+d x)}{3 d}-\frac{11 a^3 \tan ^4(c+d x)}{20 d}-\frac{\tan ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\frac{1}{5} a \int \tan ^2(c+d x) \left (-20 i a^2+20 a^2 \tan (c+d x)\right ) \, dx\\ &=\frac{2 a^3 \tan ^2(c+d x)}{d}+\frac{4 i a^3 \tan ^3(c+d x)}{3 d}-\frac{11 a^3 \tan ^4(c+d x)}{20 d}-\frac{\tan ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\frac{1}{5} a \int \tan (c+d x) \left (-20 a^2-20 i a^2 \tan (c+d x)\right ) \, dx\\ &=4 i a^3 x-\frac{4 i a^3 \tan (c+d x)}{d}+\frac{2 a^3 \tan ^2(c+d x)}{d}+\frac{4 i a^3 \tan ^3(c+d x)}{3 d}-\frac{11 a^3 \tan ^4(c+d x)}{20 d}-\frac{\tan ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\left (4 a^3\right ) \int \tan (c+d x) \, dx\\ &=4 i a^3 x+\frac{4 a^3 \log (\cos (c+d x))}{d}-\frac{4 i a^3 \tan (c+d x)}{d}+\frac{2 a^3 \tan ^2(c+d x)}{d}+\frac{4 i a^3 \tan ^3(c+d x)}{3 d}-\frac{11 a^3 \tan ^4(c+d x)}{20 d}-\frac{\tan ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}\\ \end{align*}
Mathematica [B] time = 1.73758, size = 296, normalized size = 2.35 \[ \frac{a^3 \sec (c) \sec ^5(c+d x) \left (360 i \sin (2 c+d x)-280 i \sin (2 c+3 d x)+135 i \sin (4 c+3 d x)-83 i \sin (4 c+5 d x)+150 i d x \cos (2 c+3 d x)+105 \cos (2 c+3 d x)+150 i d x \cos (4 c+3 d x)+105 \cos (4 c+3 d x)+30 i d x \cos (4 c+5 d x)+30 i d x \cos (6 c+5 d x)+75 \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+75 \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )+15 \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )+15 \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )+75 \cos (d x) \left (2 \log \left (\cos ^2(c+d x)\right )+4 i d x+3\right )+75 \cos (2 c+d x) \left (2 \log \left (\cos ^2(c+d x)\right )+4 i d x+3\right )-470 i \sin (d x)\right )}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 118, normalized size = 0.9 \begin{align*}{\frac{-4\,i{a}^{3}\tan \left ( dx+c \right ) }{d}}-{\frac{{\frac{i}{5}}{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{3\,{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{\frac{4\,i}{3}}{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+2\,{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{{a}^{3}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+{\frac{4\,i{a}^{3}\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.28724, size = 128, normalized size = 1.02 \begin{align*} -\frac{12 i \, a^{3} \tan \left (d x + c\right )^{5} + 45 \, a^{3} \tan \left (d x + c\right )^{4} - 80 i \, a^{3} \tan \left (d x + c\right )^{3} - 120 \, a^{3} \tan \left (d x + c\right )^{2} - 240 i \,{\left (d x + c\right )} a^{3} + 120 \, a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 240 i \, a^{3} \tan \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41947, size = 612, normalized size = 4.86 \begin{align*} \frac{2 \,{\left (240 \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 585 \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 695 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 385 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 83 \, a^{3} + 30 \,{\left (a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{15 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.0092, size = 216, normalized size = 1.71 \begin{align*} \frac{4 a^{3} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{\frac{32 a^{3} e^{- 2 i c} e^{8 i d x}}{d} + \frac{78 a^{3} e^{- 4 i c} e^{6 i d x}}{d} + \frac{278 a^{3} e^{- 6 i c} e^{4 i d x}}{3 d} + \frac{154 a^{3} e^{- 8 i c} e^{2 i d x}}{3 d} + \frac{166 a^{3} e^{- 10 i c}}{15 d}}{e^{10 i d x} + 5 e^{- 2 i c} e^{8 i d x} + 10 e^{- 4 i c} e^{6 i d x} + 10 e^{- 6 i c} e^{4 i d x} + 5 e^{- 8 i c} e^{2 i d x} + e^{- 10 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.98299, size = 370, normalized size = 2.94 \begin{align*} \frac{2 \,{\left (30 \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 150 \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 300 \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 300 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 150 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 240 \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 585 \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 695 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 385 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 30 \, a^{3} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 83 \, a^{3}\right )}}{15 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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