Optimal. Leaf size=160 \[ -\frac{67 a^4 \tan ^4(c+d x)}{60 d}+\frac{8 i a^4 \tan ^3(c+d x)}{3 d}+\frac{4 a^4 \tan ^2(c+d x)}{d}-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 \tan ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac{8 i a^4 \tan (c+d x)}{d}+\frac{8 a^4 \log (\cos (c+d x))}{d}+8 i a^4 x \]
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Rubi [A] time = 0.271302, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3556, 3594, 3592, 3528, 3525, 3475} \[ -\frac{67 a^4 \tan ^4(c+d x)}{60 d}+\frac{8 i a^4 \tan ^3(c+d x)}{3 d}+\frac{4 a^4 \tan ^2(c+d x)}{d}-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 \tan ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac{8 i a^4 \tan (c+d x)}{d}+\frac{8 a^4 \log (\cos (c+d x))}{d}+8 i a^4 x \]
Antiderivative was successfully verified.
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Rule 3556
Rule 3594
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))^4 \, dx &=-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}+\frac{1}{6} a \int \tan ^3(c+d x) (a+i a \tan (c+d x))^2 (10 a+14 i a \tan (c+d x)) \, dx\\ &=-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 \tan ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}+\frac{1}{30} a \int \tan ^3(c+d x) (a+i a \tan (c+d x)) \left (106 a^2+134 i a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac{67 a^4 \tan ^4(c+d x)}{60 d}-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 \tan ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}+\frac{1}{30} a \int \tan ^3(c+d x) \left (240 a^3+240 i a^3 \tan (c+d x)\right ) \, dx\\ &=\frac{8 i a^4 \tan ^3(c+d x)}{3 d}-\frac{67 a^4 \tan ^4(c+d x)}{60 d}-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 \tan ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}+\frac{1}{30} a \int \tan ^2(c+d x) \left (-240 i a^3+240 a^3 \tan (c+d x)\right ) \, dx\\ &=\frac{4 a^4 \tan ^2(c+d x)}{d}+\frac{8 i a^4 \tan ^3(c+d x)}{3 d}-\frac{67 a^4 \tan ^4(c+d x)}{60 d}-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 \tan ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}+\frac{1}{30} a \int \tan (c+d x) \left (-240 a^3-240 i a^3 \tan (c+d x)\right ) \, dx\\ &=8 i a^4 x-\frac{8 i a^4 \tan (c+d x)}{d}+\frac{4 a^4 \tan ^2(c+d x)}{d}+\frac{8 i a^4 \tan ^3(c+d x)}{3 d}-\frac{67 a^4 \tan ^4(c+d x)}{60 d}-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 \tan ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\left (8 a^4\right ) \int \tan (c+d x) \, dx\\ &=8 i a^4 x+\frac{8 a^4 \log (\cos (c+d x))}{d}-\frac{8 i a^4 \tan (c+d x)}{d}+\frac{4 a^4 \tan ^2(c+d x)}{d}+\frac{8 i a^4 \tan ^3(c+d x)}{3 d}-\frac{67 a^4 \tan ^4(c+d x)}{60 d}-\frac{\tan ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 \tan ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}\\ \end{align*}
Mathematica [B] time = 2.26305, size = 349, normalized size = 2.18 \[ \frac{a^4 \sec (c) \sec ^6(c+d x) \left (-780 i \sin (c+2 d x)+510 i \sin (3 c+2 d x)-366 i \sin (3 c+4 d x)+150 i \sin (5 c+4 d x)-86 i \sin (5 c+6 d x)+450 i d x \cos (3 c+2 d x)+345 \cos (3 c+2 d x)+180 i d x \cos (3 c+4 d x)+120 \cos (3 c+4 d x)+180 i d x \cos (5 c+4 d x)+120 \cos (5 c+4 d x)+30 i d x \cos (5 c+6 d x)+30 i d x \cos (7 c+6 d x)+225 \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )+90 \cos (3 c+4 d x) \log \left (\cos ^2(c+d x)\right )+90 \cos (5 c+4 d x) \log \left (\cos ^2(c+d x)\right )+15 \cos (5 c+6 d x) \log \left (\cos ^2(c+d x)\right )+15 \cos (7 c+6 d x) \log \left (\cos ^2(c+d x)\right )+15 \cos (c+2 d x) \left (15 \log \left (\cos ^2(c+d x)\right )+30 i d x+23\right )+10 \cos (c) \left (30 \log \left (\cos ^2(c+d x)\right )+60 i d x+49\right )+860 i \sin (c)\right )}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 134, normalized size = 0.8 \begin{align*}{\frac{-8\,i{a}^{4}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,d}}-{\frac{{\frac{4\,i}{5}}{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{7\,{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{\frac{8\,i}{3}}{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+4\,{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{{a}^{4}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+{\frac{8\,i{a}^{4}\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 = 1.67289, size = 146, normalized size = 0.91 \begin{align*} \frac{10 \, a^{4} \tan \left (d x + c\right )^{6} - 48 i \, a^{4} \tan \left (d x + c\right )^{5} - 105 \, a^{4} \tan \left (d x + c\right )^{4} + 160 i \, a^{4} \tan \left (d x + c\right )^{3} + 240 \, a^{4} \tan \left (d x + c\right )^{2} + 480 i \,{\left (d x + c\right )} a^{4} - 240 \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 480 i \, a^{4} \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 = 2.27878, size = 738, normalized size = 4.61 \begin{align*} \frac{4 \,{\left (270 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 855 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 1350 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 1125 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 486 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 86 \, a^{4} + 30 \,{\left (a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{15 \,{\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, 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 = 14.456, size = 253, normalized size = 1.58 \begin{align*} \frac{8 a^{4} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{\frac{72 a^{4} e^{- 2 i c} e^{10 i d x}}{d} + \frac{228 a^{4} e^{- 4 i c} e^{8 i d x}}{d} + \frac{360 a^{4} e^{- 6 i c} e^{6 i d x}}{d} + \frac{300 a^{4} e^{- 8 i c} e^{4 i d x}}{d} + \frac{648 a^{4} e^{- 10 i c} e^{2 i d x}}{5 d} + \frac{344 a^{4} e^{- 12 i c}}{15 d}}{e^{12 i d x} + 6 e^{- 2 i c} e^{10 i d x} + 15 e^{- 4 i c} e^{8 i d x} + 20 e^{- 6 i c} e^{6 i d x} + 15 e^{- 8 i c} e^{4 i d x} + 6 e^{- 10 i c} e^{2 i d x} + e^{- 12 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.11888, size = 440, normalized size = 2.75 \begin{align*} \frac{4 \,{\left (30 \, a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 180 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 450 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 600 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 450 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 180 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 270 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 855 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 1350 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 1125 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 486 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 30 \, a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 86 \, a^{4}\right )}}{15 \,{\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, 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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