Optimal. Leaf size=162 \[ \frac{67 a^4 \cot ^4(c+d x)}{60 d}+\frac{8 i a^4 \cot ^3(c+d x)}{3 d}-\frac{4 a^4 \cot ^2(c+d x)}{d}-\frac{8 i a^4 \cot (c+d x)}{d}-\frac{8 a^4 \log (\sin (c+d x))}{d}-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-8 i a^4 x \]
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Rubi [A] time = 0.334155, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3553, 3593, 3591, 3529, 3531, 3475} \[ \frac{67 a^4 \cot ^4(c+d x)}{60 d}+\frac{8 i a^4 \cot ^3(c+d x)}{3 d}-\frac{4 a^4 \cot ^2(c+d x)}{d}-\frac{8 i a^4 \cot (c+d x)}{d}-\frac{8 a^4 \log (\sin (c+d x))}{d}-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-8 i a^4 x \]
Antiderivative was successfully verified.
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Rule 3553
Rule 3593
Rule 3591
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{1}{6} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \left (-14 i a^2+10 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac{1}{30} \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \left (134 a^3+106 i a^3 \tan (c+d x)\right ) \, dx\\ &=\frac{67 a^4 \cot ^4(c+d x)}{60 d}-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac{1}{30} \int \cot ^4(c+d x) \left (240 i a^4-240 a^4 \tan (c+d x)\right ) \, dx\\ &=\frac{8 i a^4 \cot ^3(c+d x)}{3 d}+\frac{67 a^4 \cot ^4(c+d x)}{60 d}-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac{1}{30} \int \cot ^3(c+d x) \left (-240 a^4-240 i a^4 \tan (c+d x)\right ) \, dx\\ &=-\frac{4 a^4 \cot ^2(c+d x)}{d}+\frac{8 i a^4 \cot ^3(c+d x)}{3 d}+\frac{67 a^4 \cot ^4(c+d x)}{60 d}-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac{1}{30} \int \cot ^2(c+d x) \left (-240 i a^4+240 a^4 \tan (c+d x)\right ) \, dx\\ &=-\frac{8 i a^4 \cot (c+d x)}{d}-\frac{4 a^4 \cot ^2(c+d x)}{d}+\frac{8 i a^4 \cot ^3(c+d x)}{3 d}+\frac{67 a^4 \cot ^4(c+d x)}{60 d}-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac{1}{30} \int \cot (c+d x) \left (240 a^4+240 i a^4 \tan (c+d x)\right ) \, dx\\ &=-8 i a^4 x-\frac{8 i a^4 \cot (c+d x)}{d}-\frac{4 a^4 \cot ^2(c+d x)}{d}+\frac{8 i a^4 \cot ^3(c+d x)}{3 d}+\frac{67 a^4 \cot ^4(c+d x)}{60 d}-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\left (8 a^4\right ) \int \cot (c+d x) \, dx\\ &=-8 i a^4 x-\frac{8 i a^4 \cot (c+d x)}{d}-\frac{4 a^4 \cot ^2(c+d x)}{d}+\frac{8 i a^4 \cot ^3(c+d x)}{3 d}+\frac{67 a^4 \cot ^4(c+d x)}{60 d}-\frac{8 a^4 \log (\sin (c+d x))}{d}-\frac{\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac{7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}\\ \end{align*}
Mathematica [B] time = 1.97959, size = 363, normalized size = 2.24 \[ \frac{a^4 \csc (c) \csc ^6(c+d x) \left (-600 i d x \sin (c)-450 i d x \sin (c+2 d x)-345 \sin (c+2 d x)+450 i d x \sin (3 c+2 d x)+345 \sin (3 c+2 d x)+180 i d x \sin (3 c+4 d x)+120 \sin (3 c+4 d x)-180 i d x \sin (5 c+4 d x)-120 \sin (5 c+4 d x)-30 i d x \sin (5 c+6 d x)+30 i d x \sin (7 c+6 d x)-780 i \cos (c+2 d x)-510 i \cos (3 c+2 d x)+366 i \cos (3 c+4 d x)+150 i \cos (5 c+4 d x)-86 i \cos (5 c+6 d x)-300 \sin (c) \log \left (\sin ^2(c+d x)\right )-225 \sin (c+2 d x) \log \left (\sin ^2(c+d x)\right )+225 \sin (3 c+2 d x) \log \left (\sin ^2(c+d x)\right )+90 \sin (3 c+4 d x) \log \left (\sin ^2(c+d x)\right )-90 \sin (5 c+4 d x) \log \left (\sin ^2(c+d x)\right )-15 \sin (5 c+6 d x) \log \left (\sin ^2(c+d x)\right )+15 \sin (7 c+6 d x) \log \left (\sin ^2(c+d x)\right )-490 \sin (c)+860 i \cos (c)\right )}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 131, normalized size = 0.8 \begin{align*} -4\,{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-8\,{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-8\,i{a}^{4}x-{\frac{{\frac{4\,i}{5}}{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{8\,i{a}^{4}c}{d}}-{\frac{8\,i\cot \left ( dx+c \right ){a}^{4}}{d}}+{\frac{7\,{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{\frac{8\,i}{3}}{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{6}}{6\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66716, size = 166, normalized size = 1.02 \begin{align*} -\frac{480 i \,{\left (d x + c\right )} a^{4} - 240 \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) - \frac{-480 i \, a^{4} \tan \left (d x + c\right )^{5} - 240 \, a^{4} \tan \left (d x + c\right )^{4} + 160 i \, a^{4} \tan \left (d x + c\right )^{3} + 105 \, a^{4} \tan \left (d x + c\right )^{2} - 48 i \, a^{4} \tan \left (d x + c\right ) - 10 \, a^{4}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2347, size = 738, normalized size = 4.56 \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 = 7.85581, size = 253, normalized size = 1.56 \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 [A] time = 1.8455, size = 332, normalized size = 2.05 \begin{align*} -\frac{5 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 48 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 240 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 880 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2835 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 30720 \, a^{4} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) + 15360 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 10080 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{37632 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 10080 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2835 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 880 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 240 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 48 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 5 \, a^{4}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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