Optimal. Leaf size=101 \[ \frac {2 a^3 \cot (c+d x)}{d}-\frac {4 i a^3 \log (\sin (c+d x))}{d}+4 a^3 x-\frac {\cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {i a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.15, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3548, 3545, 3542, 3531, 3475} \[ \frac {2 a^3 \cot (c+d x)}{d}-\frac {4 i a^3 \log (\sin (c+d x))}{d}+4 a^3 x-\frac {i a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}-\frac {\cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3542
Rule 3545
Rule 3548
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {\cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+i \int \cot ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac {i a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}-\frac {\cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-(2 a) \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=\frac {2 a^3 \cot (c+d x)}{d}-\frac {i a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}-\frac {\cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-(2 a) \int \cot (c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=4 a^3 x+\frac {2 a^3 \cot (c+d x)}{d}-\frac {i a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}-\frac {\cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\left (4 i a^3\right ) \int \cot (c+d x) \, dx\\ &=4 a^3 x+\frac {2 a^3 \cot (c+d x)}{d}-\frac {4 i a^3 \log (\sin (c+d x))}{d}-\frac {i a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}-\frac {\cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\\ \end {align*}
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Mathematica [B] time = 1.33, size = 251, normalized size = 2.49 \[ \frac {a^3 \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \csc ^3(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left (-15 \sin (2 c+d x)+13 \sin (2 c+3 d x)-36 d x \cos (2 c+d x)+9 i \cos (2 c+d x)-12 d x \cos (2 c+3 d x)+12 d x \cos (4 c+3 d x)-48 \sin (c) \sin ^3(c+d x) \tan ^{-1}(\tan (4 c+d x))+9 \cos (d x) \left (-i \log \left (\sin ^2(c+d x)\right )+4 d x-i\right )+9 i \cos (2 c+d x) \log \left (\sin ^2(c+d x)\right )+3 i \cos (2 c+3 d x) \log \left (\sin ^2(c+d x)\right )-3 i \cos (4 c+3 d x) \log \left (\sin ^2(c+d x)\right )-24 \sin (d x)\right )}{24 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 138, normalized size = 1.37 \[ \frac {48 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 66 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 26 i \, a^{3} + {\left (-12 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 36 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 36 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 12 i \, a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.18, size = 146, normalized size = 1.45 \[ \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 192 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 96 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 51 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {-176 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 51 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 80, normalized size = 0.79 \[ -\frac {4 i a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}+4 a^{3} x +\frac {4 a^{3} \cot \left (d x +c \right )}{d}+\frac {4 a^{3} c}{d}-\frac {3 i a^{3} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 83, normalized size = 0.82 \[ \frac {24 \, {\left (d x + c\right )} a^{3} + 12 i \, a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 24 i \, a^{3} \log \left (\tan \left (d x + c\right )\right ) + \frac {24 \, a^{3} \tan \left (d x + c\right )^{2} - 9 i \, a^{3} \tan \left (d x + c\right ) - 2 \, a^{3}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 68, normalized size = 0.67 \[ \frac {4\,a^3\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {8\,a^3\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {a^3\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d}-\frac {a^3\,{\mathrm {cot}\left (c+d\,x\right )}^2\,3{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 136, normalized size = 1.35 \[ - \frac {4 i a^{3} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 48 i a^{3} e^{4 i c} e^{4 i d x} + 66 i a^{3} e^{2 i c} e^{2 i d x} - 26 i a^{3}}{- 3 d e^{6 i c} e^{6 i d x} + 9 d e^{4 i c} e^{4 i d x} - 9 d e^{2 i c} e^{2 i d x} + 3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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