Optimal. Leaf size=103 \[ \frac {4 a^4 \cot (c+d x)}{d}-\frac {8 i a^4 \log (\sin (c+d x))}{d}+8 a^4 x-\frac {i \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.16, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3545, 3542, 3531, 3475} \[ \frac {4 a^4 \cot (c+d x)}{d}-\frac {8 i a^4 \log (\sin (c+d x))}{d}-\frac {i \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+8 a^4 x-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3542
Rule 3545
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+(2 i a) \int \cot ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {i \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 a^2\right ) \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=\frac {4 a^4 \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {i \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 a^2\right ) \int \cot (c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=8 a^4 x+\frac {4 a^4 \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {i \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (8 i a^4\right ) \int \cot (c+d x) \, dx\\ &=8 a^4 x+\frac {4 a^4 \cot (c+d x)}{d}-\frac {8 i a^4 \log (\sin (c+d x))}{d}-\frac {a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {i \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\\ \end {align*}
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Mathematica [B] time = 1.09, size = 240, normalized size = 2.33 \[ \frac {a^4 \csc (c) \csc ^3(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left (-12 \sin (2 c+d x)+11 \sin (2 c+3 d x)-36 d x \cos (2 c+d x)+6 i \cos (2 c+d x)-12 d x \cos (2 c+3 d x)+12 d x \cos (4 c+3 d x)-48 \sin (c) \sin ^3(c+d x) \tan ^{-1}(\tan (5 c+d x))+\cos (d x) \left (-9 i \log \left (\sin ^2(c+d x)\right )+36 d x-6 i\right )+9 i \cos (2 c+d x) \log \left (\sin ^2(c+d x)\right )+3 i \cos (2 c+3 d x) \log \left (\sin ^2(c+d x)\right )-3 i \cos (4 c+3 d x) \log \left (\sin ^2(c+d x)\right )-21 \sin (d x)\right )}{6 d (\cos (d x)+i \sin (d x))^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 138, normalized size = 1.34 \[ \frac {72 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 108 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 44 i \, a^{4} + {\left (-24 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 72 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 72 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 24 i \, a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.64, size = 146, normalized size = 1.42 \[ \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 384 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 192 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 87 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {-352 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 87 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 80, normalized size = 0.78 \[ 8 a^{4} x +\frac {8 a^{4} c}{d}-\frac {8 i a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {7 a^{4} \cot \left (d x +c \right )}{d}-\frac {2 i a^{4} \left (\cot ^{2}\left (d x +c \right )\right )}{d}-\frac {a^{4} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 83, normalized size = 0.81 \[ \frac {24 \, {\left (d x + c\right )} a^{4} + 12 i \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 24 i \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {21 \, a^{4} \tan \left (d x + c\right )^{2} - 6 i \, a^{4} \tan \left (d x + c\right ) - a^{4}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.98, size = 68, normalized size = 0.66 \[ \frac {7\,a^4\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {16\,a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {a^4\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d}-\frac {a^4\,{\mathrm {cot}\left (c+d\,x\right )}^2\,2{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 136, normalized size = 1.32 \[ - \frac {8 i a^{4} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 72 i a^{4} e^{4 i c} e^{4 i d x} + 108 i a^{4} e^{2 i c} e^{2 i d x} - 44 i a^{4}}{- 3 d e^{6 i c} e^{6 i d x} + 9 d e^{4 i c} e^{4 i d x} - 9 d e^{2 i c} e^{2 i d x} + 3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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