Optimal. Leaf size=107 \[ -\frac{(A-2 i B) \left (a^3+i a^3 \tan (c+d x)\right )}{d}+\frac{a^3 (3 A-4 i B) \log (\cos (c+d x))}{d}+4 a^3 x (B+i A)+\frac{a^3 A \log (\sin (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.280963, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3594, 3589, 3475, 3531} \[ -\frac{(A-2 i B) \left (a^3+i a^3 \tan (c+d x)\right )}{d}+\frac{a^3 (3 A-4 i B) \log (\cos (c+d x))}{d}+4 a^3 x (B+i A)+\frac{a^3 A \log (\sin (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 3594
Rule 3589
Rule 3475
Rule 3531
Rubi steps
\begin{align*} \int \cot (c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=\frac{i a B (a+i a \tan (c+d x))^2}{2 d}+\frac{1}{2} \int \cot (c+d x) (a+i a \tan (c+d x))^2 (2 a A+2 a (i A+2 B) \tan (c+d x)) \, dx\\ &=\frac{i a B (a+i a \tan (c+d x))^2}{2 d}-\frac{(A-2 i B) \left (a^3+i a^3 \tan (c+d x)\right )}{d}+\frac{1}{2} \int \cot (c+d x) (a+i a \tan (c+d x)) \left (2 a^2 A+2 a^2 (3 i A+4 B) \tan (c+d x)\right ) \, dx\\ &=\frac{i a B (a+i a \tan (c+d x))^2}{2 d}-\frac{(A-2 i B) \left (a^3+i a^3 \tan (c+d x)\right )}{d}+\frac{1}{2} \int \cot (c+d x) \left (2 a^3 A+8 a^3 (i A+B) \tan (c+d x)\right ) \, dx-\left (a^3 (3 A-4 i B)\right ) \int \tan (c+d x) \, dx\\ &=4 a^3 (i A+B) x+\frac{a^3 (3 A-4 i B) \log (\cos (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^2}{2 d}-\frac{(A-2 i B) \left (a^3+i a^3 \tan (c+d x)\right )}{d}+\left (a^3 A\right ) \int \cot (c+d x) \, dx\\ &=4 a^3 (i A+B) x+\frac{a^3 (3 A-4 i B) \log (\cos (c+d x))}{d}+\frac{a^3 A \log (\sin (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^2}{2 d}-\frac{(A-2 i B) \left (a^3+i a^3 \tan (c+d x)\right )}{d}\\ \end{align*}
Mathematica [B] time = 7.65867, size = 281, normalized size = 2.63 \[ \frac{a^3 \sec (c) \sec ^2(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left (2 \cos (c) \left ((3 A-4 i B) \log \left (\cos ^2(c+d x)\right )+A \log \left (\sin ^2(c+d x)\right )+8 i A d x+8 B d x-2 i B\right )+\cos (c+2 d x) \left ((3 A-4 i B) \log \left (\cos ^2(c+d x)\right )+8 d x (B+i A)+A \log \left (\sin ^2(c+d x)\right )\right )-4 i A \sin (c+2 d x)+8 i A d x \cos (3 c+2 d x)+3 A \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )+A \cos (3 c+2 d x) \log \left (\sin ^2(c+d x)\right )+4 i A \sin (c)-12 B \sin (c+2 d x)+8 B d x \cos (3 c+2 d x)-4 i B \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )+12 B \sin (c)\right )}{8 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 135, normalized size = 1.3 \begin{align*} 4\,iAx{a}^{3}-{\frac{iA\tan \left ( dx+c \right ){a}^{3}}{d}}+{\frac{4\,iA{a}^{3}c}{d}}-{\frac{{\frac{i}{2}}B{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{4\,iB{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{A{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+4\,B{a}^{3}x-3\,{\frac{B{a}^{3}\tan \left ( dx+c \right ) }{d}}+4\,{\frac{B{a}^{3}c}{d}}+{\frac{A{a}^{3}\ln \left ( \sin \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 = 1.68772, size = 123, normalized size = 1.15 \begin{align*} -\frac{i \, B a^{3} \tan \left (d x + c\right )^{2} + 8 \,{\left (d x + c\right )}{\left (-i \, A - B\right )} a^{3} + 2 \,{\left (2 \, A - 2 i \, B\right )} a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, A a^{3} \log \left (\tan \left (d x + c\right )\right ) + 2 \,{\left (i \, A + 3 \, B\right )} a^{3} \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44725, size = 466, normalized size = 4.36 \begin{align*} \frac{2 \,{\left (A - 4 i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \,{\left (A - 3 i \, B\right )} a^{3} +{\left ({\left (3 \, A - 4 i \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \,{\left (3 \, A - 4 i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (3 \, A - 4 i \, B\right )} a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) +{\left (A a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, A a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + A a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.41282, size = 207, normalized size = 1.93 \begin{align*} \frac{\frac{\left (2 A a^{3} - 8 i B a^{3}\right ) e^{- 2 i c} e^{2 i d x}}{d} + \frac{\left (2 A a^{3} - 6 i B a^{3}\right ) e^{- 4 i c}}{d}}{e^{4 i d x} + 2 e^{- 2 i c} e^{2 i d x} + e^{- 4 i c}} + \operatorname{RootSum}{\left (z^{2} d^{2} + z \left (- 4 A a^{3} d + 4 i B a^{3} d\right ) + 3 A^{2} a^{6} - 4 i A B a^{6}, \left ( i \mapsto i \log{\left (\frac{i i d}{i A a^{3} e^{2 i c} + 2 B a^{3} e^{2 i c}} - \frac{2 i A + 2 B}{i A e^{2 i c} + 2 B e^{2 i c}} + e^{2 i d x} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.5758, size = 362, normalized size = 3.38 \begin{align*} \frac{2 \, A a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 4 \,{\left (4 \, A a^{3} - 4 i \, B a^{3}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) + 2 \,{\left (3 \, A a^{3} - 4 i \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 2 \,{\left (3 \, A a^{3} - 4 i \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{9 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 12 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 4 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 18 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 28 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, A a^{3} - 12 i \, B a^{3}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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