Optimal. Leaf size=94 \[ -\frac{a^2 (2 B+3 i A) \cot (c+d x)}{2 d}-\frac{2 a^2 (A-i B) \log (\sin (c+d x))}{d}-2 a^2 x (B+i A)-\frac{A \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d} \]
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Rubi [A] time = 0.207893, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3593, 3591, 3531, 3475} \[ -\frac{a^2 (2 B+3 i A) \cot (c+d x)}{2 d}-\frac{2 a^2 (A-i B) \log (\sin (c+d x))}{d}-2 a^2 x (B+i A)-\frac{A \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 3593
Rule 3591
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=-\frac{A \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}+\frac{1}{2} \int \cot ^2(c+d x) (a+i a \tan (c+d x)) (a (3 i A+2 B)-a (A-2 i B) \tan (c+d x)) \, dx\\ &=-\frac{a^2 (3 i A+2 B) \cot (c+d x)}{2 d}-\frac{A \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}+\frac{1}{2} \int \cot (c+d x) \left (-4 a^2 (A-i B)-4 a^2 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-2 a^2 (i A+B) x-\frac{a^2 (3 i A+2 B) \cot (c+d x)}{2 d}-\frac{A \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}-\left (2 a^2 (A-i B)\right ) \int \cot (c+d x) \, dx\\ &=-2 a^2 (i A+B) x-\frac{a^2 (3 i A+2 B) \cot (c+d x)}{2 d}-\frac{2 a^2 (A-i B) \log (\sin (c+d x))}{d}-\frac{A \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}\\ \end{align*}
Mathematica [B] time = 2.3557, size = 302, normalized size = 3.21 \[ \frac{a^2 \csc (c) \csc ^2(c+d x) (\cos (2 d x)+i \sin (2 d x)) \left (8 (B+i A) \sin (c) \sin ^2(c+d x) \tan ^{-1}(\tan (3 c+d x))+2 (B+2 i A) \cos (c)-8 i A d x \sin (c)-4 i A d x \sin (c+2 d x)+4 i A d x \sin (3 c+2 d x)-4 i A \cos (c+2 d x)-2 A \sin (c) \log \left (\sin ^2(c+d x)\right )-A \sin (c+2 d x) \log \left (\sin ^2(c+d x)\right )+A \sin (3 c+2 d x) \log \left (\sin ^2(c+d x)\right )-2 A \sin (c)-8 B d x \sin (c)-4 B d x \sin (c+2 d x)+4 B d x \sin (3 c+2 d x)-2 B \cos (c+2 d x)+2 i B \sin (c) \log \left (\sin ^2(c+d x)\right )+i B \sin (c+2 d x) \log \left (\sin ^2(c+d x)\right )-i B \sin (3 c+2 d x) \log \left (\sin ^2(c+d x)\right )\right )}{4 d (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 119, normalized size = 1.3 \begin{align*} -2\,{\frac{{a}^{2}A\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-2\,{a}^{2}Bx-2\,{\frac{B{a}^{2}c}{d}}-2\,iA{a}^{2}x-{\frac{2\,iA\cot \left ( dx+c \right ){a}^{2}}{d}}-{\frac{2\,iA{a}^{2}c}{d}}+{\frac{2\,iB{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2}A \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{\cot \left ( dx+c \right ) B{a}^{2}}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66555, size = 130, normalized size = 1.38 \begin{align*} -\frac{4 \,{\left (d x + c\right )}{\left (i \, A + B\right )} a^{2} - 2 \,{\left (A - i \, B\right )} a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \,{\left (2 \, A - 2 i \, B\right )} a^{2} \log \left (\tan \left (d x + c\right )\right ) - \frac{2 \,{\left (-2 i \, A - B\right )} a^{2} \tan \left (d x + c\right ) - A a^{2}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40485, size = 316, normalized size = 3.36 \begin{align*} \frac{2 \,{\left ({\left (3 \, A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left (2 \, A - i \, B\right )} a^{2} -{\left ({\left (A - i \, B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \,{\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (A - i \, B\right )} a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\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 [A] time = 5.36212, size = 119, normalized size = 1.27 \begin{align*} \frac{2 a^{2} \left (- A + i B\right ) \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{\left (4 A a^{2} - 2 i B a^{2}\right ) e^{- 4 i c}}{d} + \frac{\left (6 A a^{2} - 2 i B a^{2}\right ) e^{- 2 i c} e^{2 i d x}}{d}}{e^{4 i d x} - 2 e^{- 2 i c} e^{2 i d x} + e^{- 4 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.61919, size = 254, normalized size = 2.7 \begin{align*} -\frac{A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 8 i \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 16 \,{\left (2 \, A a^{2} - 2 i \, B a^{2}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) + 16 \,{\left (A a^{2} - i \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{24 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 i \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 8 i \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - A a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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