Optimal. Leaf size=157 \[ \frac{a^3 (15 A-14 i B) \cot ^2(c+d x)}{12 d}+\frac{4 a^3 (B+i A) \cot (c+d x)}{d}+\frac{4 a^3 (A-i B) \log (\sin (c+d x))}{d}-\frac{(2 B+3 i A) \cot ^3(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{6 d}+4 a^3 x (B+i A)-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d} \]
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Rubi [A] time = 0.418088, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {3593, 3591, 3529, 3531, 3475} \[ \frac{a^3 (15 A-14 i B) \cot ^2(c+d x)}{12 d}+\frac{4 a^3 (B+i A) \cot (c+d x)}{d}+\frac{4 a^3 (A-i B) \log (\sin (c+d x))}{d}-\frac{(2 B+3 i A) \cot ^3(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{6 d}+4 a^3 x (B+i A)-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 3593
Rule 3591
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 (2 a (3 i A+2 B)-2 a (A-2 i B) \tan (c+d x)) \, dx\\ &=-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d}-\frac{(3 i A+2 B) \cot ^3(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{6 d}+\frac{1}{12} \int \cot ^3(c+d x) (a+i a \tan (c+d x)) \left (-2 a^2 (15 A-14 i B)-2 a^2 (9 i A+10 B) \tan (c+d x)\right ) \, dx\\ &=\frac{a^3 (15 A-14 i B) \cot ^2(c+d x)}{12 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d}-\frac{(3 i A+2 B) \cot ^3(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{6 d}+\frac{1}{12} \int \cot ^2(c+d x) \left (-48 a^3 (i A+B)+48 a^3 (A-i B) \tan (c+d x)\right ) \, dx\\ &=\frac{4 a^3 (i A+B) \cot (c+d x)}{d}+\frac{a^3 (15 A-14 i B) \cot ^2(c+d x)}{12 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d}-\frac{(3 i A+2 B) \cot ^3(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{6 d}+\frac{1}{12} \int \cot (c+d x) \left (48 a^3 (A-i B)+48 a^3 (i A+B) \tan (c+d x)\right ) \, dx\\ &=4 a^3 (i A+B) x+\frac{4 a^3 (i A+B) \cot (c+d x)}{d}+\frac{a^3 (15 A-14 i B) \cot ^2(c+d x)}{12 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d}-\frac{(3 i A+2 B) \cot ^3(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{6 d}+\left (4 a^3 (A-i B)\right ) \int \cot (c+d x) \, dx\\ &=4 a^3 (i A+B) x+\frac{4 a^3 (i A+B) \cot (c+d x)}{d}+\frac{a^3 (15 A-14 i B) \cot ^2(c+d x)}{12 d}+\frac{4 a^3 (A-i B) \log (\sin (c+d x))}{d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^2}{4 d}-\frac{(3 i A+2 B) \cot ^3(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{6 d}\\ \end{align*}
Mathematica [B] time = 8.49607, size = 1007, normalized size = 6.41 \[ a^3 \left (\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left (A \cos \left (\frac{3 c}{2}\right )-i B \cos \left (\frac{3 c}{2}\right )-i A \sin \left (\frac{3 c}{2}\right )-B \sin \left (\frac{3 c}{2}\right )\right ) \left (-4 i \tan ^{-1}(\tan (4 c+d x)) \cos \left (\frac{3 c}{2}\right )-4 \tan ^{-1}(\tan (4 c+d x)) \sin \left (\frac{3 c}{2}\right )\right ) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left (A \cos \left (\frac{3 c}{2}\right )-i B \cos \left (\frac{3 c}{2}\right )-i A \sin \left (\frac{3 c}{2}\right )-B \sin \left (\frac{3 c}{2}\right )\right ) \left (2 \cos \left (\frac{3 c}{2}\right ) \log \left (\sin ^2(c+d x)\right )-2 i \log \left (\sin ^2(c+d x)\right ) \sin \left (\frac{3 c}{2}\right )\right ) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{x (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left (16 i A \cos ^3(c)+16 B \cos ^3(c)-4 A \cot (c) \cos ^3(c)+4 i B \cot (c) \cos ^3(c)+24 A \sin (c) \cos ^2(c)-24 i B \sin (c) \cos ^2(c)-16 i A \sin ^2(c) \cos (c)-16 B \sin ^2(c) \cos (c)-4 A \sin ^3(c)+4 i B \sin ^3(c)+(A-i B) \cot (c) (4 \cos (3 c)-4 i \sin (3 c))\right ) \sin ^4(c+d x)}{(\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(i A+B) (\cot (c+d x)+i)^3 (B+A \cot (c+d x)) (4 d x \cos (3 c)-4 i d x \sin (3 c)) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \left (\frac{1}{6} \cos (3 c)-\frac{1}{6} i \sin (3 c)\right ) (-15 i A \sin (d x)-13 B \sin (d x)) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) (-6 i A \cos (c)-2 B \cos (c)+15 A \sin (c)-9 i B \sin (c)) \left (\frac{1}{12} \cos (3 c)-\frac{1}{12} i \sin (3 c)\right ) \sin ^2(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \left (\frac{1}{6} \cos (3 c)-\frac{1}{6} i \sin (3 c)\right ) (3 i A \sin (d x)+B \sin (d x)) \sin (c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^3 (B+A \cot (c+d x)) \left (\frac{1}{4} i A \sin (3 c)-\frac{1}{4} A \cos (3 c)\right )}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 189, normalized size = 1.2 \begin{align*}{\frac{4\,iA{a}^{3}c}{d}}-{\frac{{\frac{3\,i}{2}}B{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{iA{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{4\,iA\cot \left ( dx+c \right ){a}^{3}}{d}}+2\,{\frac{A{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+4\,{\frac{A{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+4\,B{a}^{3}x+4\,{\frac{\cot \left ( dx+c \right ) B{a}^{3}}{d}}+4\,{\frac{B{a}^{3}c}{d}}-{\frac{4\,iB{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+4\,iAx{a}^{3}-{\frac{A{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{B{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.3096, size = 185, normalized size = 1.18 \begin{align*} -\frac{48 \,{\left (d x + c\right )}{\left (-i \, A - B\right )} a^{3} + 12 \,{\left (2 \, A - 2 i \, B\right )} a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \,{\left (4 \, A - 4 i \, B\right )} a^{3} \log \left (\tan \left (d x + c\right )\right ) - \frac{48 \,{\left (i \, A + B\right )} a^{3} \tan \left (d x + c\right )^{3} +{\left (24 \, A - 18 i \, B\right )} a^{3} \tan \left (d x + c\right )^{2} + 4 \,{\left (-3 i \, A - B\right )} a^{3} \tan \left (d x + c\right ) - 3 \, A a^{3}}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64047, size = 626, normalized size = 3.99 \begin{align*} -\frac{2 \,{\left (12 \,{\left (3 \, A - 2 i \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \,{\left (23 \, A - 19 i \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \,{\left (27 \, A - 23 i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left (15 \, A - 13 i \, B\right )} a^{3} - 6 \,{\left ({\left (A - i \, B\right )} a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \,{\left (A - i \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \,{\left (A - i \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \,{\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (A - i \, B\right )} a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{3 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, 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 = 34.1315, size = 221, normalized size = 1.41 \begin{align*} \frac{4 a^{3} \left (A - i B\right ) \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{\left (24 A a^{3} - 16 i B a^{3}\right ) e^{- 2 i c} e^{6 i d x}}{d} + \frac{\left (30 A a^{3} - 26 i B a^{3}\right ) e^{- 8 i c}}{3 d} + \frac{\left (46 A a^{3} - 38 i B a^{3}\right ) e^{- 4 i c} e^{4 i d x}}{d} - \frac{\left (108 A a^{3} - 92 i B a^{3}\right ) e^{- 6 i c} e^{2 i d x}}{3 d}}{e^{8 i d x} - 4 e^{- 2 i c} e^{6 i d x} + 6 e^{- 4 i c} e^{4 i d x} - 4 e^{- 6 i c} e^{2 i d x} + e^{- 8 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.76959, size = 439, normalized size = 2.8 \begin{align*} -\frac{3 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 24 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 8 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 108 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 72 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 456 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 408 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 384 \,{\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 ) - 384 \,{\left (2 \, A a^{3} - 2 i \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{1600 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1600 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 456 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 408 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 108 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 72 i \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 i \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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