Optimal. Leaf size=200 \[ \frac{a^4 (145 B+148 i A) \cot ^2(c+d x)}{60 d}-\frac{8 a^4 (A-i B) \cot (c+d x)}{d}+\frac{8 a^4 (B+i A) \log (\sin (c+d x))}{d}-\frac{(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}-8 a^4 x (A-i B)-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d} \]
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Rubi [A] time = 0.591938, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{a^4 (145 B+148 i A) \cot ^2(c+d x)}{60 d}-\frac{8 a^4 (A-i B) \cot (c+d x)}{d}+\frac{8 a^4 (B+i A) \log (\sin (c+d x))}{d}-\frac{(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}-8 a^4 x (A-i B)-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d} \]
Antiderivative was successfully verified.
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Rule 3593
Rule 3591
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac{1}{5} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 (a (8 i A+5 B)-a (2 A-5 i B) \tan (c+d x)) \, dx\\ &=-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac{(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{1}{20} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \left (-2 a^2 (28 A-25 i B)-6 a^2 (4 i A+5 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac{(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac{1}{60} \int \cot ^3(c+d x) (a+i a \tan (c+d x)) \left (-2 a^3 (148 i A+145 B)+2 a^3 (92 A-95 i B) \tan (c+d x)\right ) \, dx\\ &=\frac{a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac{(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac{1}{60} \int \cot ^2(c+d x) \left (480 a^4 (A-i B)+480 a^4 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-\frac{8 a^4 (A-i B) \cot (c+d x)}{d}+\frac{a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac{(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac{1}{60} \int \cot (c+d x) \left (480 a^4 (i A+B)-480 a^4 (A-i B) \tan (c+d x)\right ) \, dx\\ &=-8 a^4 (A-i B) x-\frac{8 a^4 (A-i B) \cot (c+d x)}{d}+\frac{a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac{(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\left (8 a^4 (i A+B)\right ) \int \cot (c+d x) \, dx\\ &=-8 a^4 (A-i B) x-\frac{8 a^4 (A-i B) \cot (c+d x)}{d}+\frac{a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}+\frac{8 a^4 (i A+B) \log (\sin (c+d x))}{d}-\frac{a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac{(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}\\ \end{align*}
Mathematica [B] time = 8.23866, size = 542, normalized size = 2.71 \[ \frac{a^4 (\cot (c+d x)+i)^4 (A \cot (c+d x)+B) \left (-8 d x (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^5(c+d x)+4 (A-i B) (\sin (4 c)+i \cos (4 c)) \sin ^5(c+d x) \log \left (\sin ^2(c+d x)\right )+8 (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^5(c+d x) \tan ^{-1}(\tan (5 c+d x))+\frac{1}{120} \csc (c) (\cos (4 c)-i \sin (4 c)) (15 (20 A d x-14 i A-20 i B d x-11 B) \cos (2 c+d x)+15 \cos (d x) (A (-20 d x+14 i)+B (11+20 i d x))+345 A \sin (2 c+d x)-275 A \sin (2 c+3 d x)-120 A \sin (4 c+3 d x)+79 A \sin (4 c+5 d x)-90 i A \cos (2 c+3 d x)+150 A d x \cos (2 c+3 d x)+90 i A \cos (4 c+3 d x)-150 A d x \cos (4 c+3 d x)-30 A d x \cos (4 c+5 d x)+30 A d x \cos (6 c+5 d x)+445 A \sin (d x)-300 i B \sin (2 c+d x)+260 i B \sin (2 c+3 d x)+90 i B \sin (4 c+3 d x)-70 i B \sin (4 c+5 d x)-60 B \cos (2 c+3 d x)-150 i B d x \cos (2 c+3 d x)+60 B \cos (4 c+3 d x)+150 i B d x \cos (4 c+3 d x)+30 i B d x \cos (4 c+5 d x)-30 i B d x \cos (6 c+5 d x)-400 i B \sin (d x))\right )}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 224, normalized size = 1.1 \begin{align*} 8\,{\frac{B{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{8\,iB{a}^{4}c}{d}}+{\frac{8\,iB\cot \left ( dx+c \right ){a}^{4}}{d}}-{\frac{iA{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{8\,iA{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{7\,A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{7\,B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-8\,{\frac{A{a}^{4}c}{d}}-8\,{\frac{A\cot \left ( dx+c \right ){a}^{4}}{d}}+8\,iBx{a}^{4}-{\frac{{\frac{4\,i}{3}}B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}-8\,A{a}^{4}x+{\frac{4\,iA{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.00317, size = 211, normalized size = 1.05 \begin{align*} -\frac{60 \,{\left (d x + c\right )}{\left (8 \, A - 8 i \, B\right )} a^{4} + 240 \,{\left (i \, A + B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \,{\left (-i \, A - B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac{{\left (480 \, A - 480 i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} - 30 \,{\left (8 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} -{\left (140 \, A - 80 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 15 \,{\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) + 12 \, A a^{4}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46433, size = 857, normalized size = 4.28 \begin{align*} \frac{{\left (-840 i \, A - 600 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (2220 i \, A + 1860 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (-2620 i \, A - 2260 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (1460 i \, A + 1280 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (-316 i \, A - 280 \, B\right )} a^{4} +{\left ({\left (120 i \, A + 120 \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} +{\left (-600 i \, A - 600 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (1200 i \, A + 1200 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (-1200 i \, A - 1200 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (600 i \, A + 600 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (-120 i \, A - 120 \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{15 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, 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 = 146.132, size = 272, normalized size = 1.36 \begin{align*} \frac{8 a^{4} \left (i A + B\right ) \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{\left (56 i A a^{4} + 40 B a^{4}\right ) e^{- 2 i c} e^{8 i d x}}{d} + \frac{\left (148 i A a^{4} + 124 B a^{4}\right ) e^{- 4 i c} e^{6 i d x}}{d} + \frac{\left (292 i A a^{4} + 256 B a^{4}\right ) e^{- 8 i c} e^{2 i d x}}{3 d} - \frac{\left (316 i A a^{4} + 280 B a^{4}\right ) e^{- 10 i c}}{15 d} - \frac{\left (524 i A a^{4} + 452 B a^{4}\right ) e^{- 6 i c} e^{4 i d x}}{3 d}}{e^{10 i d x} - 5 e^{- 2 i c} e^{8 i d x} + 10 e^{- 4 i c} e^{6 i d x} - 10 e^{- 6 i c} e^{4 i d x} + 5 e^{- 8 i c} e^{2 i d x} - e^{- 10 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.86528, size = 529, normalized size = 2.64 \begin{align*} \frac{6 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 60 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 15 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 310 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 160 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1200 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 900 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4740 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 4320 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15360 \,{\left (i \, A a^{4} + B a^{4}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) - 7680 \,{\left (-i \, A a^{4} - B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{-17536 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 17536 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4740 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 4320 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1200 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 900 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 310 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 160 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 60 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, A a^{4}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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