Optimal. Leaf size=223 \[ \frac{a^4 (92 B+93 i A) \cot ^3(c+d x)}{60 d}-\frac{4 a^4 (A-i B) \cot ^2(c+d x)}{d}-\frac{8 a^4 (B+i A) \cot (c+d x)}{d}-\frac{8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac{(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-8 a^4 x (B+i A)-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]
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Rubi [A] time = 0.645658, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{a^4 (92 B+93 i A) \cot ^3(c+d x)}{60 d}-\frac{4 a^4 (A-i B) \cot ^2(c+d x)}{d}-\frac{8 a^4 (B+i A) \cot (c+d x)}{d}-\frac{8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac{(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-8 a^4 x (B+i A)-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 3593
Rule 3591
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac{1}{6} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^3 (3 a (3 i A+2 B)-3 a (A-2 i B) \tan (c+d x)) \, dx\\ &=-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{1}{30} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \left (-6 a^2 (13 A-12 i B)-6 a^2 (7 i A+8 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{1}{120} \int \cot ^4(c+d x) (a+i a \tan (c+d x)) \left (-6 a^3 (93 i A+92 B)+6 a^3 (67 A-68 i B) \tan (c+d x)\right ) \, dx\\ &=\frac{a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{1}{120} \int \cot ^3(c+d x) \left (960 a^4 (A-i B)+960 a^4 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-\frac{4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac{a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{1}{120} \int \cot ^2(c+d x) \left (960 a^4 (i A+B)-960 a^4 (A-i B) \tan (c+d x)\right ) \, dx\\ &=-\frac{8 a^4 (i A+B) \cot (c+d x)}{d}-\frac{4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac{a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{1}{120} \int \cot (c+d x) \left (-960 a^4 (A-i B)-960 a^4 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-8 a^4 (i A+B) x-\frac{8 a^4 (i A+B) \cot (c+d x)}{d}-\frac{4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac{a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-\left (8 a^4 (A-i B)\right ) \int \cot (c+d x) \, dx\\ &=-8 a^4 (i A+B) x-\frac{8 a^4 (i A+B) \cot (c+d x)}{d}-\frac{4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac{a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac{8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac{a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}\\ \end{align*}
Mathematica [B] time = 9.4112, size = 1009, normalized size = 4.52 \[ a^4 \left (\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (A \cos (2 c)-i B \cos (2 c)-i A \sin (2 c)-B \sin (2 c)) \left (8 i \tan ^{-1}(\tan (5 c+d x)) \cos (2 c)+8 \tan ^{-1}(\tan (5 c+d x)) \sin (2 c)\right ) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (A \cos (2 c)-i B \cos (2 c)-i A \sin (2 c)-B \sin (2 c)) \left (4 i \log \left (\sin ^2(c+d x)\right ) \sin (2 c)-4 \cos (2 c) \log \left (\sin ^2(c+d x)\right )\right ) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{x (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left (-40 i A \cos ^4(c)-40 B \cos ^4(c)+8 A \cot (c) \cos ^4(c)-8 i B \cot (c) \cos ^4(c)-80 A \sin (c) \cos ^3(c)+80 i B \sin (c) \cos ^3(c)+80 i A \sin ^2(c) \cos ^2(c)+80 B \sin ^2(c) \cos ^2(c)+40 A \sin ^3(c) \cos (c)-40 i B \sin ^3(c) \cos (c)-8 i A \sin ^4(c)-8 B \sin ^4(c)+(A-i B) \cot (c) (8 i \sin (4 c)-8 \cos (4 c))\right ) \sin ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) \csc (c+d x) \left (\frac{1}{240} \cos (4 c)-\frac{1}{240} i \sin (4 c)\right ) (860 i A \cos (c)+790 B \cos (c)-780 i A \cos (c+2 d x)-720 B \cos (c+2 d x)-510 i A \cos (3 c+2 d x)-465 B \cos (3 c+2 d x)+366 i A \cos (3 c+4 d x)+354 B \cos (3 c+4 d x)+150 i A \cos (5 c+4 d x)+120 B \cos (5 c+4 d x)-86 i A \cos (5 c+6 d x)-79 B \cos (5 c+6 d x)-490 A \sin (c)+420 i B \sin (c)-600 i A d x \sin (c)-600 B d x \sin (c)-345 A \sin (c+2 d x)+300 i B \sin (c+2 d x)-450 i A d x \sin (c+2 d x)-450 B d x \sin (c+2 d x)+345 A \sin (3 c+2 d x)-300 i B \sin (3 c+2 d x)+450 i A d x \sin (3 c+2 d x)+450 B d x \sin (3 c+2 d x)+120 A \sin (3 c+4 d x)-90 i B \sin (3 c+4 d x)+180 i A d x \sin (3 c+4 d x)+180 B d x \sin (3 c+4 d x)-120 A \sin (5 c+4 d x)+90 i B \sin (5 c+4 d x)-180 i A d x \sin (5 c+4 d x)-180 B d x \sin (5 c+4 d x)-30 i A d x \sin (5 c+6 d x)-30 B d x \sin (5 c+6 d x)+30 i A d x \sin (7 c+6 d x)+30 B d x \sin (7 c+6 d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 259, normalized size = 1.2 \begin{align*}{\frac{4\,iB{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{{\frac{8\,i}{3}}A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}-8\,{\frac{A{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-4\,{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-8\,{\frac{B{a}^{4}c}{d}}-8\,{\frac{\cot \left ( dx+c \right ) B{a}^{4}}{d}}-{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{7\,A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{7\,B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{\frac{4\,i}{5}}A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{8\,iA\cot \left ( dx+c \right ){a}^{4}}{d}}+{\frac{8\,iB{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-8\,B{a}^{4}x-{\frac{8\,iA{a}^{4}c}{d}}-{\frac{iB{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}-8\,iAx{a}^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.04492, size = 239, normalized size = 1.07 \begin{align*} -\frac{480 \,{\left (d x + c\right )}{\left (i \, A + B\right )} a^{4} - 60 \,{\left (4 \, A - 4 i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \,{\left (8 \, A - 8 i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) - \frac{480 \,{\left (-i \, A - B\right )} a^{4} \tan \left (d x + c\right )^{5} -{\left (240 \, A - 240 i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} + 20 \,{\left (8 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} +{\left (105 \, A - 60 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 12 \,{\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) - 10 \, A a^{4}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36707, size = 948, normalized size = 4.25 \begin{align*} \frac{4 \,{\left (30 \,{\left (9 \, A - 7 i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 \,{\left (19 \, A - 17 i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \,{\left (135 \, A - 121 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 15 \,{\left (75 \, A - 68 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \,{\left (81 \, A - 74 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left (86 \, A - 79 i \, B\right )} a^{4} - 30 \,{\left ({\left (A - i \, B\right )} a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \,{\left (A - i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \,{\left (A - i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \,{\left (A - i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \,{\left (A - i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \,{\left (A - i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (A - i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \,{\left (d e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.11067, size = 624, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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