Optimal. Leaf size=183 \[ -\frac {(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac {(-B+3 i A) \log (\sin (c+d x))}{a^3 d}+\frac {(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {x (25 A+7 i B)}{8 a^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.53, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3596, 3529, 3531, 3475} \[ -\frac {(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac {(-B+3 i A) \log (\sin (c+d x))}{a^3 d}+\frac {(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {x (25 A+7 i B)}{8 a^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3596
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx &=\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {\int \frac {\cot ^2(c+d x) (a (7 A+i B)-4 a (i A-B) \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\cot ^2(c+d x) \left (3 a^2 (13 A+3 i B)-3 a^2 (11 i A-5 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\int \cot ^2(c+d x) \left (6 a^3 (25 A+7 i B)-48 a^3 (3 i A-B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac {(25 A+7 i B) \cot (c+d x)}{8 a^3 d}+\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\int \cot (c+d x) \left (-48 a^3 (3 i A-B)-6 a^3 (25 A+7 i B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac {(25 A+7 i B) x}{8 a^3}-\frac {(25 A+7 i B) \cot (c+d x)}{8 a^3 d}+\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {(3 i A-B) \int \cot (c+d x) \, dx}{a^3}\\ &=-\frac {(25 A+7 i B) x}{8 a^3}-\frac {(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac {(3 i A-B) \log (\sin (c+d x))}{a^3 d}+\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 7.39, size = 1282, normalized size = 7.01 \[ \frac {\csc \left (\frac {c}{2}\right ) \csc (c+d x) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (\frac {1}{2} i A \cos (3 c-d x)-\frac {1}{2} i A \cos (3 c+d x)-\frac {1}{2} A \sin (3 c-d x)+\frac {1}{2} A \sin (3 c+d x)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{2 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {(5 B-7 i A) \cos (4 d x) \sec ^2(c+d x) \left (\frac {\cos (c)}{32}-\frac {1}{32} i \sin (c)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {(11 B-23 i A) \cos (2 d x) \sec ^2(c+d x) \left (\frac {\cos (c)}{16}+\frac {1}{16} i \sin (c)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {\sec ^2(c+d x) \left (-3 i A \cos \left (\frac {3 c}{2}\right )+B \cos \left (\frac {3 c}{2}\right )+3 A \sin \left (\frac {3 c}{2}\right )+i B \sin \left (\frac {3 c}{2}\right )\right ) \left (\tan ^{-1}(\tan (d x)) \sin \left (\frac {3 c}{2}\right )-i \tan ^{-1}(\tan (d x)) \cos \left (\frac {3 c}{2}\right )\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {\sec ^2(c+d x) \left (-3 i A \cos \left (\frac {3 c}{2}\right )+B \cos \left (\frac {3 c}{2}\right )+3 A \sin \left (\frac {3 c}{2}\right )+i B \sin \left (\frac {3 c}{2}\right )\right ) \left (\frac {1}{2} \cos \left (\frac {3 c}{2}\right ) \log \left (\sin ^2(c+d x)\right )+\frac {1}{2} i \sin \left (\frac {3 c}{2}\right ) \log \left (\sin ^2(c+d x)\right )\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {x \sec ^2(c+d x) (-6 A \cos (c)-2 i B \cos (c)+3 i A \cot (c) \cos (c)-B \cot (c) \cos (c)-3 i A \sin (c)+B \sin (c)+(B-3 i A) \cot (c) (\cos (3 c)+i \sin (3 c))) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {(B-i A) \cos (6 d x) \sec ^2(c+d x) \left (\frac {1}{48} \cos (3 c)-\frac {1}{48} i \sin (3 c)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {(25 A+7 i B) \sec ^2(c+d x) \left (-\frac {1}{8} d x \cos (3 c)-\frac {1}{8} i d x \sin (3 c)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {(23 A+11 i B) \sec ^2(c+d x) \left (-\frac {\cos (c)}{16}-\frac {1}{16} i \sin (c)\right ) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {(7 A+5 i B) \sec ^2(c+d x) \left (\frac {1}{32} i \sin (c)-\frac {\cos (c)}{32}\right ) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac {(A+i B) \sec ^2(c+d x) \left (\frac {1}{48} i \sin (3 c)-\frac {1}{48} \cos (3 c)\right ) \sin (6 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 175, normalized size = 0.96 \[ -\frac {12 \, {\left (49 \, A + 15 i \, B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} - {\left (12 \, {\left (49 \, A + 15 i \, B\right )} d x - 330 i \, A + 66 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (117 i \, A - 51 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (19 i \, A - 13 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - {\left ({\left (-288 i \, A + 96 \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (288 i \, A - 96 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 2 i \, A + 2 \, B}{96 \, {\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.28, size = 186, normalized size = 1.02 \[ -\frac {\frac {6 \, {\left (-49 i \, A + 15 \, B\right )} \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {6 \, {\left (i \, A + B\right )} \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} + \frac {96 \, {\left (3 i \, A - B\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}} + \frac {96 \, {\left (-3 i \, A \tan \left (d x + c\right ) + B \tan \left (d x + c\right ) + A\right )}}{a^{3} \tan \left (d x + c\right )} + \frac {539 \, A \tan \left (d x + c\right )^{3} + 165 i \, B \tan \left (d x + c\right )^{3} - 1821 i \, A \tan \left (d x + c\right )^{2} + 579 \, B \tan \left (d x + c\right )^{2} - 2085 \, A \tan \left (d x + c\right ) - 699 i \, B \tan \left (d x + c\right ) + 819 i \, A - 301 \, B}{a^{3} {\left (i \, \tan \left (d x + c\right ) + 1\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 252, normalized size = 1.38 \[ -\frac {B \ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}-\frac {i A \ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}-\frac {3 i A \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}+\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {A}{a^{3} d \tan \left (d x +c \right )}+\frac {5 i A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {3 B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {17 A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}-\frac {7 i B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}+\frac {49 i \ln \left (\tan \left (d x +c \right )-i\right ) A}{16 d \,a^{3}}-\frac {15 \ln \left (\tan \left (d x +c \right )-i\right ) B}{16 d \,a^{3}}+\frac {A}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {i B}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.88, size = 197, normalized size = 1.08 \[ -\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {25\,A}{8\,a^3}+\frac {B\,7{}\mathrm {i}}{8\,a^3}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {17\,B}{8\,a^3}+\frac {A\,63{}\mathrm {i}}{8\,a^3}\right )+\frac {A\,1{}\mathrm {i}}{a^3}-\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {71\,A}{12\,a^3}+\frac {B\,17{}\mathrm {i}}{12\,a^3}\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,3{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-B+A\,3{}\mathrm {i}\right )}{a^3\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{16\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-15\,B+A\,49{}\mathrm {i}\right )}{16\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 342, normalized size = 1.87 \[ - \frac {2 i A}{a^{3} d e^{2 i c} e^{2 i d x} - a^{3} d} + \begin {cases} - \frac {\left (\left (512 i A a^{6} d^{2} e^{6 i c} - 512 B a^{6} d^{2} e^{6 i c}\right ) e^{- 6 i d x} + \left (5376 i A a^{6} d^{2} e^{8 i c} - 3840 B a^{6} d^{2} e^{8 i c}\right ) e^{- 4 i d x} + \left (35328 i A a^{6} d^{2} e^{10 i c} - 16896 B a^{6} d^{2} e^{10 i c}\right ) e^{- 2 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac {- 49 A - 15 i B}{8 a^{3}} + \frac {i \left (49 i A e^{6 i c} + 23 i A e^{4 i c} + 7 i A e^{2 i c} + i A - 15 B e^{6 i c} - 11 B e^{4 i c} - 5 B e^{2 i c} - B\right ) e^{- 6 i c}}{8 a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {x \left (49 A + 15 i B\right )}{8 a^{3}} - \frac {i \left (3 A + i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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