Integrand size = 34, antiderivative size = 183 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {(25 A+7 i B) x}{8 a^3}-\frac {(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac {(3 i A-B) \log (\sin (c+d x))}{a^3 d}+\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )} \]
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Time = 0.72 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3677, 3610, 3612, 3556} \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac {(-B+3 i A) \log (\sin (c+d x))}{a^3 d}+\frac {(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {x (25 A+7 i B)}{8 a^3}+\frac {(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3} \]
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Rule 3556
Rule 3610
Rule 3612
Rule 3677
Rubi steps \begin{align*} \text {integral}& = \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*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.58 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {\frac {4 (A+i B) \cot ^4(c+d x)}{(i+\cot (c+d x))^3}+\frac {(11 A+5 i B) \cot ^3(c+d x)}{(i+\cot (c+d x))^2}+\frac {12 (3 A+i B) \cot ^2(c+d x)}{i+\cot (c+d x)}-3 \left ((25 A+7 i B) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )+8 (3 i A-B) (\log (\cos (c+d x))+\log (\tan (c+d x)))\right )}{24 a^3 d} \]
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Time = 0.18 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.17
method | result | size |
risch | \(-\frac {15 i x B}{8 a^{3}}-\frac {49 x A}{8 a^{3}}+\frac {11 \,{\mathrm e}^{-2 i \left (d x +c \right )} B}{16 a^{3} d}-\frac {23 i {\mathrm e}^{-2 i \left (d x +c \right )} A}{16 a^{3} d}+\frac {5 \,{\mathrm e}^{-4 i \left (d x +c \right )} B}{32 a^{3} d}-\frac {7 i {\mathrm e}^{-4 i \left (d x +c \right )} A}{32 a^{3} d}+\frac {{\mathrm e}^{-6 i \left (d x +c \right )} B}{48 a^{3} d}-\frac {i {\mathrm e}^{-6 i \left (d x +c \right )} A}{48 a^{3} d}-\frac {2 i B c}{a^{3} d}-\frac {6 A c}{a^{3} d}-\frac {2 i A}{a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{a^{3} d}-\frac {3 i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A}{a^{3} d}\) | \(214\) |
derivativedivides | \(-\frac {A \cot \left (d x +c \right )}{a^{3} d}+\frac {3 i A \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2 a^{3} d}+\frac {25 A \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )}{8 a^{3} d}-\frac {B \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2 a^{3} d}+\frac {7 i B \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )}{8 a^{3} d}-\frac {31 A}{8 a^{3} d \left (i+\cot \left (d x +c \right )\right )}-\frac {17 i B}{8 a^{3} d \left (i+\cot \left (d x +c \right )\right )}+\frac {9 i A}{8 a^{3} d \left (i+\cot \left (d x +c \right )\right )^{2}}-\frac {7 B}{8 a^{3} d \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {A}{6 a^{3} d \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {i B}{6 a^{3} d \left (i+\cot \left (d x +c \right )\right )^{3}}\) | \(226\) |
default | \(-\frac {A \cot \left (d x +c \right )}{a^{3} d}+\frac {3 i A \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2 a^{3} d}+\frac {25 A \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )}{8 a^{3} d}-\frac {B \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2 a^{3} d}+\frac {7 i B \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )}{8 a^{3} d}-\frac {31 A}{8 a^{3} d \left (i+\cot \left (d x +c \right )\right )}-\frac {17 i B}{8 a^{3} d \left (i+\cot \left (d x +c \right )\right )}+\frac {9 i A}{8 a^{3} d \left (i+\cot \left (d x +c \right )\right )^{2}}-\frac {7 B}{8 a^{3} d \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {A}{6 a^{3} d \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {i B}{6 a^{3} d \left (i+\cot \left (d x +c \right )\right )^{3}}\) | \(226\) |
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Time = 0.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {12 \, {\left (49 \, A + 15 i \, B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} - 6 \, {\left (2 \, {\left (49 \, A + 15 i \, B\right )} d x - 55 i \, A + 11 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (-39 i \, A + 17 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (19 i \, A - 13 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 96 \, {\left ({\left (3 i \, A - B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (-3 i \, A + B\right )} e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 2 i \, A + 2 \, B}{96 \, {\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \]
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Time = 0.53 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=- \frac {2 i A}{a^{3} d e^{2 i c} e^{2 i d x} - a^{3} d} + \begin {cases} \frac {\left (\left (- 512 i A a^{6} d^{2} e^{6 i c} + 512 B a^{6} d^{2} e^{6 i c}\right ) e^{- 6 i d x} + \left (- 5376 i A a^{6} d^{2} e^{8 i c} + 3840 B a^{6} d^{2} e^{8 i c}\right ) e^{- 4 i d x} + \left (- 35328 i A a^{6} d^{2} e^{10 i c} + 16896 B a^{6} d^{2} e^{10 i c}\right ) e^{- 2 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac {- 49 A - 15 i B}{8 a^{3}} + \frac {\left (- 49 A e^{6 i c} - 23 A e^{4 i c} - 7 A e^{2 i c} - A - 15 i B e^{6 i c} - 11 i B e^{4 i c} - 5 i B e^{2 i c} - i B\right ) e^{- 6 i c}}{8 a^{3}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (- 49 A - 15 i B\right )}{8 a^{3}} - \frac {i \left (3 A + i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{3} d} \]
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Exception generated. \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.91 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {\frac {6 \, {\left (i \, A + B\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{a^{3}} + \frac {6 \, {\left (-49 i \, A + 15 \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} + \frac {96 \, {\left (3 i \, A - B\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}} + \frac {96 \, {\left (-3 i \, A \tan \left (d x + c\right ) + B \tan \left (d x + c\right ) + A\right )}}{a^{3} \tan \left (d x + c\right )} + \frac {539 \, A \tan \left (d x + c\right )^{3} + 165 i \, B \tan \left (d x + c\right )^{3} - 1821 i \, A \tan \left (d x + c\right )^{2} + 579 \, B \tan \left (d x + c\right )^{2} - 2085 \, A \tan \left (d x + c\right ) - 699 i \, B \tan \left (d x + c\right ) + 819 i \, A - 301 \, B}{a^{3} {\left (i \, \tan \left (d x + c\right ) + 1\right )}^{3}}}{96 \, d} \]
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Time = 7.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.08 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {25\,A}{8\,a^3}+\frac {B\,7{}\mathrm {i}}{8\,a^3}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {17\,B}{8\,a^3}+\frac {A\,63{}\mathrm {i}}{8\,a^3}\right )+\frac {A\,1{}\mathrm {i}}{a^3}-\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {71\,A}{12\,a^3}+\frac {B\,17{}\mathrm {i}}{12\,a^3}\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,3{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-B+A\,3{}\mathrm {i}\right )}{a^3\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{16\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-15\,B+A\,49{}\mathrm {i}\right )}{16\,a^3\,d} \]
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