Integrand size = 24, antiderivative size = 159 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {65 x}{16 a^4}-\frac {65 \cot (c+d x)}{16 a^4 d}-\frac {4 i \log (\sin (c+d x))}{a^4 d}+\frac {31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {2 \cot (c+d x)}{a^4 d (1+i \tan (c+d x))}+\frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3} \]
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Time = 0.51 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3640, 3677, 3610, 3612, 3556} \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {65 \cot (c+d x)}{16 a^4 d}-\frac {4 i \log (\sin (c+d x))}{a^4 d}+\frac {2 \cot (c+d x)}{a^4 d (1+i \tan (c+d x))}+\frac {31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac {65 x}{16 a^4}+\frac {7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4} \]
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Rule 3556
Rule 3610
Rule 3612
Rule 3640
Rule 3677
Rubi steps \begin{align*} \text {integral}& = \frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {\int \frac {\cot ^2(c+d x) (9 a-5 i a \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx}{8 a^2} \\ & = \frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {\int \frac {\cot ^2(c+d x) \left (68 a^2-56 i a^2 \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{48 a^4} \\ & = \frac {31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {\int \frac {\cot ^2(c+d x) \left (396 a^3-372 i a^3 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{192 a^6} \\ & = \frac {31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {2 \cot (c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\int \cot ^2(c+d x) \left (1560 a^4-1536 i a^4 \tan (c+d x)\right ) \, dx}{384 a^8} \\ & = -\frac {65 \cot (c+d x)}{16 a^4 d}+\frac {31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {2 \cot (c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\int \cot (c+d x) \left (-1536 i a^4-1560 a^4 \tan (c+d x)\right ) \, dx}{384 a^8} \\ & = -\frac {65 x}{16 a^4}-\frac {65 \cot (c+d x)}{16 a^4 d}+\frac {31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {2 \cot (c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {(4 i) \int \cot (c+d x) \, dx}{a^4} \\ & = -\frac {65 x}{16 a^4}-\frac {65 \cot (c+d x)}{16 a^4 d}-\frac {4 i \log (\sin (c+d x))}{a^4 d}+\frac {31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac {\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac {2 \cot (c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.76 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {\frac {14 \cot ^4(c+d x)}{a^4 (i+\cot (c+d x))^3}+\frac {\frac {31 \cot ^3(c+d x)}{(i+\cot (c+d x))^2}+\frac {96 \cot ^2(c+d x)}{i+\cot (c+d x)}-195 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )-192 i (\log (\cos (c+d x))+\log (\tan (c+d x)))}{a^4}+\frac {6 \cot (c+d x)}{(a+i a \tan (c+d x))^4}}{48 d} \]
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Time = 0.56 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {129 x}{16 a^{4}}-\frac {9 i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{4} d}-\frac {15 i {\mathrm e}^{-4 i \left (d x +c \right )}}{32 a^{4} d}-\frac {i {\mathrm e}^{-6 i \left (d x +c \right )}}{12 a^{4} d}-\frac {i {\mathrm e}^{-8 i \left (d x +c \right )}}{128 a^{4} d}-\frac {8 c}{a^{4} d}-\frac {2 i}{d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {4 i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}\) | \(132\) |
derivativedivides | \(\frac {2 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \,a^{4}}-\frac {65 \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}-\frac {i}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {17 i}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {5}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {49}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}-\frac {4 i \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}-\frac {1}{a^{4} d \tan \left (d x +c \right )}\) | \(147\) |
default | \(\frac {2 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \,a^{4}}-\frac {65 \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}-\frac {i}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {17 i}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {5}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {49}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}-\frac {4 i \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}-\frac {1}{a^{4} d \tan \left (d x +c \right )}\) | \(147\) |
norman | \(\frac {-\frac {7 i \left (\tan ^{5}\left (d x +c \right )\right )}{a d}-\frac {1}{a d}-\frac {949 \left (\tan ^{4}\left (d x +c \right )\right )}{48 a d}-\frac {715 \left (\tan ^{6}\left (d x +c \right )\right )}{48 a d}-\frac {65 \left (\tan ^{8}\left (d x +c \right )\right )}{16 a d}-\frac {65 x \tan \left (d x +c \right )}{16 a}-\frac {65 x \left (\tan ^{3}\left (d x +c \right )\right )}{4 a}-\frac {195 x \left (\tan ^{5}\left (d x +c \right )\right )}{8 a}-\frac {65 x \left (\tan ^{7}\left (d x +c \right )\right )}{4 a}-\frac {65 x \left (\tan ^{9}\left (d x +c \right )\right )}{16 a}-\frac {175 \left (\tan ^{2}\left (d x +c \right )\right )}{16 a d}-\frac {2 i \left (\tan ^{7}\left (d x +c \right )\right )}{a d}-\frac {14 i \tan \left (d x +c \right )}{3 d a}-\frac {26 i \left (\tan ^{3}\left (d x +c \right )\right )}{3 d a}}{\tan \left (d x +c \right ) a^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{4}}-\frac {4 i \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {2 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \,a^{4}}\) | \(269\) |
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Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {3096 \, d x e^{\left (10 i \, d x + 10 i \, c\right )} - 24 \, {\left (129 \, d x - 68 i\right )} e^{\left (8 i \, d x + 8 i \, c\right )} + 1536 \, {\left (i \, e^{\left (10 i \, d x + 10 i \, c\right )} - i \, e^{\left (8 i \, d x + 8 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 684 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 148 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 29 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i}{384 \, {\left (a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} - a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )}\right )}} \]
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Time = 0.36 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.58 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\begin {cases} \frac {\left (- 442368 i a^{12} d^{3} e^{18 i c} e^{- 2 i d x} - 92160 i a^{12} d^{3} e^{16 i c} e^{- 4 i d x} - 16384 i a^{12} d^{3} e^{14 i c} e^{- 6 i d x} - 1536 i a^{12} d^{3} e^{12 i c} e^{- 8 i d x}\right ) e^{- 20 i c}}{196608 a^{16} d^{4}} & \text {for}\: a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (\frac {\left (- 129 e^{8 i c} - 72 e^{6 i c} - 30 e^{4 i c} - 8 e^{2 i c} - 1\right ) e^{- 8 i c}}{16 a^{4}} + \frac {129}{16 a^{4}}\right ) & \text {otherwise} \end {cases} - \frac {2 i}{a^{4} d e^{2 i c} e^{2 i d x} - a^{4} d} - \frac {129 x}{16 a^{4}} - \frac {4 i \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{4} d} \]
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Exception generated. \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 1.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.77 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {\frac {12 i \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} - \frac {1548 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} + \frac {1536 i \, \log \left (\tan \left (d x + c\right )\right )}{a^{4}} + \frac {384 \, {\left (-4 i \, \tan \left (d x + c\right ) + 1\right )}}{a^{4} \tan \left (d x + c\right )} + \frac {3225 i \, \tan \left (d x + c\right )^{4} + 14076 \, \tan \left (d x + c\right )^{3} - 23286 i \, \tan \left (d x + c\right )^{2} - 17404 \, \tan \left (d x + c\right ) + 5017 i}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \]
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Time = 4.62 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.04 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,129{}\mathrm {i}}{32\,a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^4\,d}-\frac {\frac {1}{a^4}-\frac {851\,{\mathrm {tan}\left (c+d\,x\right )}^2}{48\,a^4}+\frac {65\,{\mathrm {tan}\left (c+d\,x\right )}^4}{16\,a^4}+\frac {\mathrm {tan}\left (c+d\,x\right )\,26{}\mathrm {i}}{3\,a^4}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,57{}\mathrm {i}}{4\,a^4}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5-{\mathrm {tan}\left (c+d\,x\right )}^4\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (c+d\,x\right )}^3+{\mathrm {tan}\left (c+d\,x\right )}^2\,4{}\mathrm {i}+\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,4{}\mathrm {i}}{a^4\,d} \]
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