Integrand size = 34, antiderivative size = 191 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {(7 A+25 i B) x}{8 a^3}-\frac {(i A-3 B) \log (\cos (c+d x))}{a^3 d}+\frac {(7 A+25 i B) \tan (c+d x)}{8 a^3 d}+\frac {(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}-\frac {(i A-3 B) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )} \]
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Time = 0.58 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {3676, 3606, 3556} \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {(-3 B+i A) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {(7 A+25 i B) \tan (c+d x)}{8 a^3 d}-\frac {(-3 B+i A) \log (\cos (c+d x))}{a^3 d}-\frac {x (7 A+25 i B)}{8 a^3}+\frac {(-B+i A) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2} \]
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Rule 3556
Rule 3606
Rule 3676
Rubi steps \begin{align*} \text {integral}& = \frac {(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac {\int \frac {\tan ^3(c+d x) (4 a (i A-B)+a (A+7 i B) \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2} \\ & = \frac {(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\tan ^2(c+d x) \left (-3 a^2 (5 A+11 i B)+3 a^2 (3 i A-13 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4} \\ & = \frac {(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}-\frac {(i A-3 B) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {\int \tan (c+d x) \left (-48 a^3 (i A-3 B)-6 a^3 (7 A+25 i B) \tan (c+d x)\right ) \, dx}{48 a^6} \\ & = -\frac {(7 A+25 i B) x}{8 a^3}+\frac {(7 A+25 i B) \tan (c+d x)}{8 a^3 d}+\frac {(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}-\frac {(i A-3 B) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {(i A-3 B) \int \tan (c+d x) \, dx}{a^3} \\ & = -\frac {(7 A+25 i B) x}{8 a^3}-\frac {(i A-3 B) \log (\cos (c+d x))}{a^3 d}+\frac {(7 A+25 i B) \tan (c+d x)}{8 a^3 d}+\frac {(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}-\frac {(i A-3 B) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )} \\ \end{align*}
Time = 1.78 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.41 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {-3 ((15 A+49 i B) \log (i-\tan (c+d x))+(A-i B) \log (i+\tan (c+d x)))+3 (14 i A-50 B+3 (-15 i A+49 B) \log (i-\tan (c+d x))+(-3 i A-3 B) \log (i+\tan (c+d x))) \tan (c+d x)+3 (-2 (17 A+63 i B)+3 (15 A+49 i B) \log (i-\tan (c+d x))+3 (A-i B) \log (i+\tan (c+d x))) \tan ^2(c+d x)+(-68 i A+284 B+(45 i A-147 B) \log (i-\tan (c+d x))+3 (i A+B) \log (i+\tan (c+d x))) \tan ^3(c+d x)+48 i B \tan ^4(c+d x)}{48 a^3 d (-i+\tan (c+d x))^3} \]
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Time = 0.12 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {49 i x B}{8 a^{3}}-\frac {15 x A}{8 a^{3}}-\frac {23 \,{\mathrm e}^{-2 i \left (d x +c \right )} B}{16 a^{3} d}+\frac {11 i {\mathrm e}^{-2 i \left (d x +c \right )} A}{16 a^{3} d}+\frac {7 \,{\mathrm e}^{-4 i \left (d x +c \right )} B}{32 a^{3} d}-\frac {5 i {\mathrm e}^{-4 i \left (d x +c \right )} A}{32 a^{3} d}-\frac {{\mathrm e}^{-6 i \left (d x +c \right )} B}{48 a^{3} d}+\frac {i {\mathrm e}^{-6 i \left (d x +c \right )} A}{48 a^{3} d}-\frac {6 i B c}{a^{3} d}-\frac {2 A c}{a^{3} d}-\frac {2 B}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{a^{3} d}-\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{a^{3} d}\) | \(214\) |
derivativedivides | \(\frac {i B \tan \left (d x +c \right )}{d \,a^{3}}-\frac {A}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {i B}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {3 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{3}}-\frac {25 i B \arctan \left (\tan \left (d x +c \right )\right )}{8 d \,a^{3}}+\frac {i A \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{3}}-\frac {7 A \arctan \left (\tan \left (d x +c \right )\right )}{8 d \,a^{3}}+\frac {17 A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}+\frac {31 i B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}+\frac {7 i A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {9 B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}\) | \(215\) |
default | \(\frac {i B \tan \left (d x +c \right )}{d \,a^{3}}-\frac {A}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {i B}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {3 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{3}}-\frac {25 i B \arctan \left (\tan \left (d x +c \right )\right )}{8 d \,a^{3}}+\frac {i A \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{3}}-\frac {7 A \arctan \left (\tan \left (d x +c \right )\right )}{8 d \,a^{3}}+\frac {17 A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}+\frac {31 i B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}+\frac {7 i A}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {9 B}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}\) | \(215\) |
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Time = 0.26 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.91 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {12 \, {\left (15 \, A + 49 i \, B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} + 6 \, {\left (2 \, {\left (15 \, A + 49 i \, B\right )} d x - 11 i \, A + 55 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (-17 i \, A + 39 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (-13 i \, A + 19 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 96 \, {\left ({\left (i \, A - 3 \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (i \, A - 3 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 2 i \, A + 2 \, B}{96 \, {\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \]
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Time = 0.60 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.76 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=- \frac {2 B}{a^{3} d e^{2 i c} e^{2 i d x} + a^{3} d} + \begin {cases} \frac {\left (\left (512 i A a^{6} d^{2} e^{6 i c} - 512 B a^{6} d^{2} e^{6 i c}\right ) e^{- 6 i d x} + \left (- 3840 i A a^{6} d^{2} e^{8 i c} + 5376 B a^{6} d^{2} e^{8 i c}\right ) e^{- 4 i d x} + \left (16896 i A a^{6} d^{2} e^{10 i c} - 35328 B a^{6} d^{2} e^{10 i c}\right ) e^{- 2 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac {- 15 A - 49 i B}{8 a^{3}} + \frac {\left (- 15 A e^{6 i c} + 11 A e^{4 i c} - 5 A e^{2 i c} + A - 49 i B e^{6 i c} + 23 i B e^{4 i c} - 7 i B e^{2 i c} + i B\right ) e^{- 6 i c}}{8 a^{3}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (- 15 A - 49 i B\right )}{8 a^{3}} - \frac {i \left (A + 3 i B\right ) \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{a^{3} d} \]
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Exception generated. \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 1.08 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.74 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {\frac {6 \, {\left (i \, A + B\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{a^{3}} - \frac {6 \, {\left (-15 i \, A + 49 \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} + \frac {96 i \, B \tan \left (d x + c\right )}{a^{3}} - \frac {165 i \, A \tan \left (d x + c\right )^{3} - 539 \, B \tan \left (d x + c\right )^{3} + 291 \, A \tan \left (d x + c\right )^{2} + 1245 i \, B \tan \left (d x + c\right )^{2} - 171 i \, A \tan \left (d x + c\right ) + 981 \, B \tan \left (d x + c\right ) - 29 \, A - 259 i \, B}{a^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \]
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Time = 8.03 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.96 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,7{}\mathrm {i}}{2\,a^3}+\frac {\left (-3\,B+A\,1{}\mathrm {i}\right )\,27{}\mathrm {i}}{8\,a^3}\right )+\frac {4\,B}{3\,a^3}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {5\,B}{2\,a^3}+\frac {17\,\left (-3\,B+A\,1{}\mathrm {i}\right )}{8\,a^3}\right )+\frac {17\,\left (-3\,B+A\,1{}\mathrm {i}\right )}{12\,a^3}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+1\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{16\,a^3\,d}+\frac {B\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-49\,B+A\,15{}\mathrm {i}\right )}{16\,a^3\,d} \]
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