Integrand size = 31, antiderivative size = 233 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\left (\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x\right )-\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \log (\sin (c+d x))}{d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
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Time = 0.57 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3686, 3716, 3709, 3610, 3612, 3556} \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {a \left (5 a^2 A-15 a b B-12 A b^2\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (5 a B+7 A b) \cot ^4(c+d x)}{20 d}+\frac {\left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \log (\sin (c+d x))}{d}-x \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
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Rule 3556
Rule 3610
Rule 3612
Rule 3686
Rule 3709
Rule 3716
Rubi steps \begin{align*} \text {integral}& = -\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (a (7 A b+5 a B)-5 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-b (3 a A-5 b B) \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^4(c+d x) \left (-a \left (5 a^2 A-12 A b^2-15 a b B\right )-5 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-b^2 (3 a A-5 b B) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^3(c+d x) \left (-5 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+5 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^2(c+d x) \left (5 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+5 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)\right ) \, dx \\ & = -\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot (c+d x) \left (5 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-5 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)\right ) \, dx \\ & = -\left (\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x\right )-\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \int \cot (c+d x) \, dx \\ & = -\left (\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x\right )-\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \log (\sin (c+d x))}{d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.02 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {-60 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)+30 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)+20 a \left (a^2 A-3 A b^2-3 a b B\right ) \cot ^3(c+d x)-15 a^2 (3 A b+a B) \cot ^4(c+d x)-12 a^3 A \cot ^5(c+d x)+30 i (a+i b)^3 (A+i B) \log (i-\tan (c+d x))+60 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \log (\tan (c+d x))+30 (i a+b)^3 (A-i B) \log (i+\tan (c+d x))}{60 d} \]
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Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {\left (-90 A \,a^{2} b +30 A \,b^{3}-30 B \,a^{3}+90 B a \,b^{2}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (180 A \,a^{2} b -60 A \,b^{3}+60 B \,a^{3}-180 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-12 A \left (\cot ^{5}\left (d x +c \right )\right ) a^{3}+\left (-45 A \,a^{2} b -15 B \,a^{3}\right ) \left (\cot ^{4}\left (d x +c \right )\right )+20 a \left (\cot ^{3}\left (d x +c \right )\right ) \left (A \,a^{2}-3 A \,b^{2}-3 B a b \right )+\left (90 A \,a^{2} b -30 A \,b^{3}+30 B \,a^{3}-90 B a \,b^{2}\right ) \left (\cot ^{2}\left (d x +c \right )\right )+\left (-60 A \,a^{3}+180 A a \,b^{2}+180 B \,a^{2} b -60 B \,b^{3}\right ) \cot \left (d x +c \right )-60 d x \left (A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right )}{60 d}\) | \(243\) |
derivativedivides | \(\frac {\frac {\left (-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A \,a^{3}+3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}}{2 \tan \left (d x +c \right )^{2}}+\left (3 A \,a^{2} b -A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}}{\tan \left (d x +c \right )}-\frac {A \,a^{3}}{5 \tan \left (d x +c \right )^{5}}-\frac {a^{2} \left (3 A b +B a \right )}{4 \tan \left (d x +c \right )^{4}}+\frac {a \left (A \,a^{2}-3 A \,b^{2}-3 B a b \right )}{3 \tan \left (d x +c \right )^{3}}}{d}\) | \(248\) |
default | \(\frac {\frac {\left (-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A \,a^{3}+3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}}{2 \tan \left (d x +c \right )^{2}}+\left (3 A \,a^{2} b -A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}}{\tan \left (d x +c \right )}-\frac {A \,a^{3}}{5 \tan \left (d x +c \right )^{5}}-\frac {a^{2} \left (3 A b +B a \right )}{4 \tan \left (d x +c \right )^{4}}+\frac {a \left (A \,a^{2}-3 A \,b^{2}-3 B a b \right )}{3 \tan \left (d x +c \right )^{3}}}{d}\) | \(248\) |
norman | \(\frac {\left (-A \,a^{3}+3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )-\frac {A \,a^{3}}{5 d}+\frac {\left (3 A \,a^{2} b -A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}-\frac {\left (A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\frac {a \left (A \,a^{2}-3 A \,b^{2}-3 B a b \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a^{2} \left (3 A b +B a \right ) \tan \left (d x +c \right )}{4 d}}{\tan \left (d x +c \right )^{5}}+\frac {\left (3 A \,a^{2} b -A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (3 A \,a^{2} b -A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(267\) |
risch | \(-A \,a^{3} x +3 A a \,b^{2} x +3 B \,a^{2} b x -B \,b^{3} x +\frac {2 i A \,b^{3} c}{d}-3 i A \,a^{2} b x -i B \,a^{3} x +3 i B a \,b^{2} x -\frac {2 i \left (-60 A a \,b^{2}-60 B \,a^{2} b +23 A \,a^{3}+15 B \,b^{3}-180 i A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+135 i B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-90 i A \,a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+45 i B a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-45 i B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+90 i A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+180 i A \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-135 i B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+45 A \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+15 B \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-90 A \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-60 B \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+140 A \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+90 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-70 A \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-60 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+45 i A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-60 i B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-15 i A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-45 i A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+60 i B \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+30 i B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-30 i B \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+15 i A \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+210 A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+210 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-90 A a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-90 B \,a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+270 A a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+270 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-330 A a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-330 B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {6 i A \,a^{2} b c}{d}+i A \,b^{3} x -\frac {2 i a^{3} B c}{d}+\frac {6 i B a \,b^{2} c}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A b}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A \,b^{3}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B a \,b^{2}}{d}\) | \(755\) |
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Time = 0.27 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.14 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} + 15 \, {\left (3 \, B a^{3} + 9 \, A a^{2} b - 6 \, B a b^{2} - 2 \, A b^{3} - 4 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )^{4} - 12 \, A a^{3} + 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )^{3} + 20 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} \tan \left (d x + c\right )^{2} - 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \tan \left (d x + c\right )}{60 \, d \tan \left (d x + c\right )^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (231) = 462\).
Time = 4.17 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.02 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a^{3} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{3} \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a^{3} x & \text {for}\: c = - d x \\- A a^{3} x - \frac {A a^{3}}{d \tan {\left (c + d x \right )}} + \frac {A a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {A a^{3}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {3 A a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 A a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {3 A a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {3 A a^{2} b}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 A a b^{2} x + \frac {3 A a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {A a b^{2}}{d \tan ^{3}{\left (c + d x \right )}} + \frac {A b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A b^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {B a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {B a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {B a^{3}}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 B a^{2} b x + \frac {3 B a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {B a^{2} b}{d \tan ^{3}{\left (c + d x \right )}} + \frac {3 B a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 B a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 B a b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - B b^{3} x - \frac {B b^{3}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.07 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} {\left (d x + c\right )} + 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )^{4} + 12 \, A a^{3} - 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )^{3} - 20 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} \tan \left (d x + c\right )^{2} + 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (225) = 450\).
Time = 1.12 (sec) , antiderivative size = 670, normalized size of antiderivative = 2.88 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 540 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 960 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} {\left (d x + c\right )} - 960 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 960 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2192 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6576 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6576 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2192 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 660 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 540 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 360 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 70 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
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Time = 8.43 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.02 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )+\frac {A\,a^3}{5}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {A\,a^3}{3}+B\,a^2\,b+A\,a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (A\,a^3-3\,B\,a^2\,b-3\,A\,a\,b^2+B\,b^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-\frac {B\,a^3}{2}-\frac {3\,A\,a^2\,b}{2}+\frac {3\,B\,a\,b^2}{2}+\frac {A\,b^3}{2}\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-B\,a^3-3\,A\,a^2\,b+3\,B\,a\,b^2+A\,b^3\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \]
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