Integrand size = 31, antiderivative size = 352 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^2 A-6 A b^2+3 a b B\right ) \log (\sin (c+d x))}{a^5 d}-\frac {b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^3 d}+\frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Time = 1.44 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3690, 3730, 3732, 3611, 3556} \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {\left (a^2 A+3 a b B-6 A b^2\right ) \log (\sin (c+d x))}{a^5 d}+\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}+\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b^3 \left (-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2} \]
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Rule 3556
Rule 3611
Rule 3690
Rule 3730
Rule 3732
Rubi steps \begin{align*} \text {integral}& = -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot ^2(c+d x) \left (2 (2 A b-a B)+2 a A \tan (c+d x)+4 A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{2 a} \\ & = \frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (-2 \left (a^2 A-6 A b^2+3 a b B\right )-2 a^2 B \tan (c+d x)+6 b (2 A b-a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{2 a^2} \\ & = \frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (-4 \left (a^2+b^2\right ) \left (a^2 A-6 A b^2+3 a b B\right )+4 a^3 (A b-a B) \tan (c+d x)+4 b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{4 a^3 \left (a^2+b^2\right )} \\ & = \frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (-4 \left (a^2+b^2\right )^2 \left (a^2 A-6 A b^2+3 a b B\right )+4 a^4 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+4 b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{4 a^4 \left (a^2+b^2\right )^2} \\ & = \frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (a^2 A-6 A b^2+3 a b B\right ) \int \cot (c+d x) \, dx}{a^5}-\frac {\left (b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^5 \left (a^2+b^2\right )^3} \\ & = \frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^2 A-6 A b^2+3 a b B\right ) \log (\sin (c+d x))}{a^5 d}-\frac {b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^3 d}+\frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.69 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {(3 A b-a B) \cot (c+d x)}{a^4 d}-\frac {A \cot ^2(c+d x)}{2 a^3 d}+\frac {(A+i B) \log (i-\tan (c+d x))}{2 (a+i b)^3 d}-\frac {\left (a^2 A-6 A b^2+3 a b B\right ) \log (\tan (c+d x))}{a^5 d}+\frac {(A-i B) \log (i+\tan (c+d x))}{2 (a-i b)^3 d}-\frac {b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right ) \log (a+b \tan (c+d x))}{a^5 \left (a^2+b^2\right )^3 d}+\frac {b^3 (A b-a B)}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^3 \left (5 a^2 A b+3 A b^3-4 a^3 B-2 a b^2 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Time = 0.71 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {A}{2 a^{3} \tan \left (d x +c \right )^{2}}-\frac {-3 A b +B a}{a^{4} \tan \left (d x +c \right )}+\frac {\left (-A \,a^{2}+6 A \,b^{2}-3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}+\frac {b^{3} \left (5 A \,a^{2} b +3 A \,b^{3}-4 B \,a^{3}-2 B a \,b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{4} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (15 A \,a^{4} b +17 A \,a^{2} b^{3}+6 A \,b^{5}-10 B \,a^{5}-9 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{5}}+\frac {\left (A b -B a \right ) b^{3}}{2 \left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(320\) |
default | \(\frac {\frac {\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {A}{2 a^{3} \tan \left (d x +c \right )^{2}}-\frac {-3 A b +B a}{a^{4} \tan \left (d x +c \right )}+\frac {\left (-A \,a^{2}+6 A \,b^{2}-3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}+\frac {b^{3} \left (5 A \,a^{2} b +3 A \,b^{3}-4 B \,a^{3}-2 B a \,b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{4} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (15 A \,a^{4} b +17 A \,a^{2} b^{3}+6 A \,b^{5}-10 B \,a^{5}-9 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{5}}+\frac {\left (A b -B a \right ) b^{3}}{2 \left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(320\) |
parallelrisch | \(\frac {-30 b^{3} \left (A \,a^{4} b +\frac {17}{15} A \,a^{2} b^{3}+\frac {2}{5} A \,b^{5}-\frac {2}{3} B \,a^{5}-\frac {3}{5} B \,a^{3} b^{2}-\frac {1}{5} B a \,b^{4}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (a +b \tan \left (d x +c \right )\right )+a^{5} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )-2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (A \,a^{2}-6 A \,b^{2}+3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )+6 b^{2} \left (-\frac {B \,a^{8} d x}{3}+b \left (A d x +\frac {2 B}{3}\right ) a^{7}-\frac {11 \left (-\frac {6 B d x}{11}+A \right ) b^{2} a^{6}}{6}-\frac {\left (A d x -\frac {21 B}{2}\right ) b^{3} a^{5}}{3}-\frac {22 A \,a^{4} b^{4}}{3}+\frac {13 B \,a^{3} b^{5}}{3}-\frac {17 A \,a^{2} b^{6}}{2}+\frac {3 B a \,b^{7}}{2}-3 A \,b^{8}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+12 a \left (-\frac {B \,a^{8} d x}{3}+a^{7} \left (A d x +\frac {B}{2}\right ) b -\frac {4 \left (-\frac {3 B d x}{4}+A \right ) b^{2} a^{6}}{3}-\frac {b^{3} \left (A d x -7 B \right ) a^{5}}{3}-5 A \,a^{4} b^{4}+\frac {17 B \,a^{3} b^{5}}{6}-\frac {17 A \,a^{2} b^{6}}{3}+B a \,b^{7}-2 A \,b^{8}\right ) b \tan \left (d x +c \right )-a^{3} \left (A a \left (a^{2}+b^{2}\right )^{3} \left (\cot ^{2}\left (d x +c \right )\right )-4 \left (a^{2}+b^{2}\right )^{3} \left (A b -\frac {B a}{2}\right ) \cot \left (d x +c \right )-6 d \left (A \,a^{2} b -\frac {1}{3} A \,b^{3}-\frac {1}{3} B \,a^{3}+B a \,b^{2}\right ) a^{4} x \right )}{2 \left (a^{2}+b^{2}\right )^{3} a^{5} d \left (a +b \tan \left (d x +c \right )\right )^{2}}\) | \(492\) |
norman | \(\frac {\frac {\left (2 A b -B a \right ) \tan \left (d x +c \right )}{a^{2} d}+\frac {b^{2} \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) x \left (\tan ^{4}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {A}{2 a d}-\frac {b^{2} \left (11 A \,a^{4} b^{2}+33 A \,a^{2} b^{4}+18 A \,b^{6}-4 B \,a^{5} b -17 B \,a^{3} b^{3}-9 B a \,b^{5}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{2 a^{5} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (8 A \,a^{4} b^{2}+22 A \,a^{2} b^{4}+12 A \,b^{6}-3 B \,a^{5} b -11 B \,a^{3} b^{3}-6 B a \,b^{5}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d \,a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {\left (A \,a^{2}-6 A \,b^{2}+3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5} d}+\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{3} \left (15 A \,a^{4} b +17 A \,a^{2} b^{3}+6 A \,b^{5}-10 B \,a^{5}-9 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{5} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(610\) |
risch | \(\text {Expression too large to display}\) | \(2061\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1065 vs. \(2 (346) = 692\).
Time = 0.40 (sec) , antiderivative size = 1065, normalized size of antiderivative = 3.03 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {A a^{10} + 3 \, A a^{8} b^{2} + 3 \, A a^{6} b^{4} + A a^{4} b^{6} + {\left (A a^{8} b^{2} + 3 \, A a^{6} b^{4} - 9 \, B a^{5} b^{5} + 14 \, A a^{4} b^{6} - 3 \, B a^{3} b^{7} + 6 \, A a^{2} b^{8} + 2 \, {\left (B a^{8} b^{2} - 3 \, A a^{7} b^{3} - 3 \, B a^{6} b^{4} + A a^{5} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (A a^{9} b + B a^{8} b^{2} - 2 \, B a^{6} b^{4} + 6 \, B a^{4} b^{6} - 11 \, A a^{3} b^{7} + 3 \, B a^{2} b^{8} - 6 \, A a b^{9} + 2 \, {\left (B a^{9} b - 3 \, A a^{8} b^{2} - 3 \, B a^{7} b^{3} + A a^{6} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} + {\left (A a^{10} + 4 \, B a^{9} b - 8 \, A a^{8} b^{2} + 12 \, B a^{7} b^{3} - 30 \, A a^{6} b^{4} + 23 \, B a^{5} b^{5} - 45 \, A a^{4} b^{6} + 9 \, B a^{3} b^{7} - 18 \, A a^{2} b^{8} + 2 \, {\left (B a^{10} - 3 \, A a^{9} b - 3 \, B a^{8} b^{2} + A a^{7} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left ({\left (A a^{8} b^{2} + 3 \, B a^{7} b^{3} - 3 \, A a^{6} b^{4} + 9 \, B a^{5} b^{5} - 15 \, A a^{4} b^{6} + 9 \, B a^{3} b^{7} - 17 \, A a^{2} b^{8} + 3 \, B a b^{9} - 6 \, A b^{10}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (A a^{9} b + 3 \, B a^{8} b^{2} - 3 \, A a^{7} b^{3} + 9 \, B a^{6} b^{4} - 15 \, A a^{5} b^{5} + 9 \, B a^{4} b^{6} - 17 \, A a^{3} b^{7} + 3 \, B a^{2} b^{8} - 6 \, A a b^{9}\right )} \tan \left (d x + c\right )^{3} + {\left (A a^{10} + 3 \, B a^{9} b - 3 \, A a^{8} b^{2} + 9 \, B a^{7} b^{3} - 15 \, A a^{6} b^{4} + 9 \, B a^{5} b^{5} - 17 \, A a^{4} b^{6} + 3 \, B a^{3} b^{7} - 6 \, A a^{2} b^{8}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left ({\left (10 \, B a^{5} b^{5} - 15 \, A a^{4} b^{6} + 9 \, B a^{3} b^{7} - 17 \, A a^{2} b^{8} + 3 \, B a b^{9} - 6 \, A b^{10}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (10 \, B a^{6} b^{4} - 15 \, A a^{5} b^{5} + 9 \, B a^{4} b^{6} - 17 \, A a^{3} b^{7} + 3 \, B a^{2} b^{8} - 6 \, A a b^{9}\right )} \tan \left (d x + c\right )^{3} + {\left (10 \, B a^{7} b^{3} - 15 \, A a^{6} b^{4} + 9 \, B a^{5} b^{5} - 17 \, A a^{4} b^{6} + 3 \, B a^{3} b^{7} - 6 \, A a^{2} b^{8}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (B a^{10} - 2 \, A a^{9} b + 3 \, B a^{8} b^{2} - 6 \, A a^{7} b^{3} + 3 \, B a^{6} b^{4} - 6 \, A a^{5} b^{5} + B a^{4} b^{6} - 2 \, A a^{3} b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{11} b^{2} + 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} + a^{5} b^{8}\right )} d \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{12} b + 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} + a^{6} b^{7}\right )} d \tan \left (d x + c\right )^{3} + {\left (a^{13} + 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} + a^{7} b^{6}\right )} d \tan \left (d x + c\right )^{2}\right )}} \]
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Exception generated. \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
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Time = 0.40 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.54 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (10 \, B a^{5} b^{3} - 15 \, A a^{4} b^{4} + 9 \, B a^{3} b^{5} - 17 \, A a^{2} b^{6} + 3 \, B a b^{7} - 6 \, A b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {A a^{7} + 2 \, A a^{5} b^{2} + A a^{3} b^{4} + 2 \, {\left (B a^{5} b^{2} - 3 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 11 \, A a^{2} b^{5} + 3 \, B a b^{6} - 6 \, A b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, B a^{6} b - 11 \, A a^{5} b^{2} + 17 \, B a^{4} b^{3} - 33 \, A a^{3} b^{4} + 9 \, B a^{2} b^{5} - 18 \, A a b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{7} - 2 \, A a^{6} b + 2 \, B a^{5} b^{2} - 4 \, A a^{4} b^{3} + B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{8} b^{2} + 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{10} + 2 \, a^{8} b^{2} + a^{6} b^{4}\right )} \tan \left (d x + c\right )^{2}} + \frac {2 \, {\left (A a^{2} + 3 \, B a b - 6 \, A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 812 vs. \(2 (346) = 692\).
Time = 1.24 (sec) , antiderivative size = 812, normalized size of antiderivative = 2.31 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {4 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, {\left (10 \, B a^{5} b^{4} - 15 \, A a^{4} b^{5} + 9 \, B a^{3} b^{6} - 17 \, A a^{2} b^{7} + 3 \, B a b^{8} - 6 \, A b^{9}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}} - \frac {3 \, A a^{7} b^{2} \tan \left (d x + c\right )^{4} + 9 \, B a^{6} b^{3} \tan \left (d x + c\right )^{4} - 9 \, A a^{5} b^{4} \tan \left (d x + c\right )^{4} - 3 \, B a^{4} b^{5} \tan \left (d x + c\right )^{4} + 6 \, A a^{8} b \tan \left (d x + c\right )^{3} + 14 \, B a^{7} b^{2} \tan \left (d x + c\right )^{3} - 6 \, A a^{6} b^{3} \tan \left (d x + c\right )^{3} - 34 \, B a^{5} b^{4} \tan \left (d x + c\right )^{3} + 56 \, A a^{4} b^{5} \tan \left (d x + c\right )^{3} - 36 \, B a^{3} b^{6} \tan \left (d x + c\right )^{3} + 68 \, A a^{2} b^{7} \tan \left (d x + c\right )^{3} - 12 \, B a b^{8} \tan \left (d x + c\right )^{3} + 24 \, A b^{9} \tan \left (d x + c\right )^{3} + 3 \, A a^{9} \tan \left (d x + c\right )^{2} + B a^{8} b \tan \left (d x + c\right )^{2} + 13 \, A a^{7} b^{2} \tan \left (d x + c\right )^{2} - 45 \, B a^{6} b^{3} \tan \left (d x + c\right )^{2} + 88 \, A a^{5} b^{4} \tan \left (d x + c\right )^{2} - 52 \, B a^{4} b^{5} \tan \left (d x + c\right )^{2} + 102 \, A a^{3} b^{6} \tan \left (d x + c\right )^{2} - 18 \, B a^{2} b^{7} \tan \left (d x + c\right )^{2} + 36 \, A a b^{8} \tan \left (d x + c\right )^{2} - 4 \, B a^{9} \tan \left (d x + c\right ) + 8 \, A a^{8} b \tan \left (d x + c\right ) - 12 \, B a^{7} b^{2} \tan \left (d x + c\right ) + 24 \, A a^{6} b^{3} \tan \left (d x + c\right ) - 12 \, B a^{5} b^{4} \tan \left (d x + c\right ) + 24 \, A a^{4} b^{5} \tan \left (d x + c\right ) - 4 \, B a^{3} b^{6} \tan \left (d x + c\right ) + 8 \, A a^{2} b^{7} \tan \left (d x + c\right ) - 2 \, A a^{9} - 6 \, A a^{7} b^{2} - 6 \, A a^{5} b^{4} - 2 \, A a^{3} b^{6}}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} {\left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right )\right )}^{2}} + \frac {4 \, {\left (A a^{2} + 3 \, B a b - 6 \, A b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}}}{4 \, d} \]
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Time = 14.25 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.23 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,A\,b-B\,a\right )}{a^2}-\frac {A}{2\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-B\,a^5\,b^2+3\,A\,a^4\,b^3-6\,B\,a^3\,b^4+11\,A\,a^2\,b^5-3\,B\,a\,b^6+6\,A\,b^7\right )}{a^4\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-4\,B\,a^5\,b+11\,A\,a^4\,b^2-17\,B\,a^3\,b^3+33\,A\,a^2\,b^4-9\,B\,a\,b^5+18\,A\,b^6\right )}{2\,a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {A}{a^3}-\frac {A\,a+3\,B\,b}{{\left (a^2+b^2\right )}^2}-\frac {6\,A\,b^2}{a^5}+\frac {3\,B\,b}{a^4}+\frac {4\,b^2\,\left (A\,a+B\,b\right )}{{\left (a^2+b^2\right )}^3}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,a^2+3\,B\,a\,b-6\,A\,b^2\right )}{a^5\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )} \]
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