Integrand size = 40, antiderivative size = 215 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {B \log (\sin (c+d x))}{a^3 d}-\frac {b \left (6 a^4 b B+3 a^2 b^3 B+b^5 B-3 a^5 C+a^3 b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b (b B-a C)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 b B+b^3 B-2 a^3 C\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Time = 0.80 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3713, 3690, 3730, 3732, 3611, 3556} \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {B \log (\sin (c+d x))}{a^3 d}+\frac {b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right )}{a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{\left (a^2+b^2\right )^3}-\frac {b \left (-3 a^5 C+6 a^4 b B+a^3 b^2 C+3 a^2 b^3 B+b^5 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^3} \]
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Rule 3556
Rule 3611
Rule 3690
Rule 3713
Rule 3730
Rule 3732
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot (c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx \\ & = \frac {b (b B-a C)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (2 \left (a^2+b^2\right ) B-2 a (b B-a C) \tan (c+d x)+2 b (b B-a C) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )} \\ & = \frac {b (b B-a C)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 b B+b^3 B-2 a^3 C\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (2 \left (a^2+b^2\right )^2 B-2 a^2 \left (2 a b B-a^2 C+b^2 C\right ) \tan (c+d x)+2 b \left (3 a^2 b B+b^3 B-2 a^3 C\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^2 \left (a^2+b^2\right )^2} \\ & = -\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {b (b B-a C)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 b B+b^3 B-2 a^3 C\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {B \int \cot (c+d x) \, dx}{a^3}-\frac {\left (b \left (6 a^4 b B+3 a^2 b^3 B+b^5 B-3 a^5 C+a^3 b^2 C\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )^3} \\ & = -\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {B \log (\sin (c+d x))}{a^3 d}-\frac {b \left (6 a^4 b B+3 a^2 b^3 B+b^5 B-3 a^5 C+a^3 b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b (b B-a C)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 b B+b^3 B-2 a^3 C\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.31 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.04 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {-\frac {(B+i C) \log (i-\tan (c+d x))}{(a+i b)^3}+\frac {2 B \log (\tan (c+d x))}{a^3}-\frac {(B-i C) \log (i+\tan (c+d x))}{(a-i b)^3}-\frac {2 b \left (6 a^4 b B+3 a^2 b^3 B+b^5 B-3 a^5 C+a^3 b^2 C\right ) \log (a+b \tan (c+d x))}{a^3 \left (a^2+b^2\right )^3}+\frac {b (b B-a C)}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 b \left (3 a^2 b B+b^3 B-2 a^3 C\right )}{a^2 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 d} \]
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Time = 0.66 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {\frac {\left (-B \,a^{3}+3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b \left (3 B \,a^{2} b +B \,b^{3}-2 C \,a^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (6 B \,a^{4} b +3 B \,a^{2} b^{3}+B \,b^{5}-3 C \,a^{5}+C \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\left (B b -C a \right ) b}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(243\) |
default | \(\frac {\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {\frac {\left (-B \,a^{3}+3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 B \,a^{2} b +B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b \left (3 B \,a^{2} b +B \,b^{3}-2 C \,a^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (6 B \,a^{4} b +3 B \,a^{2} b^{3}+B \,b^{5}-3 C \,a^{5}+C \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\left (B b -C a \right ) b}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(243\) |
parallelrisch | \(\frac {-12 b \left (B \,a^{4} b +\frac {1}{2} B \,a^{2} b^{3}+\frac {1}{6} B \,b^{5}-\frac {1}{2} C \,a^{5}+\frac {1}{6} C \,a^{3} b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (a +b \tan \left (d x +c \right )\right )-a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+2 B \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\tan \left (d x +c \right )\right )-6 \left (-\frac {C \,a^{6} d x}{3}+b \left (B d x -\frac {C}{3}\right ) a^{5}+\frac {b^{2} \left (2 C d x +B \right ) a^{4}}{2}-\frac {b^{3} \left (B d x +C \right ) a^{3}}{3}+\frac {2 B \,a^{2} b^{4}}{3}+\frac {B \,b^{6}}{6}\right ) b^{2} \tan \left (d x +c \right )^{2}-12 a^{4} d b \left (B \,a^{2} b -\frac {1}{3} B \,b^{3}-\frac {1}{3} C \,a^{3}+C a \,b^{2}\right ) x \tan \left (d x +c \right )-6 a^{2} \left (-\frac {C \,a^{6} d x}{3}+b \left (B d x +\frac {C}{2}\right ) a^{5}-\frac {2 b^{2} \left (-\frac {3 C d x}{2}+B \right ) a^{4}}{3}-\frac {b^{3} \left (B d x -2 C \right ) a^{3}}{3}-B \,a^{2} b^{4}+\frac {C a \,b^{5}}{6}-\frac {B \,b^{6}}{3}\right )}{2 \left (a^{2}+b^{2}\right )^{3} a^{3} d \left (a +b \tan \left (d x +c \right )\right )^{2}}\) | \(375\) |
norman | \(\frac {-\frac {b^{2} \left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) x \tan \left (d x +c \right )^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b \left (4 B \,a^{2} b^{2}+2 B \,b^{4}-3 C \,a^{3} b -C a \,b^{3}\right ) \tan \left (d x +c \right )^{2}}{d \,a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{2} \left (7 B \,a^{2} b^{2}+3 B \,b^{4}-5 C \,a^{3} b -C a \,b^{3}\right ) \tan \left (d x +c \right )^{3}}{2 d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 b \left (3 B \,a^{2} b -B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) a x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {\left (B \,a^{3}-3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (6 B \,a^{4} b +3 B \,a^{2} b^{3}+B \,b^{5}-3 C \,a^{5}+C \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}\) | \(515\) |
risch | \(\frac {2 i C \,b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {x C}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {6 i b^{4} B x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a}+\frac {2 i C \,b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {6 i b^{4} B c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a d}+\frac {2 i b^{6} B x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3}}-\frac {2 i B x}{a^{3}}+\frac {12 i a \,b^{2} B c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {6 i a^{2} b C c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {2 i \left (3 i B \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 i C \,a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-2 B a \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+C \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-4 i B \,a^{2} b^{4}+3 i C \,a^{3} b^{3}-B a \,b^{5}+i B \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-4 B \,a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 C \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-i B \,b^{6}-4 B \,a^{3} b^{3}+3 C \,a^{4} b^{2}\right )}{\left (i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} a^{2} d \left (-i b +a \right )^{3}}+\frac {12 i a \,b^{2} B x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 i a^{2} b C x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {i x B}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {2 i b^{6} B c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}-\frac {2 i B c}{a^{3} d}-\frac {6 a \,b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a d}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C \,b^{3}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {B \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\) | \(1021\) |
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Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (213) = 426\).
Time = 0.33 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.18 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=-\frac {7 \, C a^{5} b^{3} - 9 \, B a^{4} b^{4} + C a^{3} b^{5} - 3 \, B a^{2} b^{6} - 2 \, {\left (C a^{8} - 3 \, B a^{7} b - 3 \, C a^{6} b^{2} + B a^{5} b^{3}\right )} d x - {\left (5 \, C a^{5} b^{3} - 7 \, B a^{4} b^{4} - C a^{3} b^{5} - B a^{2} b^{6} + 2 \, {\left (C a^{6} b^{2} - 3 \, B a^{5} b^{3} - 3 \, C a^{4} b^{4} + B a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left (B a^{8} + 3 \, B a^{6} b^{2} + 3 \, B a^{4} b^{4} + B a^{2} b^{6} + {\left (B a^{6} b^{2} + 3 \, B a^{4} b^{4} + 3 \, B a^{2} b^{6} + B b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{7} b + 3 \, B a^{5} b^{3} + 3 \, B a^{3} b^{5} + B a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (3 \, C a^{7} b - 6 \, B a^{6} b^{2} - C a^{5} b^{3} - 3 \, B a^{4} b^{4} - B a^{2} b^{6} + {\left (3 \, C a^{5} b^{3} - 6 \, B a^{4} b^{4} - C a^{3} b^{5} - 3 \, B a^{2} b^{6} - B b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, C a^{6} b^{2} - 6 \, B a^{5} b^{3} - C a^{4} b^{4} - 3 \, B a^{3} b^{5} - B a b^{7}\right )} \tan \left (d x + c\right )\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 (3 \, C a^{6} b^{2} - 4 \, B a^{5} b^{3} - 3 \, C a^{4} b^{4} + 3 \, B a^{3} b^{5} + B a b^{7} + 2 \, {\left (C a^{7} b - 3 \, B a^{6} b^{2} - 3 \, C a^{5} b^{3} + B a^{4} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \]
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Exception generated. \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
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Time = 0.34 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.73 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, C a^{5} b - 6 \, B a^{4} b^{2} - C a^{3} b^{3} - 3 \, B a^{2} b^{4} - B b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}} - \frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, C a^{4} b - 7 \, B a^{3} b^{2} + C a^{2} b^{3} - 3 \, B a b^{4} + 2 \, {\left (2 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - B b^{5}\right )} \tan \left (d x + c\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )} + \frac {2 \, B \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (213) = 426\).
Time = 1.24 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.23 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, C a^{5} b^{2} - 6 \, B a^{4} b^{3} - C a^{3} b^{4} - 3 \, B a^{2} b^{5} - B b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}} + \frac {2 \, B \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {9 \, C a^{5} b^{3} \tan \left (d x + c\right )^{2} - 18 \, B a^{4} b^{4} \tan \left (d x + c\right )^{2} - 3 \, C a^{3} b^{5} \tan \left (d x + c\right )^{2} - 9 \, B a^{2} b^{6} \tan \left (d x + c\right )^{2} - 3 \, B b^{8} \tan \left (d x + c\right )^{2} + 22 \, C a^{6} b^{2} \tan \left (d x + c\right ) - 42 \, B a^{5} b^{3} \tan \left (d x + c\right ) - 2 \, C a^{4} b^{4} \tan \left (d x + c\right ) - 26 \, B a^{3} b^{5} \tan \left (d x + c\right ) - 8 \, B a b^{7} \tan \left (d x + c\right ) + 14 \, C a^{7} b - 25 \, B a^{6} b^{2} + 3 \, C a^{5} b^{3} - 19 \, B a^{4} b^{4} + C a^{3} b^{5} - 6 \, B a^{2} b^{6}}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
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Time = 10.87 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {-5\,C\,a^3\,b+7\,B\,a^2\,b^2-C\,a\,b^3+3\,B\,b^4}{2\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2\,C\,a^3\,b^2+3\,B\,a^2\,b^3+B\,b^5\right )}{a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,C\,a^5+6\,B\,a^4\,b+C\,a^3\,b^2+3\,B\,a^2\,b^3+B\,b^5\right )}{a^3\,d\,{\left (a^2+b^2\right )}^3} \]
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