Integrand size = 19, antiderivative size = 226 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\log (\sin (c+d x))}{a^4 d}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^4 d}+\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]
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Time = 0.75 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3650, 3730, 3732, 3611, 3556} \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\log (\sin (c+d x))}{a^4 d}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^4}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))} \]
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Rule 3556
Rule 3611
Rule 3650
Rule 3730
Rule 3732
Rubi steps \begin{align*} \text {integral}& = \frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {\cot (c+d x) \left (3 \left (a^2+b^2\right )-3 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a \left (a^2+b^2\right )} \\ & = \frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (6 \left (a^2+b^2\right )^2-12 a^3 b \tan (c+d x)+6 b^2 \left (3 a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^2 \left (a^2+b^2\right )^2} \\ & = \frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (6 \left (a^2+b^2\right )^3-6 a^3 b \left (3 a^2-b^2\right ) \tan (c+d x)+6 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^3 \left (a^2+b^2\right )^3} \\ & = -\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \cot (c+d x) \, dx}{a^4}-\frac {\left (b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \left (a^2+b^2\right )^4} \\ & = -\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\log (\sin (c+d x))}{a^4 d}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^4 d}+\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.11 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.08 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \left (-a^4 (a-i b)^4 \log (i-\tan (c+d x))+2 \left (a^2+b^2\right )^4 \log (\tan (c+d x))-a^4 (a+i b)^4 \log (i+\tan (c+d x))-2 b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a+b \tan (c+d x))\right )}{a^2 \left (a^2+b^2\right )^2}+\frac {2 a b^2 \left (a^2+b^2\right )}{(a+b \tan (c+d x))^3}+\frac {3 \left (3 a^2 b^2+b^4\right )}{(a+b \tan (c+d x))^2}+\frac {6 \left (6 a^4 b^2+3 a^2 b^4+b^6\right )}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{6 a^2 \left (a^2+b^2\right )^2 d} \]
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Time = 0.89 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {b^{2}}{3 a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{2 a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (6 a^{4}+3 a^{2} b^{2}+b^{4}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (10 a^{6}+5 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{4}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\) | \(246\) |
default | \(\frac {\frac {b^{2}}{3 a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{2 a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (6 a^{4}+3 a^{2} b^{2}+b^{4}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (10 a^{6}+5 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{4}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\) | \(246\) |
parallelrisch | \(\frac {-20 \left (a +b \tan \left (d x +c \right )\right )^{3} \left (a^{6}+\frac {1}{2} a^{4} b^{2}+\frac {2}{5} a^{2} b^{4}+\frac {1}{10} b^{6}\right ) b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )-a^{4} \left (a^{2}+2 a b -b^{2}\right ) \left (a^{2}-2 a b -b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\sec ^{2}\left (d x +c \right )\right )+2 \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\tan \left (d x +c \right )\right )-8 \left (b^{3} \left (\frac {47}{24} a^{6} b +\frac {27}{8} a^{4} b^{3}+\frac {15}{8} b^{5} a^{2}+\frac {11}{24} b^{7}+a^{7} d x -b^{2} a^{5} x d \right ) \left (\tan ^{3}\left (d x +c \right )\right )+3 a \left (a^{7} d x -b^{2} a^{5} x d +\frac {35}{24} a^{6} b +\frac {21}{8} a^{4} b^{3}+\frac {37}{24} b^{5} a^{2}+\frac {3}{8} b^{7}\right ) b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+3 a^{2} \left (a^{7} d x -b^{2} a^{5} x d +\frac {5}{6} a^{6} b +\frac {19}{12} a^{4} b^{3}+b^{5} a^{2}+\frac {1}{4} b^{7}\right ) b \tan \left (d x +c \right )+a^{8} d x \left (a -b \right ) \left (a +b \right )\right ) b}{2 \left (a^{2}+b^{2}\right )^{4} d \,a^{4} \left (a +b \tan \left (d x +c \right )\right )^{3}}\) | \(353\) |
norman | \(\frac {\frac {b \left (-10 a^{4} b^{2}-9 a^{2} b^{4}-3 b^{6}\right ) \tan \left (d x +c \right )}{d \,a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {4 \left (a^{2}-b^{2}\right ) b \,a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {b^{2} \left (-35 a^{4} b^{2}-28 a^{2} b^{4}-9 b^{6}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{3} \left (-47 a^{4} b^{2}-34 a^{2} b^{4}-11 b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{6 d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {b^{2} \left (10 a^{6}+5 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\) | \(589\) |
risch | \(\frac {2 i b^{8} c}{a^{4} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {20 i a^{2} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}+\frac {2 i b^{3} \left (30 i a^{3} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+45 i a^{5} b \,{\mathrm e}^{4 i \left (d x +c \right )}-30 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a \,b^{5}+9 i a \,b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-60 i a^{5} b -60 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-84 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-30 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-22 i a^{3} b^{3}-15 i a^{5} b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i a \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-30 a^{6}+19 a^{4} b^{2}+8 a^{2} b^{4}+3 b^{6}\right )}{3 \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} a^{3} \left (-i a +b \right )^{3} d \left (i a +b \right )^{4}}+\frac {2 i b^{8} x}{a^{4} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {8 i b^{6} x}{a^{2} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {8 i b^{6} c}{a^{2} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {10 i b^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {2 i x}{a^{4}}+\frac {10 i b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {20 i a^{2} b^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {2 i c}{d \,a^{4}}-\frac {10 a^{2} b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {5 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {4 b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{2} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {b^{8} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{4} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}\) | \(1035\) |
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Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (222) = 444\).
Time = 0.31 (sec) , antiderivative size = 793, normalized size of antiderivative = 3.51 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {75 \, a^{7} b^{4} + 42 \, a^{5} b^{6} + 11 \, a^{3} b^{8} - {\left (47 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + 24 \, {\left (a^{7} b^{4} - a^{5} b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 24 \, {\left (a^{10} b - a^{8} b^{3}\right )} d x - 3 \, {\left (35 \, a^{7} b^{4} - 12 \, a^{5} b^{6} - 5 \, a^{3} b^{8} - 2 \, a b^{10} + 24 \, {\left (a^{8} b^{3} - a^{6} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8} + {\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\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 ) - 3 \, {\left (10 \, a^{9} b^{2} + 5 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8} + {\left (10 \, a^{6} b^{5} + 5 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (10 \, a^{7} b^{4} + 5 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (10 \, a^{8} b^{3} + 5 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\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 ) - 3 \, {\left (20 \, a^{8} b^{3} - 37 \, a^{6} b^{5} - 18 \, a^{4} b^{7} - 5 \, a^{2} b^{9} + 24 \, {\left (a^{9} b^{2} - a^{7} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{12} b^{3} + 4 \, a^{10} b^{5} + 6 \, a^{8} b^{7} + 4 \, a^{6} b^{9} + a^{4} b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{13} b^{2} + 4 \, a^{11} b^{4} + 6 \, a^{9} b^{6} + 4 \, a^{7} b^{8} + a^{5} b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{14} b + 4 \, a^{12} b^{3} + 6 \, a^{10} b^{5} + 4 \, a^{8} b^{7} + a^{6} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{15} + 4 \, a^{13} b^{2} + 6 \, a^{11} b^{4} + 4 \, a^{9} b^{6} + a^{7} b^{8}\right )} d\right )}} \]
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Exception generated. \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]
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none
Time = 0.43 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.96 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, a^{6} b^{2} + 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {47 \, a^{6} b^{2} + 34 \, a^{4} b^{4} + 11 \, a^{2} b^{6} + 6 \, {\left (6 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (27 \, a^{5} b^{3} + 16 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \tan \left (d x + c\right )}{a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6} + {\left (a^{9} b^{3} + 3 \, a^{7} b^{5} + 3 \, a^{5} b^{7} + a^{3} b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} \tan \left (d x + c\right )} - \frac {6 \, \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (222) = 444\).
Time = 0.84 (sec) , antiderivative size = 476, normalized size of antiderivative = 2.11 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, a^{6} b^{3} + 5 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{12} b + 4 \, a^{10} b^{3} + 6 \, a^{8} b^{5} + 4 \, a^{6} b^{7} + a^{4} b^{9}} - \frac {110 \, a^{6} b^{5} \tan \left (d x + c\right )^{3} + 55 \, a^{4} b^{7} \tan \left (d x + c\right )^{3} + 44 \, a^{2} b^{9} \tan \left (d x + c\right )^{3} + 11 \, b^{11} \tan \left (d x + c\right )^{3} + 366 \, a^{7} b^{4} \tan \left (d x + c\right )^{2} + 219 \, a^{5} b^{6} \tan \left (d x + c\right )^{2} + 156 \, a^{3} b^{8} \tan \left (d x + c\right )^{2} + 39 \, a b^{10} \tan \left (d x + c\right )^{2} + 411 \, a^{8} b^{3} \tan \left (d x + c\right ) + 294 \, a^{6} b^{5} \tan \left (d x + c\right ) + 195 \, a^{4} b^{7} \tan \left (d x + c\right ) + 48 \, a^{2} b^{9} \tan \left (d x + c\right ) + 157 \, a^{9} b^{2} + 136 \, a^{7} b^{4} + 89 \, a^{5} b^{6} + 22 \, a^{3} b^{8}}{{\left (a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}} - \frac {6 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}}}{6 \, d} \]
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Time = 5.03 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.70 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {47\,a^4\,b^2+34\,a^2\,b^4+11\,b^6}{6\,a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (27\,a^4\,b^3+16\,a^2\,b^5+5\,b^7\right )}{2\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (6\,a^4\,b^4+3\,a^2\,b^6+b^8\right )}{a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^4\,d}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (10\,a^6+5\,a^4\,b^2+4\,a^2\,b^4+b^6\right )}{a^4\,d\,{\left (a^2+b^2\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \]
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