Integrand size = 31, antiderivative size = 187 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x+\frac {a^2 \left (a^2 A-3 A b^2-3 a b B\right ) \cot (c+d x)}{d}-\frac {b^4 B \log (\cos (c+d x))}{d}-\frac {a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \log (\sin (c+d x))}{d}-\frac {a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d} \]
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Time = 0.83 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3686, 3726, 3716, 3705, 3556} \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}-\frac {a \left (a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right ) \log (\sin (c+d x))}{d}+x \left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right )-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}-\frac {b^4 B \log (\cos (c+d x))}{d} \]
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Rule 3556
Rule 3686
Rule 3705
Rule 3716
Rule 3726
Rubi steps \begin{align*} \text {integral}& = -\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 \left (3 a (2 A b+a B)-3 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+3 b^2 B \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}+\frac {1}{6} \int \cot ^2(c+d x) (a+b \tan (c+d x)) \left (-6 a \left (a^2 A-3 A b^2-3 a b B\right )-6 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+6 b^3 B \tan ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 \left (a^2 A-3 A b^2-3 a b B\right ) \cot (c+d x)}{d}-\frac {a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}+\frac {1}{6} \int \cot (c+d x) \left (-6 a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right )+6 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \tan (c+d x)+6 b^4 B \tan ^2(c+d x)\right ) \, dx \\ & = \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x+\frac {a^2 \left (a^2 A-3 A b^2-3 a b B\right ) \cot (c+d x)}{d}-\frac {a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}+\left (b^4 B\right ) \int \tan (c+d x) \, dx-\left (a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right )\right ) \int \cot (c+d x) \, dx \\ & = \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x+\frac {a^2 \left (a^2 A-3 A b^2-3 a b B\right ) \cot (c+d x)}{d}-\frac {b^4 B \log (\cos (c+d x))}{d}-\frac {a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \log (\sin (c+d x))}{d}-\frac {a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.89 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 a^2 \left (a^2 A-6 A b^2-4 a b B\right ) \cot (c+d x)-3 a^3 (4 A b+a B) \cot ^2(c+d x)-2 a^4 A \cot ^3(c+d x)+3 (a+i b)^4 (-i A+B) \log (i-\tan (c+d x))-6 a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \log (\tan (c+d x))+3 (a-i b)^4 (i A+B) \log (i+\tan (c+d x))}{6 d} \]
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Time = 0.24 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}}{3 \tan \left (d x +c \right )^{3}}-\frac {a^{3} \left (4 A b +B a \right )}{2 \tan \left (d x +c \right )^{2}}-a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{\tan \left (d x +c \right )}}{d}\) | \(195\) |
default | \(\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}}{3 \tan \left (d x +c \right )^{3}}-\frac {a^{3} \left (4 A b +B a \right )}{2 \tan \left (d x +c \right )^{2}}-a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{\tan \left (d x +c \right )}}{d}\) | \(195\) |
parallelrisch | \(\frac {3 \left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+6 \left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-2 A \left (\cot ^{3}\left (d x +c \right )\right ) a^{4}+3 \left (-4 A \,a^{3} b -B \,a^{4}\right ) \left (\cot ^{2}\left (d x +c \right )\right )+6 a^{2} \cot \left (d x +c \right ) \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )+6 d x \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right )}{6 d}\) | \(195\) |
norman | \(\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {A \,a^{4}}{3 d}-\frac {a^{3} \left (4 A b +B a \right ) \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{3}}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(208\) |
risch | \(-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,b^{4}}{d}+A \,a^{4} x -\frac {2 i a^{2} \left (-6 A \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+18 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}+12 i A a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-24 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-12 i A a b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 A \,a^{2}+18 A \,b^{2}+12 B a b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {2 i a^{4} B c}{d}-\frac {12 i B \,a^{2} b^{2} c}{d}+i B \,b^{4} x +\frac {2 i B \,b^{4} c}{d}+\frac {4 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A \,b^{3}}{d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B \,b^{2}}{d}+A \,b^{4} x +i B \,a^{4} x -6 i B \,a^{2} b^{2} x -4 i A a \,b^{3} x -\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A b}{d}-\frac {8 i A a \,b^{3} c}{d}+4 i A \,a^{3} b x +\frac {8 i A \,a^{3} b c}{d}+4 B a \,b^{3} x -6 A \,a^{2} b^{2} x -4 B \,a^{3} b x\) | \(445\) |
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Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.19 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {3 \, B b^{4} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 2 \, A a^{4} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b - 2 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \]
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Time = 2.29 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.97 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a^{4} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a^{4} x & \text {for}\: c = - d x \\A a^{4} x + \frac {A a^{4}}{d \tan {\left (c + d x \right )}} - \frac {A a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {2 A a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {4 A a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {2 A a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - 6 A a^{2} b^{2} x - \frac {6 A a^{2} b^{2}}{d \tan {\left (c + d x \right )}} - \frac {2 A a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 A a b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + A b^{4} x + \frac {B a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - 4 B a^{3} b x - \frac {4 B a^{3} b}{d \tan {\left (c + d x \right )}} - \frac {3 B a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {6 B a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 4 B a b^{3} x + \frac {B b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.08 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {2 \, A a^{4} - 6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
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Time = 1.51 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.50 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + \frac {11 \, B a^{4} \tan \left (d x + c\right )^{3} + 44 \, A a^{3} b \tan \left (d x + c\right )^{3} - 66 \, B a^{2} b^{2} \tan \left (d x + c\right )^{3} - 44 \, A a b^{3} \tan \left (d x + c\right )^{3} + 6 \, A a^{4} \tan \left (d x + c\right )^{2} - 24 \, B a^{3} b \tan \left (d x + c\right )^{2} - 36 \, A a^{2} b^{2} \tan \left (d x + c\right )^{2} - 3 \, B a^{4} \tan \left (d x + c\right ) - 12 \, A a^{3} b \tan \left (d x + c\right ) - 2 \, A a^{4}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
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Time = 8.02 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^4+4\,A\,a^3\,b-6\,B\,a^2\,b^2-4\,A\,a\,b^3\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{2}+2\,A\,b\,a^3\right )+\frac {A\,a^4}{3}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-A\,a^4+4\,B\,a^3\,b+6\,A\,a^2\,b^2\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d} \]
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