Integrand size = 31, antiderivative size = 88 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\left (b^2 B-a (2 A b+a B)\right ) x-\frac {a (2 A b+a B) \cot (c+d x)}{d}-\frac {a^2 A \cot ^2(c+d x)}{2 d}-\frac {\left (a^2 A-A b^2-2 a b B\right ) \log (\sin (c+d x))}{d} \]
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Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3685, 3709, 3612, 3556} \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {\left (a^2 A-2 a b B-A b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 A \cot ^2(c+d x)}{2 d}+x \left (b^2 B-a (a B+2 A b)\right )-\frac {a (a B+2 A b) \cot (c+d x)}{d} \]
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Rule 3556
Rule 3612
Rule 3685
Rule 3709
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 A \cot ^2(c+d x)}{2 d}+\int \cot ^2(c+d x) \left (a (2 A b+a B)-\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+b^2 B \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {a (2 A b+a B) \cot (c+d x)}{d}-\frac {a^2 A \cot ^2(c+d x)}{2 d}+\int \cot (c+d x) \left (-a^2 A+A b^2+2 a b B+\left (b^2 B-a (2 A b+a B)\right ) \tan (c+d x)\right ) \, dx \\ & = \left (b^2 B-a (2 A b+a B)\right ) x-\frac {a (2 A b+a B) \cot (c+d x)}{d}-\frac {a^2 A \cot ^2(c+d x)}{2 d}+\left (-a^2 A+A b^2+2 a b B\right ) \int \cot (c+d x) \, dx \\ & = \left (b^2 B-a (2 A b+a B)\right ) x-\frac {a (2 A b+a B) \cot (c+d x)}{d}-\frac {a^2 A \cot ^2(c+d x)}{2 d}-\frac {\left (a^2 A-A b^2-2 a b B\right ) \log (\sin (c+d x))}{d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.40 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {-2 a (2 A b+a B) \cot (c+d x)-a^2 A \cot ^2(c+d x)+(a+i b)^2 (A+i B) \log (i-\tan (c+d x))-2 \left (a^2 A-A b^2-2 a b B\right ) \log (\tan (c+d x))+(a-i b)^2 (A-i B) \log (i+\tan (c+d x))}{2 d} \]
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Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {A \,a^{2} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+B \,a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+2 A a b \left (-\cot \left (d x +c \right )-d x -c \right )+2 B a b \ln \left (\sin \left (d x +c \right )\right )+A \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+B \,b^{2} \left (d x +c \right )}{d}\) | \(107\) |
default | \(\frac {A \,a^{2} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+B \,a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+2 A a b \left (-\cot \left (d x +c \right )-d x -c \right )+2 B a b \ln \left (\sin \left (d x +c \right )\right )+A \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+B \,b^{2} \left (d x +c \right )}{d}\) | \(107\) |
parallelrisch | \(\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (-2 A \,a^{2}+2 A \,b^{2}+4 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )-A \left (\cot ^{2}\left (d x +c \right )\right ) a^{2}+\left (-4 A a b -2 B \,a^{2}\right ) \cot \left (d x +c \right )-4 d \left (A a b +\frac {1}{2} B \,a^{2}-\frac {1}{2} B \,b^{2}\right ) x}{2 d}\) | \(114\) |
norman | \(\frac {\left (-2 A a b -B \,a^{2}+B \,b^{2}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )-\frac {A \,a^{2}}{2 d}-\frac {a \left (2 A b +B a \right ) \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{2}}-\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(130\) |
risch | \(-2 A a b x -B \,a^{2} x +B \,b^{2} x -\frac {2 i A \,b^{2} c}{d}-i A \,b^{2} x +i A \,a^{2} x -\frac {4 i B a b c}{d}-\frac {2 i a \left (i A a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 A b \,{\mathrm e}^{2 i \left (d x +c \right )}+B a \,{\mathrm e}^{2 i \left (d x +c \right )}-2 A b -B a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-2 i B a b x +\frac {2 i a^{2} A c}{d}-\frac {A \,a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A \,b^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B a b}{d}\) | \(205\) |
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Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.39 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + A a^{2} + {\left (A a^{2} + 2 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{2 \, d \tan \left (d x + c\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (78) = 156\).
Time = 0.82 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.43 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a^{2} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{2} \cot ^{3}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a^{2} x & \text {for}\: c = - d x \\\frac {A a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - 2 A a b x - \frac {2 A a b}{d \tan {\left (c + d x \right )}} - \frac {A b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - B a^{2} x - \frac {B a^{2}}{d \tan {\left (c + d x \right )}} - \frac {B a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 B a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + B b^{2} x & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {2 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )} - {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {A a^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (86) = 172\).
Time = 1.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.69 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )} - 8 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 8 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {12 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 8.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.44 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-A\,a^2+2\,B\,a\,b+A\,b^2\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {A\,a^2}{2}+\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^2+2\,A\,b\,a\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}{2\,d} \]
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