Integrand size = 31, antiderivative size = 127 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\left (\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x\right )-\frac {a^2 (2 A b+a B) \cot (c+d x)}{d}-\frac {b^3 B \log (\cos (c+d x))}{d}-\frac {a \left (a^2 A-3 A b^2-3 a b B\right ) \log (\sin (c+d x))}{d}-\frac {a A \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d} \]
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Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3686, 3716, 3705, 3556} \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {a \left (a^2 A-3 a b B-3 A b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 (a B+2 A b) \cot (c+d x)}{d}-x \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right )-\frac {a A \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {b^3 B \log (\cos (c+d x))}{d} \]
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Rule 3556
Rule 3686
Rule 3705
Rule 3716
Rubi steps \begin{align*} \text {integral}& = -\frac {a A \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot ^2(c+d x) (a+b \tan (c+d x)) \left (2 a (2 A b+a B)-2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+2 b^2 B \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {a^2 (2 A b+a B) \cot (c+d x)}{d}-\frac {a A \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot (c+d x) \left (-2 a \left (a^2 A-3 A b^2-3 a b B\right )-2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+2 b^3 B \tan ^2(c+d x)\right ) \, dx \\ & = -\left (\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x\right )-\frac {a^2 (2 A b+a B) \cot (c+d x)}{d}-\frac {a A \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+\left (b^3 B\right ) \int \tan (c+d x) \, dx-\left (a \left (a^2 A-3 A b^2-3 a b B\right )\right ) \int \cot (c+d x) \, dx \\ & = -\left (\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x\right )-\frac {a^2 (2 A b+a B) \cot (c+d x)}{d}-\frac {b^3 B \log (\cos (c+d x))}{d}-\frac {a \left (a^2 A-3 A b^2-3 a b B\right ) \log (\sin (c+d x))}{d}-\frac {a A \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.99 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {-2 a^2 (3 A b+a B) \cot (c+d x)-a^3 A \cot ^2(c+d x)+(a+i b)^3 (A+i B) \log (i-\tan (c+d x))-2 a \left (a^2 A-3 A b^2-3 a b B\right ) \log (\tan (c+d x))+(a-i b)^3 (A-i B) \log (i+\tan (c+d x))}{2 d} \]
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Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\frac {\left (A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (-2 A \,a^{3}+6 A a \,b^{2}+6 B \,a^{2} b \right ) \ln \left (\tan \left (d x +c \right )\right )-A \left (\cot ^{2}\left (d x +c \right )\right ) a^{3}+\left (-6 A \,a^{2} b -2 B \,a^{3}\right ) \cot \left (d x +c \right )-6 \left (A \,a^{2} b -\frac {1}{3} A \,b^{3}+\frac {1}{3} B \,a^{3}-B a \,b^{2}\right ) d x}{2 d}\) | \(136\) |
derivativedivides | \(\frac {\frac {\left (A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{3}}{2 \tan \left (d x +c \right )^{2}}-\frac {a^{2} \left (3 A b +B a \right )}{\tan \left (d x +c \right )}-a \left (A \,a^{2}-3 A \,b^{2}-3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(140\) |
default | \(\frac {\frac {\left (A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{3}}{2 \tan \left (d x +c \right )^{2}}-\frac {a^{2} \left (3 A b +B a \right )}{\tan \left (d x +c \right )}-a \left (A \,a^{2}-3 A \,b^{2}-3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(140\) |
norman | \(\frac {\left (-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )-\frac {A \,a^{3}}{2 d}-\frac {a^{2} \left (3 A b +B a \right ) \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{2}}+\frac {\left (A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \left (A \,a^{2}-3 A \,b^{2}-3 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(150\) |
risch | \(-\frac {6 i B \,a^{2} b c}{d}-3 i B \,a^{2} b x +\frac {2 i B \,b^{3} c}{d}-\frac {6 i A a \,b^{2} c}{d}-3 A \,a^{2} b x +A \,b^{3} x -B \,a^{3} x +3 B a \,b^{2} x -\frac {2 i a^{2} \left (i A a \,{\mathrm e}^{2 i \left (d x +c \right )}+3 A b \,{\mathrm e}^{2 i \left (d x +c \right )}+B a \,{\mathrm e}^{2 i \left (d x +c \right )}-3 A b -B a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+i A \,a^{3} x -3 i A a \,b^{2} x +i B \,b^{3} x +\frac {2 i a^{3} A c}{d}-\frac {A \,a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A \,b^{2}}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B b}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,b^{3}}{d}\) | \(267\) |
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Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.28 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {B b^{3} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + A a^{3} + {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + {\left (A a^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \tan \left (d x + c\right )}{2 \, d \tan \left (d x + c\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (121) = 242\).
Time = 1.08 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.06 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a^{3} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{3} \cot ^{3}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a^{3} x & \text {for}\: c = - d x \\\frac {A a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - 3 A a^{2} b x - \frac {3 A a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {3 A a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 A a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + A b^{3} x - B a^{3} x - \frac {B a^{3}}{d \tan {\left (c + d x \right )}} - \frac {3 B a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 B a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 3 B a b^{2} x + \frac {B b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {2 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} {\left (d x + c\right )} - {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {A a^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
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Time = 1.40 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.52 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {2 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} {\left (d x + c\right )} - {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac {3 \, A a^{3} \tan \left (d x + c\right )^{2} - 9 \, B a^{2} b \tan \left (d x + c\right )^{2} - 9 \, A a b^{2} \tan \left (d x + c\right )^{2} - 2 \, B a^{3} \tan \left (d x + c\right ) - 6 \, A a^{2} b \tan \left (d x + c\right ) - A a^{3}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
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Time = 8.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-A\,a^3+3\,B\,a^2\,b+3\,A\,a\,b^2\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^3+3\,A\,b\,a^2\right )+\frac {A\,a^3}{2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \]
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