Integrand size = 40, antiderivative size = 88 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\left (b^2 C-a (2 b B+a C)\right ) x-\frac {a (2 b B+a C) \cot (c+d x)}{d}-\frac {a^2 B \cot ^2(c+d x)}{2 d}-\frac {\left (a^2 B-b^2 B-2 a b C\right ) \log (\sin (c+d x))}{d} \]
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Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3713, 3685, 3709, 3612, 3556} \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {\left (a^2 B-2 a b C-b^2 B\right ) \log (\sin (c+d x))}{d}-\frac {a^2 B \cot ^2(c+d x)}{2 d}+x \left (b^2 C-a (a C+2 b B)\right )-\frac {a (a C+2 b B) \cot (c+d x)}{d} \]
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Rule 3556
Rule 3612
Rule 3685
Rule 3709
Rule 3713
Rubi steps \begin{align*} \text {integral}& = \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (B+C \tan (c+d x)) \, dx \\ & = -\frac {a^2 B \cot ^2(c+d x)}{2 d}+\int \cot ^2(c+d x) \left (a (2 b B+a C)-\left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)+b^2 C \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {a (2 b B+a C) \cot (c+d x)}{d}-\frac {a^2 B \cot ^2(c+d x)}{2 d}+\int \cot (c+d x) \left (-a^2 B+b^2 B+2 a b C+\left (b^2 C-a (2 b B+a C)\right ) \tan (c+d x)\right ) \, dx \\ & = \left (b^2 C-a (2 b B+a C)\right ) x-\frac {a (2 b B+a C) \cot (c+d x)}{d}-\frac {a^2 B \cot ^2(c+d x)}{2 d}+\left (-a^2 B+b^2 B+2 a b C\right ) \int \cot (c+d x) \, dx \\ & = \left (b^2 C-a (2 b B+a C)\right ) x-\frac {a (2 b B+a C) \cot (c+d x)}{d}-\frac {a^2 B \cot ^2(c+d x)}{2 d}-\frac {\left (a^2 B-b^2 B-2 a b C\right ) \log (\sin (c+d x))}{d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.40 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {-2 a (2 b B+a C) \cot (c+d x)-a^2 B \cot ^2(c+d x)+(a+i b)^2 (B+i C) \log (i-\tan (c+d x))-2 \left (a^2 B-b^2 B-2 a b C\right ) \log (\tan (c+d x))+(a-i b)^2 (B-i C) \log (i+\tan (c+d x))}{2 d} \]
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Time = 0.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {B \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+C \,b^{2} \left (d x +c \right )+2 B a b \left (-\cot \left (d x +c \right )-d x -c \right )+2 C a b \ln \left (\sin \left (d x +c \right )\right )+B \,a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+C \,a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(107\) |
default | \(\frac {B \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+C \,b^{2} \left (d x +c \right )+2 B a b \left (-\cot \left (d x +c \right )-d x -c \right )+2 C a b \ln \left (\sin \left (d x +c \right )\right )+B \,a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+C \,a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(107\) |
parallelrisch | \(\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+\left (-2 B \,a^{2}+2 B \,b^{2}+4 C a b \right ) \ln \left (\tan \left (d x +c \right )\right )-B \,a^{2} \cot \left (d x +c \right )^{2}+\left (-4 B a b -2 C \,a^{2}\right ) \cot \left (d x +c \right )-4 d x \left (B a b +\frac {1}{2} C \,a^{2}-\frac {1}{2} C \,b^{2}\right )}{2 d}\) | \(114\) |
norman | \(\frac {\left (-2 B a b -C \,a^{2}+C \,b^{2}\right ) x \tan \left (d x +c \right )^{3}-\frac {B \,a^{2} \tan \left (d x +c \right )}{2 d}-\frac {a \left (2 B b +C a \right ) \tan \left (d x +c \right )^{2}}{d}}{\tan \left (d x +c \right )^{3}}-\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(138\) |
risch | \(-i B \,b^{2} x -\frac {2 i B \,b^{2} c}{d}+\frac {2 i B \,a^{2} c}{d}-2 B a b x -C \,a^{2} x +C \,b^{2} x -\frac {4 i C a b c}{d}-\frac {2 i a \left (2 B b \,{\mathrm e}^{2 i \left (d x +c \right )}+C a \,{\mathrm e}^{2 i \left (d x +c \right )}+i B a \,{\mathrm e}^{2 i \left (d x +c \right )}-2 B b -C a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-2 i C a b x +i B \,a^{2} x -\frac {B \,a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B \,b^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C a b}{d}\) | \(205\) |
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Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.39 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {{\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + B a^{2} + {\left (B a^{2} + 2 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (C a^{2} + 2 \, B a b\right )} \tan \left (d x + c\right )}{2 \, d \tan \left (d x + c\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (78) = 156\).
Time = 1.76 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.34 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\begin {cases} \text {NaN} & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\\text {NaN} & \text {for}\: c = - d x \\\frac {B a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - 2 B a b x - \frac {2 B a b}{d \tan {\left (c + d x \right )}} - \frac {B b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - C a^{2} x - \frac {C a^{2}}{d \tan {\left (c + d x \right )}} - \frac {C a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 C a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + C b^{2} x & \text {otherwise} \end {cases} \]
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Time = 0.38 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )} - {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {B a^{2} + 2 \, {\left (C a^{2} + 2 \, B a b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (86) = 172\).
Time = 0.87 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.69 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )} - 8 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 8 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {12 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 8.70 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.44 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-B\,a^2+2\,C\,a\,b+B\,b^2\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {B\,a^2}{2}+\mathrm {tan}\left (c+d\,x\right )\,\left (C\,a^2+2\,B\,b\,a\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}{2\,d} \]
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