Integrand size = 40, antiderivative size = 70 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\left (2 a b B+a^2 C-b^2 C\right ) x-\frac {b (b B+2 a C) \log (\cos (c+d x))}{d}+\frac {a^2 B \log (\sin (c+d x))}{d}+\frac {b^2 C \tan (c+d x)}{d} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3713, 3687, 3705, 3556} \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=x \left (a^2 C+2 a b B-b^2 C\right )+\frac {a^2 B \log (\sin (c+d x))}{d}-\frac {b (2 a C+b B) \log (\cos (c+d x))}{d}+\frac {b^2 C \tan (c+d x)}{d} \]
[In]
[Out]
Rule 3556
Rule 3687
Rule 3705
Rule 3713
Rubi steps \begin{align*} \text {integral}& = \int \cot (c+d x) (a+b \tan (c+d x))^2 (B+C \tan (c+d x)) \, dx \\ & = \frac {b^2 C \tan (c+d x)}{d}+\int \cot (c+d x) \left (a^2 B+\left (2 a b B+\left (a^2-b^2\right ) C\right ) \tan (c+d x)+\left (b^2 B+2 a b C\right ) \tan ^2(c+d x)\right ) \, dx \\ & = \left (2 a b B+a^2 C-b^2 C\right ) x+\frac {b^2 C \tan (c+d x)}{d}+\left (a^2 B\right ) \int \cot (c+d x) \, dx+(b (b B+2 a C)) \int \tan (c+d x) \, dx \\ & = \left (2 a b B+a^2 C-b^2 C\right ) x-\frac {b (b B+2 a C) \log (\cos (c+d x))}{d}+\frac {a^2 B \log (\sin (c+d x))}{d}+\frac {b^2 C \tan (c+d x)}{d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.30 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {(a+i b)^2 (B+i C) \log (i-\tan (c+d x))-2 a^2 B \log (\tan (c+d x))+(a-i b)^2 (B-i C) \log (i+\tan (c+d x))-2 b^2 C \tan (c+d x)}{2 d} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(\frac {\left (-B \,a^{2}+B \,b^{2}+2 C a b \right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+2 B \,a^{2} \ln \left (\tan \left (d x +c \right )\right )+2 C \,b^{2} \tan \left (d x +c \right )+4 d x \left (B a b +\frac {1}{2} C \,a^{2}-\frac {1}{2} C \,b^{2}\right )}{2 d}\) | \(80\) |
derivativedivides | \(-\frac {\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (\cot \left (d x +c \right )^{2}+1\right )}{2}+\left (2 B a b +C \,a^{2}-C \,b^{2}\right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )-\frac {C \,b^{2}}{\cot \left (d x +c \right )}+b \left (B b +2 C a \right ) \ln \left (\cot \left (d x +c \right )\right )}{d}\) | \(99\) |
default | \(-\frac {\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (\cot \left (d x +c \right )^{2}+1\right )}{2}+\left (2 B a b +C \,a^{2}-C \,b^{2}\right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )-\frac {C \,b^{2}}{\cot \left (d x +c \right )}+b \left (B b +2 C a \right ) \ln \left (\cot \left (d x +c \right )\right )}{d}\) | \(99\) |
norman | \(\frac {\left (2 B a b +C \,a^{2}-C \,b^{2}\right ) x \tan \left (d x +c \right )+\frac {C \,b^{2} \tan \left (d x +c \right )^{2}}{d}}{\tan \left (d x +c \right )}+\frac {B \,a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(101\) |
risch | \(i B \,b^{2} x +\frac {4 i C a b c}{d}+\frac {2 i C \,b^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+2 B a b x +C \,a^{2} x -C \,b^{2} x +2 i C a b x -\frac {2 i B \,a^{2} c}{d}+\frac {2 i B \,b^{2} c}{d}-i B \,a^{2} x -\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,b^{2}}{d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C a b}{d}+\frac {B \,a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(160\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.31 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {B a^{2} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, C b^{2} \tan \left (d x + c\right ) + 2 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} d x - {\left (2 \, C a b + B b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (66) = 132\).
Time = 0.75 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.94 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\begin {cases} - \frac {B a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 2 B a b x + \frac {B b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + C a^{2} x + \frac {C a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - C b^{2} x + \frac {C b^{2} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.21 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {2 \, B a^{2} \log \left (\tan \left (d x + c\right )\right ) + 2 \, C b^{2} \tan \left (d x + c\right ) + 2 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )} - {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
[In]
[Out]
none
Time = 1.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.23 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {2 \, B a^{2} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 2 \, C b^{2} \tan \left (d x + c\right ) + 2 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )} - {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
[In]
[Out]
Time = 8.59 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.29 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {B\,a^2\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d}+\frac {C\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}{2\,d} \]
[In]
[Out]