Integrand size = 40, antiderivative size = 117 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x-\frac {b \left (3 a b B+3 a^2 C-b^2 C\right ) \log (\cos (c+d x))}{d}+\frac {a^3 B \log (\sin (c+d x))}{d}+\frac {b^2 (b B+2 a C) \tan (c+d x)}{d}+\frac {b C (a+b \tan (c+d x))^2}{2 d} \]
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Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3713, 3688, 3718, 3705, 3556} \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {a^3 B \log (\sin (c+d x))}{d}-\frac {b \left (3 a^2 C+3 a b B-b^2 C\right ) \log (\cos (c+d x))}{d}+x \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right )+\frac {b^2 (2 a C+b B) \tan (c+d x)}{d}+\frac {b C (a+b \tan (c+d x))^2}{2 d} \]
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Rule 3556
Rule 3688
Rule 3705
Rule 3713
Rule 3718
Rubi steps \begin{align*} \text {integral}& = \int \cot (c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx \\ & = \frac {b C (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot (c+d x) (a+b \tan (c+d x)) \left (2 a^2 B+2 \left (2 a b B+a^2 C-b^2 C\right ) \tan (c+d x)+2 b (b B+2 a C) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {b^2 (b B+2 a C) \tan (c+d x)}{d}+\frac {b C (a+b \tan (c+d x))^2}{2 d}-\frac {1}{2} \int \cot (c+d x) \left (-2 a^3 B-2 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)-2 b \left (3 a b B+3 a^2 C-b^2 C\right ) \tan ^2(c+d x)\right ) \, dx \\ & = \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x+\frac {b^2 (b B+2 a C) \tan (c+d x)}{d}+\frac {b C (a+b \tan (c+d x))^2}{2 d}+\left (a^3 B\right ) \int \cot (c+d x) \, dx+\left (b \left (3 a b B+3 a^2 C-b^2 C\right )\right ) \int \tan (c+d x) \, dx \\ & = \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x-\frac {b \left (3 a b B+3 a^2 C-b^2 C\right ) \log (\cos (c+d x))}{d}+\frac {a^3 B \log (\sin (c+d x))}{d}+\frac {b^2 (b B+2 a C) \tan (c+d x)}{d}+\frac {b C (a+b \tan (c+d x))^2}{2 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {-(a+i b)^3 (B+i C) \log (i-\tan (c+d x))+2 a^3 B \log (\tan (c+d x))-(a-i b)^3 (B-i C) \log (i+\tan (c+d x))+2 b^2 (b B+3 a C) \tan (c+d x)+b^3 C \tan ^2(c+d x)}{2 d} \]
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Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(\frac {\left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+2 B \,a^{3} \ln \left (\tan \left (d x +c \right )\right )+C \,b^{3} \tan \left (d x +c \right )^{2}+\left (2 B \,b^{3}+6 C a \,b^{2}\right ) \tan \left (d x +c \right )+6 d \left (B \,a^{2} b -\frac {1}{3} B \,b^{3}+\frac {1}{3} C \,a^{3}-C a \,b^{2}\right ) x}{2 d}\) | \(121\) |
derivativedivides | \(\frac {\frac {C \,b^{3} \tan \left (d x +c \right )^{2}}{2}+B \,b^{3} \tan \left (d x +c \right )+3 C a \,b^{2} \tan \left (d x +c \right )+B \,a^{3} \ln \left (\tan \left (d x +c \right )\right )+\frac {\left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(130\) |
default | \(\frac {\frac {C \,b^{3} \tan \left (d x +c \right )^{2}}{2}+B \,b^{3} \tan \left (d x +c \right )+3 C a \,b^{2} \tan \left (d x +c \right )+B \,a^{3} \ln \left (\tan \left (d x +c \right )\right )+\frac {\left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(130\) |
norman | \(\frac {\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) x \tan \left (d x +c \right )+\frac {b^{2} \left (B b +3 C a \right ) \tan \left (d x +c \right )^{2}}{d}+\frac {C \,b^{3} \tan \left (d x +c \right )^{3}}{2 d}}{\tan \left (d x +c \right )}+\frac {B \,a^{3} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(142\) |
risch | \(\frac {6 i B a \,b^{2} c}{d}-i B \,a^{3} x +3 i C \,a^{2} b x +\frac {2 i b^{2} \left (B b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 C a \,{\mathrm e}^{2 i \left (d x +c \right )}-i C b \,{\mathrm e}^{2 i \left (d x +c \right )}+B b +3 C a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+3 B \,a^{2} b x -B \,b^{3} x +C \,a^{3} x -3 C a \,b^{2} x +3 i B a \,b^{2} x -i C \,b^{3} x -\frac {2 i B \,a^{3} c}{d}-\frac {2 i C \,b^{3} c}{d}+\frac {6 i C \,a^{2} b c}{d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B a \,b^{2}}{d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C \,a^{2} b}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C \,b^{3}}{d}+\frac {B \,a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(264\) |
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Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {C b^{3} \tan \left (d x + c\right )^{2} + B a^{3} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} d x - {\left (3 \, C a^{2} b + 3 \, B a b^{2} - C b^{3}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )}{2 \, d} \]
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Time = 0.98 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.80 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\begin {cases} - \frac {B a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 3 B a^{2} b x + \frac {3 B a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - B b^{3} x + \frac {B b^{3} \tan {\left (c + d x \right )}}{d} + C a^{3} x + \frac {3 C a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 3 C a b^{2} x + \frac {3 C a b^{2} \tan {\left (c + d x \right )}}{d} - \frac {C b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C b^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right )^{3} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.33 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {C b^{3} \tan \left (d x + c\right )^{2} + 2 \, B a^{3} \log \left (\tan \left (d x + c\right )\right ) + 2 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )} - {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )}{2 \, d} \]
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Time = 1.62 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.10 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {C b^{3} \tan \left (d x + c\right )^{2} + 2 \, B a^{3} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 6 \, C a b^{2} \tan \left (d x + c\right ) + 2 \, B b^{3} \tan \left (d x + c\right ) + 2 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )} - {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
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Time = 9.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,b^3+3\,C\,a\,b^2\right )}{d}+\frac {B\,a^3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {C\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \]
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