Integrand size = 32, antiderivative size = 112 \[ \int (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\left (\left (2 a b B+a^2 C-b^2 C\right ) x\right )-\frac {\left (a^2 B-b^2 B-2 a b C\right ) \log (\cos (c+d x))}{d}+\frac {b (a B-b C) \tan (c+d x)}{d}+\frac {B (a+b \tan (c+d x))^2}{2 d}+\frac {C (a+b \tan (c+d x))^3}{3 b d} \]
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Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3711, 3609, 3606, 3556} \[ \int (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {\left (a^2 B-2 a b C-b^2 B\right ) \log (\cos (c+d x))}{d}-x \left (a^2 C+2 a b B-b^2 C\right )+\frac {b (a B-b C) \tan (c+d x)}{d}+\frac {B (a+b \tan (c+d x))^2}{2 d}+\frac {C (a+b \tan (c+d x))^3}{3 b d} \]
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Rule 3556
Rule 3606
Rule 3609
Rule 3711
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+b \tan (c+d x))^3}{3 b d}+\int (a+b \tan (c+d x))^2 (-C+B \tan (c+d x)) \, dx \\ & = \frac {B (a+b \tan (c+d x))^2}{2 d}+\frac {C (a+b \tan (c+d x))^3}{3 b d}+\int (a+b \tan (c+d x)) (-b B-a C+(a B-b C) \tan (c+d x)) \, dx \\ & = -\left (\left (2 a b B+a^2 C-b^2 C\right ) x\right )+\frac {b (a B-b C) \tan (c+d x)}{d}+\frac {B (a+b \tan (c+d x))^2}{2 d}+\frac {C (a+b \tan (c+d x))^3}{3 b d}+\left (a^2 B-b^2 B-2 a b C\right ) \int \tan (c+d x) \, dx \\ & = -\left (\left (2 a b B+a^2 C-b^2 C\right ) x\right )-\frac {\left (a^2 B-b^2 B-2 a b C\right ) \log (\cos (c+d x))}{d}+\frac {b (a B-b C) \tan (c+d x)}{d}+\frac {B (a+b \tan (c+d x))^2}{2 d}+\frac {C (a+b \tan (c+d x))^3}{3 b d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.96 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.54 \[ \int (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {2 C (a+b \tan (c+d x))^3+3 (a B+b C) \left (i \left ((a+i b)^2 \log (i-\tan (c+d x))-(a-i b)^2 \log (i+\tan (c+d x))\right )-2 b^2 \tan (c+d x)\right )+3 B \left ((i a-b)^3 \log (i-\tan (c+d x))-(i a+b)^3 \log (i+\tan (c+d x))+6 a b^2 \tan (c+d x)+b^3 \tan ^2(c+d x)\right )}{6 b d} \]
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Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07
method | result | size |
norman | \(\left (-2 B a b -C \,a^{2}+C \,b^{2}\right ) x +\frac {\left (2 B a b +C \,a^{2}-C \,b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{2} \tan \left (d x +c \right )^{3}}{3 d}+\frac {b \left (B b +2 C a \right ) \tan \left (d x +c \right )^{2}}{2 d}+\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(120\) |
parts | \(\frac {\left (B \,b^{2}+2 C a b \right ) \left (\frac {\tan \left (d x +c \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {\left (2 B a b +C \,a^{2}\right ) \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {B \,a^{2} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}+\frac {C \,b^{2} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(126\) |
derivativedivides | \(\frac {\frac {C \,b^{2} \tan \left (d x +c \right )^{3}}{3}+\frac {B \,b^{2} \tan \left (d x +c \right )^{2}}{2}+C a b \tan \left (d x +c \right )^{2}+2 B a b \tan \left (d x +c \right )+C \,a^{2} \tan \left (d x +c \right )-C \,b^{2} \tan \left (d x +c \right )+\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-2 B a b -C \,a^{2}+C \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(135\) |
default | \(\frac {\frac {C \,b^{2} \tan \left (d x +c \right )^{3}}{3}+\frac {B \,b^{2} \tan \left (d x +c \right )^{2}}{2}+C a b \tan \left (d x +c \right )^{2}+2 B a b \tan \left (d x +c \right )+C \,a^{2} \tan \left (d x +c \right )-C \,b^{2} \tan \left (d x +c \right )+\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-2 B a b -C \,a^{2}+C \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(135\) |
parallelrisch | \(\frac {2 C \,b^{2} \tan \left (d x +c \right )^{3}-12 B a b d x +3 B \,b^{2} \tan \left (d x +c \right )^{2}-6 C \,a^{2} d x +6 C \,b^{2} d x +6 C a b \tan \left (d x +c \right )^{2}+3 B \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{2}-3 B \ln \left (1+\tan \left (d x +c \right )^{2}\right ) b^{2}+12 B a b \tan \left (d x +c \right )-6 C \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a b +6 C \,a^{2} \tan \left (d x +c \right )-6 C \,b^{2} \tan \left (d x +c \right )}{6 d}\) | \(156\) |
risch | \(-i B \,b^{2} x +\frac {2 i B \,a^{2} c}{d}+\frac {2 i \left (-3 i B \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 i C a b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 C \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-3 i B \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 i C a b \,{\mathrm e}^{2 i \left (d x +c \right )}+12 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 C \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 B a b +3 C \,a^{2}-4 C \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-2 B a b x -C \,a^{2} x +C \,b^{2} x -\frac {2 i B \,b^{2} c}{d}-\frac {4 i C a b c}{d}-2 i C a b x +i B \,a^{2} x -\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,a^{2}}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,b^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C a b}{d}\) | \(324\) |
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Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.06 \[ \int (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {2 \, C b^{2} \tan \left (d x + c\right )^{3} - 6 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} d x + 3 \, {\left (2 \, C a b + B b^{2}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 6 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \tan \left (d x + c\right )}{6 \, d} \]
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Time = 0.13 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.73 \[ \int (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\begin {cases} \frac {B a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 2 B a b x + \frac {2 B a b \tan {\left (c + d x \right )}}{d} - \frac {B b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B b^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} - C a^{2} x + \frac {C a^{2} \tan {\left (c + d x \right )}}{d} - \frac {C a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {C a b \tan ^{2}{\left (c + d x \right )}}{d} + C b^{2} x + \frac {C b^{2} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {C b^{2} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07 \[ \int (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {2 \, C b^{2} \tan \left (d x + c\right )^{3} + 3 \, {\left (2 \, C a b + B b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )} + 3 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \tan \left (d x + c\right )}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (108) = 216\).
Time = 1.19 (sec) , antiderivative size = 1389, normalized size of antiderivative = 12.40 \[ \int (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 8.43 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08 \[ \int (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {B\,b^2}{2}+C\,a\,b\right )}{d}-x\,\left (C\,a^2+2\,B\,a\,b-C\,b^2\right )+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (C\,a^2+2\,B\,a\,b-C\,b^2\right )}{d}-\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (-\frac {B\,a^2}{2}+C\,a\,b+\frac {B\,b^2}{2}\right )}{d}+\frac {C\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d} \]
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