Integrand size = 29, antiderivative size = 112 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\left (\left (2 a A b+a^2 B-b^2 B\right ) x\right )-\frac {\left (a^2 A-A b^2-2 a b B\right ) \log (\cos (c+d x))}{d}+\frac {b (a A-b B) \tan (c+d x)}{d}+\frac {A (a+b \tan (c+d x))^2}{2 d}+\frac {B (a+b \tan (c+d x))^3}{3 b d} \]
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Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3673, 3609, 3606, 3556} \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {\left (a^2 A-2 a b B-A b^2\right ) \log (\cos (c+d x))}{d}-x \left (a^2 B+2 a A b-b^2 B\right )+\frac {b (a A-b B) \tan (c+d x)}{d}+\frac {A (a+b \tan (c+d x))^2}{2 d}+\frac {B (a+b \tan (c+d x))^3}{3 b d} \]
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Rule 3556
Rule 3606
Rule 3609
Rule 3673
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b \tan (c+d x))^3}{3 b d}+\int (-B+A \tan (c+d x)) (a+b \tan (c+d x))^2 \, dx \\ & = \frac {A (a+b \tan (c+d x))^2}{2 d}+\frac {B (a+b \tan (c+d x))^3}{3 b d}+\int (a+b \tan (c+d x)) (-A b-a B+(a A-b B) \tan (c+d x)) \, dx \\ & = -\left (\left (2 a A b+a^2 B-b^2 B\right ) x\right )+\frac {b (a A-b B) \tan (c+d x)}{d}+\frac {A (a+b \tan (c+d x))^2}{2 d}+\frac {B (a+b \tan (c+d x))^3}{3 b d}+\left (a^2 A-A b^2-2 a b B\right ) \int \tan (c+d x) \, dx \\ & = -\left (\left (2 a A b+a^2 B-b^2 B\right ) x\right )-\frac {\left (a^2 A-A b^2-2 a b B\right ) \log (\cos (c+d x))}{d}+\frac {b (a A-b B) \tan (c+d x)}{d}+\frac {A (a+b \tan (c+d x))^2}{2 d}+\frac {B (a+b \tan (c+d x))^3}{3 b d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.54 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {2 B (a+b \tan (c+d x))^3+3 (a A+b B) \left (i \left ((a+i b)^2 \log (i-\tan (c+d x))-(a-i b)^2 \log (i+\tan (c+d x))\right )-2 b^2 \tan (c+d x)\right )+3 A \left ((i a-b)^3 \log (i-\tan (c+d x))-(i a+b)^3 \log (i+\tan (c+d x))+6 a b^2 \tan (c+d x)+b^3 \tan ^2(c+d x)\right )}{6 b d} \]
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Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07
method | result | size |
norman | \(\left (-2 A a b -B \,a^{2}+B \,b^{2}\right ) x +\frac {\left (2 A a b +B \,a^{2}-B \,b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b \left (A b +2 B a \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(120\) |
parts | \(\frac {\left (A \,b^{2}+2 B a b \right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (2 A a b +B \,a^{2}\right ) \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2}}{2 d}+\frac {B \,b^{2} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(126\) |
derivativedivides | \(\frac {\frac {B \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {A \,b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+B a b \left (\tan ^{2}\left (d x +c \right )\right )+2 A a b \tan \left (d x +c \right )+B \tan \left (d x +c \right ) a^{2}-B \,b^{2} \tan \left (d x +c \right )+\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-2 A a b -B \,a^{2}+B \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(135\) |
default | \(\frac {\frac {B \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {A \,b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+B a b \left (\tan ^{2}\left (d x +c \right )\right )+2 A a b \tan \left (d x +c \right )+B \tan \left (d x +c \right ) a^{2}-B \,b^{2} \tan \left (d x +c \right )+\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-2 A a b -B \,a^{2}+B \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(135\) |
parallelrisch | \(\frac {2 B \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-12 A a b d x +3 A \,b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-6 B x \,a^{2} d +6 B \,b^{2} d x +6 B a b \left (\tan ^{2}\left (d x +c \right )\right )+3 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2}-3 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{2}+12 A a b \tan \left (d x +c \right )-6 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a b +6 B \tan \left (d x +c \right ) a^{2}-6 B \,b^{2} \tan \left (d x +c \right )}{6 d}\) | \(156\) |
risch | \(-2 A a b x -B \,a^{2} x +B \,b^{2} x +\frac {2 i \left (-3 i A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 i B a b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 A a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 B \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-3 i A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 i B a b \,{\mathrm e}^{2 i \left (d x +c \right )}+12 A a b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 B \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 A a b +3 B \,a^{2}-4 B \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-i A \,b^{2} x -\frac {4 i B a b c}{d}+i A \,a^{2} x -\frac {2 i A \,b^{2} c}{d}-2 i B a b x +\frac {2 i a^{2} A c}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A \,b^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B a b}{d}\) | \(324\) |
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Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.06 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {2 \, B b^{2} \tan \left (d x + c\right )^{3} - 6 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} d x + 3 \, {\left (2 \, B a b + A b^{2}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 6 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \tan \left (d x + c\right )}{6 \, d} \]
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Time = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.71 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\begin {cases} \frac {A a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 2 A a b x + \frac {2 A a b \tan {\left (c + d x \right )}}{d} - \frac {A b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A b^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} - B a^{2} x + \frac {B a^{2} \tan {\left (c + d x \right )}}{d} - \frac {B a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {B a b \tan ^{2}{\left (c + d x \right )}}{d} + B b^{2} x + \frac {B b^{2} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {B b^{2} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{2} \tan {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {2 \, B b^{2} \tan \left (d x + c\right )^{3} + 3 \, {\left (2 \, B a b + A b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )} + 3 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \tan \left (d x + c\right )}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (108) = 216\).
Time = 1.05 (sec) , antiderivative size = 1389, normalized size of antiderivative = 12.40 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
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Time = 7.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )}{d}-x\,\left (B\,a^2+2\,A\,a\,b-B\,b^2\right )+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^2+2\,A\,a\,b-B\,b^2\right )}{d}-\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (-\frac {A\,a^2}{2}+B\,a\,b+\frac {A\,b^2}{2}\right )}{d}+\frac {B\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d} \]
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