Integrand size = 29, antiderivative size = 87 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-((a A-b B) x)+\frac {(A b+a B) \log (\cos (c+d x))}{d}+\frac {(a A-b B) \tan (c+d x)}{d}+\frac {(A b+a B) \tan ^2(c+d x)}{2 d}+\frac {b B \tan ^3(c+d x)}{3 d} \]
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Time = 0.15 (sec) , antiderivative size = 87, 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} \[ \int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {(a B+A b) \tan ^2(c+d x)}{2 d}+\frac {(a A-b B) \tan (c+d x)}{d}+\frac {(a B+A b) \log (\cos (c+d x))}{d}-x (a A-b B)+\frac {b B \tan ^3(c+d x)}{3 d} \]
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Rule 3556
Rule 3606
Rule 3609
Rule 3673
Rubi steps \begin{align*} \text {integral}& = \frac {b B \tan ^3(c+d x)}{3 d}+\int \tan ^2(c+d x) (a A-b B+(A b+a B) \tan (c+d x)) \, dx \\ & = \frac {(A b+a B) \tan ^2(c+d x)}{2 d}+\frac {b B \tan ^3(c+d x)}{3 d}+\int \tan (c+d x) (-A b-a B+(a A-b B) \tan (c+d x)) \, dx \\ & = -((a A-b B) x)+\frac {(a A-b B) \tan (c+d x)}{d}+\frac {(A b+a B) \tan ^2(c+d x)}{2 d}+\frac {b B \tan ^3(c+d x)}{3 d}+(-A b-a B) \int \tan (c+d x) \, dx \\ & = -((a A-b B) x)+\frac {(A b+a B) \log (\cos (c+d x))}{d}+\frac {(a A-b B) \tan (c+d x)}{d}+\frac {(A b+a B) \tan ^2(c+d x)}{2 d}+\frac {b B \tan ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {(-6 a A+6 b B) \arctan (\tan (c+d x))+6 (A b+a B) \log (\cos (c+d x))+6 (a A-b B) \tan (c+d x)+3 (A b+a B) \tan ^2(c+d x)+2 b B \tan ^3(c+d x)}{6 d} \]
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Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01
method | result | size |
norman | \(\left (-a A +B b \right ) x +\frac {\left (a A -B b \right ) \tan \left (d x +c \right )}{d}+\frac {\left (A b +B a \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b B \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {\left (A b +B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(88\) |
parts | \(\frac {\left (A b +B a \right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {B b \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {a A \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(91\) |
derivativedivides | \(\frac {\frac {b B \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {A b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {B \left (\tan ^{2}\left (d x +c \right )\right ) a}{2}+A \tan \left (d x +c \right ) a -b B \tan \left (d x +c \right )+\frac {\left (-A b -B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a A +B b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(99\) |
default | \(\frac {\frac {b B \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {A b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {B \left (\tan ^{2}\left (d x +c \right )\right ) a}{2}+A \tan \left (d x +c \right ) a -b B \tan \left (d x +c \right )+\frac {\left (-A b -B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a A +B b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(99\) |
parallelrisch | \(-\frac {-2 b B \left (\tan ^{3}\left (d x +c \right )\right )+6 A x a d -3 A b \left (\tan ^{2}\left (d x +c \right )\right )-6 B b d x -3 B \left (\tan ^{2}\left (d x +c \right )\right ) a +3 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b -6 A \tan \left (d x +c \right ) a +3 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a +6 b B \tan \left (d x +c \right )}{6 d}\) | \(105\) |
risch | \(-i A b x -i B a x -A a x +B b x -\frac {2 i A b c}{d}-\frac {2 i a B c}{d}+\frac {2 i \left (-3 i A b \,{\mathrm e}^{4 i \left (d x +c \right )}-3 i B a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 A a \,{\mathrm e}^{4 i \left (d x +c \right )}-6 B b \,{\mathrm e}^{4 i \left (d x +c \right )}-3 i A b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i B a \,{\mathrm e}^{2 i \left (d x +c \right )}+6 A a \,{\mathrm e}^{2 i \left (d x +c \right )}-6 B b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 a A -4 B b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A b}{d}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}\) | \(213\) |
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Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {2 \, B b \tan \left (d x + c\right )^{3} - 6 \, {\left (A a - B b\right )} d x + 3 \, {\left (B a + A b\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a + A b\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 6 \, {\left (A a - B b\right )} \tan \left (d x + c\right )}{6 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.56 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\begin {cases} - A a x + \frac {A a \tan {\left (c + d x \right )}}{d} - \frac {A b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A b \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac {B a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a \tan ^{2}{\left (c + d x \right )}}{2 d} + B b x + \frac {B b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {B b \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right ) \tan ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.37 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {2 \, B b \tan \left (d x + c\right )^{3} + 3 \, {\left (B a + A b\right )} \tan \left (d x + c\right )^{2} - 6 \, {\left (A a - B b\right )} {\left (d x + c\right )} - 3 \, {\left (B a + A b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, {\left (A a - B b\right )} \tan \left (d x + c\right )}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (83) = 166\).
Time = 0.85 (sec) , antiderivative size = 937, normalized size of antiderivative = 10.77 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
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Time = 7.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (A\,a-B\,b\right )-\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )-d\,x\,\left (A\,a-B\,b\right )+\frac {B\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}}{d} \]
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