Integrand size = 25, antiderivative size = 215 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=-\left (\left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right ) x\right )-\frac {\left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}+\frac {2 b (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f} \]
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Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3624, 3609, 3606, 3556} \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=-\frac {\left (2 a^3 c d+3 a^2 b \left (c^2-d^2\right )-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}-x \left (-\left (a^3 \left (c^2-d^2\right )\right )+6 a^2 b c d+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {2 b (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f} \]
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Rule 3556
Rule 3606
Rule 3609
Rule 3624
Rubi steps \begin{align*} \text {integral}& = \frac {d^2 (a+b \tan (e+f x))^4}{4 b f}+\int (a+b \tan (e+f x))^3 \left (c^2-d^2+2 c d \tan (e+f x)\right ) \, dx \\ & = \frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}+\int (a+b \tan (e+f x))^2 \left (-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)\right ) \, dx \\ & = \frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}+\int (a+b \tan (e+f x)) ((a c-b c-a d-b d) (a c+b c+a d-b d)+2 (b c+a d) (a c-b d) \tan (e+f x)) \, dx \\ & = -\left (\left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right ) x\right )+\frac {2 b (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f}+\left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \int \tan (e+f x) \, dx \\ & = -\left (\left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right ) x\right )-\frac {\left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}+\frac {2 b (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac {2 c d (a+b \tan (e+f x))^3}{3 f}+\frac {d^2 (a+b \tan (e+f x))^4}{4 b f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.61 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.03 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\frac {3 d^2 (a+b \tan (e+f x))^4-6 \left (2 a c d+b \left (-c^2+d^2\right )\right ) \left ((i a-b)^3 \log (i-\tan (e+f x))-(i a+b)^3 \log (i+\tan (e+f x))+6 a b^2 \tan (e+f x)+b^3 \tan ^2(e+f x)\right )-4 c d \left (3 i (a+i b)^4 \log (i-\tan (e+f x))-3 i (a-i b)^4 \log (i+\tan (e+f x))+6 b^2 \left (-6 a^2+b^2\right ) \tan (e+f x)-12 a b^3 \tan ^2(e+f x)-2 b^4 \tan ^3(e+f x)\right )}{12 b f} \]
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Time = 0.27 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.08
method | result | size |
parts | \(a^{3} c^{2} x +\frac {\left (3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {\left (3 a^{2} b \,d^{2}+6 a \,b^{2} c d +b^{3} c^{2}\right ) \left (\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}+\frac {\left (a^{3} d^{2}+6 a^{2} b c d +3 a \,b^{2} c^{2}\right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {b^{3} d^{2} \left (\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}\) | \(232\) |
norman | \(\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) x +\frac {\left (a^{3} d^{2}+6 a^{2} b c d +3 a \,b^{2} c^{2}-3 a \,b^{2} d^{2}-2 b^{3} c d \right ) \tan \left (f x +e \right )}{f}+\frac {b \left (3 a^{2} d^{2}+6 a b c d +b^{2} c^{2}-b^{2} d^{2}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {b^{3} d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}+\frac {b^{2} d \left (3 a d +2 b c \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}-3 a^{2} b \,d^{2}-6 a \,b^{2} c d -b^{3} c^{2}+b^{3} d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}\) | \(258\) |
derivativedivides | \(\frac {\frac {b^{3} d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{4}+a \,b^{2} d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+\frac {2 b^{3} c d \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {3 a^{2} b \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+3 a \,b^{2} c d \left (\tan ^{2}\left (f x +e \right )\right )+\frac {b^{3} c^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {b^{3} d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+a^{3} d^{2} \tan \left (f x +e \right )+6 a^{2} b c d \tan \left (f x +e \right )+3 a \,b^{2} c^{2} \tan \left (f x +e \right )-3 a \,b^{2} d^{2} \tan \left (f x +e \right )-2 b^{3} c d \tan \left (f x +e \right )+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}-3 a^{2} b \,d^{2}-6 a \,b^{2} c d -b^{3} c^{2}+b^{3} d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(307\) |
default | \(\frac {\frac {b^{3} d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{4}+a \,b^{2} d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+\frac {2 b^{3} c d \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {3 a^{2} b \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+3 a \,b^{2} c d \left (\tan ^{2}\left (f x +e \right )\right )+\frac {b^{3} c^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {b^{3} d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+a^{3} d^{2} \tan \left (f x +e \right )+6 a^{2} b c d \tan \left (f x +e \right )+3 a \,b^{2} c^{2} \tan \left (f x +e \right )-3 a \,b^{2} d^{2} \tan \left (f x +e \right )-2 b^{3} c d \tan \left (f x +e \right )+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}-3 a^{2} b \,d^{2}-6 a \,b^{2} c d -b^{3} c^{2}+b^{3} d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(307\) |
parallelrisch | \(\frac {-6 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) b^{3} c^{2}+6 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) b^{3} d^{2}+3 b^{3} d^{2} \left (\tan ^{4}\left (f x +e \right )\right )+6 b^{3} c^{2} \left (\tan ^{2}\left (f x +e \right )\right )-6 b^{3} d^{2} \left (\tan ^{2}\left (f x +e \right )\right )+12 a^{3} d^{2} \tan \left (f x +e \right )+12 a^{3} c^{2} f x -12 a^{3} d^{2} f x +12 a \,b^{2} d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+8 b^{3} c d \left (\tan ^{3}\left (f x +e \right )\right )+18 a^{2} b \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )+36 a \,b^{2} c^{2} \tan \left (f x +e \right )-36 a \,b^{2} d^{2} \tan \left (f x +e \right )-24 b^{3} c d \tan \left (f x +e \right )+72 a^{2} b c d \tan \left (f x +e \right )+36 a \,b^{2} c d \left (\tan ^{2}\left (f x +e \right )\right )-36 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a \,b^{2} c d -36 a \,b^{2} c^{2} f x +36 a \,b^{2} d^{2} f x +24 b^{3} c d f x +12 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{3} c d +18 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{2} b \,c^{2}-18 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{2} b \,d^{2}-72 a^{2} b c d f x}{12 f}\) | \(367\) |
risch | \(a^{3} c^{2} x -a^{3} d^{2} x -\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) b^{3} d^{2}}{f}-i b^{3} c^{2} x -3 a \,b^{2} c^{2} x +3 a \,b^{2} d^{2} x +2 b^{3} c d x +i b^{3} d^{2} x +\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) b^{3} c^{2}}{f}-6 a^{2} b c d x +\frac {6 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a \,b^{2} c d}{f}-6 i a \,b^{2} c d x +\frac {4 i a^{3} c d e}{f}+\frac {6 i a^{2} b \,c^{2} e}{f}-\frac {6 i a^{2} b \,d^{2} e}{f}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{3} c d}{f}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{2} b \,c^{2}}{f}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{2} b \,d^{2}}{f}+2 i a^{3} c d x +3 i a^{2} b \,c^{2} x -3 i a^{2} b \,d^{2} x +\frac {2 i \left (9 a \,b^{2} c^{2}-12 a \,b^{2} d^{2}-20 b^{3} c d \,{\mathrm e}^{2 i \left (f x +e \right )}+27 a \,b^{2} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-30 a \,b^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-36 a \,b^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-24 b^{3} c d \,{\mathrm e}^{4 i \left (f x +e \right )}-12 b^{3} c d \,{\mathrm e}^{6 i \left (f x +e \right )}+27 a \,b^{2} c^{2} {\mathrm e}^{4 i \left (f x +e \right )}+9 a \,b^{2} c^{2} {\mathrm e}^{6 i \left (f x +e \right )}-18 a \,b^{2} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}+6 i b^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-3 i b^{3} c^{2} {\mathrm e}^{6 i \left (f x +e \right )}-6 i b^{3} c^{2} {\mathrm e}^{4 i \left (f x +e \right )}+3 a^{3} d^{2}-8 b^{3} c d +18 a^{2} b c d +9 a^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3 a^{3} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}+9 a^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-18 i a \,b^{2} c d \,{\mathrm e}^{2 i \left (f x +e \right )}+18 a^{2} b c d \,{\mathrm e}^{6 i \left (f x +e \right )}+54 a^{2} b c d \,{\mathrm e}^{4 i \left (f x +e \right )}+54 a^{2} b c d \,{\mathrm e}^{2 i \left (f x +e \right )}+6 i b^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-3 i b^{3} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+6 i b^{3} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-9 i a^{2} b \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-18 i a^{2} b \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-9 i a^{2} b \,d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-36 i a \,b^{2} c d \,{\mathrm e}^{4 i \left (f x +e \right )}-18 i a \,b^{2} c d \,{\mathrm e}^{6 i \left (f x +e \right )}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}-\frac {2 i b^{3} c^{2} e}{f}+\frac {2 i b^{3} d^{2} e}{f}-\frac {12 i a \,b^{2} c d e}{f}\) | \(872\) |
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Time = 0.26 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.17 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\frac {3 \, b^{3} d^{2} \tan \left (f x + e\right )^{4} + 4 \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \tan \left (f x + e\right )^{3} + 12 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} - 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d - {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} f x + 6 \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} - 6 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d - {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 12 \, {\left (3 \, a b^{2} c^{2} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d + {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (190) = 380\).
Time = 0.17 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.07 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\begin {cases} a^{3} c^{2} x + \frac {a^{3} c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - a^{3} d^{2} x + \frac {a^{3} d^{2} \tan {\left (e + f x \right )}}{f} + \frac {3 a^{2} b c^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 6 a^{2} b c d x + \frac {6 a^{2} b c d \tan {\left (e + f x \right )}}{f} - \frac {3 a^{2} b d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 a^{2} b d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} - 3 a b^{2} c^{2} x + \frac {3 a b^{2} c^{2} \tan {\left (e + f x \right )}}{f} - \frac {3 a b^{2} c d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {3 a b^{2} c d \tan ^{2}{\left (e + f x \right )}}{f} + 3 a b^{2} d^{2} x + \frac {a b^{2} d^{2} \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 a b^{2} d^{2} \tan {\left (e + f x \right )}}{f} - \frac {b^{3} c^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {b^{3} c^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} + 2 b^{3} c d x + \frac {2 b^{3} c d \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 b^{3} c d \tan {\left (e + f x \right )}}{f} + \frac {b^{3} d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {b^{3} d^{2} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {b^{3} d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \left (a + b \tan {\left (e \right )}\right )^{3} \left (c + d \tan {\left (e \right )}\right )^{2} & \text {otherwise} \end {cases} \]
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Time = 0.63 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.18 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\frac {3 \, b^{3} d^{2} \tan \left (f x + e\right )^{4} + 4 \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \tan \left (f x + e\right )^{3} + 6 \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} + 12 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} - 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d - {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} {\left (f x + e\right )} + 6 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d - {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 12 \, {\left (3 \, a b^{2} c^{2} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d + {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3977 vs. \(2 (209) = 418\).
Time = 3.00 (sec) , antiderivative size = 3977, normalized size of antiderivative = 18.50 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=\text {Too large to display} \]
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Time = 6.59 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.20 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx=x\,\left (a^3\,c^2-a^3\,d^2-6\,a^2\,b\,c\,d-3\,a\,b^2\,c^2+3\,a\,b^2\,d^2+2\,b^3\,c\,d\right )+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^3\,d^2-b^2\,d\,\left (3\,a\,d+2\,b\,c\right )+3\,a\,b^2\,c^2+6\,a^2\,b\,c\,d\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (-a^3\,c\,d-\frac {3\,a^2\,b\,c^2}{2}+\frac {3\,a^2\,b\,d^2}{2}+3\,a\,b^2\,c\,d+\frac {b^3\,c^2}{2}-\frac {b^3\,d^2}{2}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {3\,a^2\,b\,d^2}{2}+3\,a\,b^2\,c\,d+\frac {b^3\,c^2}{2}-\frac {b^3\,d^2}{2}\right )}{f}+\frac {b^3\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^4}{4\,f}+\frac {b^2\,d\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (3\,a\,d+2\,b\,c\right )}{3\,f} \]
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