Integrand size = 25, antiderivative size = 219 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx=-\left (\left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x\right )-\frac {\left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \log (\cos (e+f x))}{f}+\frac {2 d (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f} \]
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Time = 0.31 (sec) , antiderivative size = 219, 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} \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx=-\frac {\left (a^2 \left (3 c^2 d-d^3\right )+2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}-x \left (-\left (a^2 \left (c^3-3 c d^2\right )\right )+2 a b d \left (3 c^2-d^2\right )+b^2 c \left (c^2-3 d^2\right )\right )+\frac {\left (a^2 d+2 a b c-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {2 d (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f} \]
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Rule 3556
Rule 3606
Rule 3609
Rule 3624
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 (c+d \tan (e+f x))^4}{4 d f}+\int \left (a^2-b^2+2 a b \tan (e+f x)\right ) (c+d \tan (e+f x))^3 \, dx \\ & = \frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f}+\int (c+d \tan (e+f x))^2 \left (a^2 c-b^2 c-2 a b d+\left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)\right ) \, dx \\ & = \frac {\left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f}+\int (c+d \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 (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x\right )+\frac {2 d (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f}+\left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \int \tan (e+f x) \, dx \\ & = -\left (\left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x\right )-\frac {\left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \log (\cos (e+f x))}{f}+\frac {2 d (b c+a d) (a c-b d) \tan (e+f x)}{f}+\frac {\left (2 a b c+a^2 d-b^2 d\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {2 a b (c+d \tan (e+f x))^3}{3 f}+\frac {b^2 (c+d \tan (e+f x))^4}{4 d f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.39 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.01 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx=\frac {3 b^2 (c+d \tan (e+f x))^4-6 \left (2 a b c-a^2 d+b^2 d\right ) \left ((i c-d)^3 \log (i-\tan (e+f x))-(i c+d)^3 \log (i+\tan (e+f x))+6 c d^2 \tan (e+f x)+d^3 \tan ^2(e+f x)\right )-4 a b \left (3 i (c+i d)^4 \log (i-\tan (e+f x))-3 i (c-i d)^4 \log (i+\tan (e+f x))+6 d^2 \left (-6 c^2+d^2\right ) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-2 d^4 \tan ^3(e+f x)\right )}{12 d f} \]
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Time = 0.25 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.06
method | result | size |
parts | \(a^{2} c^{3} x +\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) \left (\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}+\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {b^{2} d^{3} \left (\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}\) | \(232\) |
norman | \(\left (a^{2} c^{3}-3 a^{2} c \,d^{2}-6 a b \,c^{2} d +2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) x +\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d -2 a b \,d^{3}+b^{2} c^{3}-3 b^{2} c \,d^{2}\right ) \tan \left (f x +e \right )}{f}+\frac {b^{2} d^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}+\frac {d \left (a^{2} d^{2}+6 a b c d +3 b^{2} c^{2}-b^{2} d^{2}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {b \,d^{2} \left (2 a d +3 b c \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {\left (3 a^{2} c^{2} d -a^{2} d^{3}+2 a b \,c^{3}-6 a b c \,d^{2}-3 b^{2} c^{2} d +b^{2} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}\) | \(258\) |
derivativedivides | \(\frac {\frac {b^{2} d^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {2 a b \,d^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3}+b^{2} c \,d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+\frac {\left (\tan ^{2}\left (f x +e \right )\right ) a^{2} d^{3}}{2}+3 a b c \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )+\frac {3 b^{2} c^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {b^{2} d^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+3 \tan \left (f x +e \right ) a^{2} c \,d^{2}+6 a b \,c^{2} d \tan \left (f x +e \right )-2 a b \,d^{3} \tan \left (f x +e \right )+b^{2} c^{3} \tan \left (f x +e \right )-3 b^{2} c \,d^{2} \tan \left (f x +e \right )+\frac {\left (3 a^{2} c^{2} d -a^{2} d^{3}+2 a b \,c^{3}-6 a b c \,d^{2}-3 b^{2} c^{2} d +b^{2} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{2} c^{3}-3 a^{2} c \,d^{2}-6 a b \,c^{2} d +2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(307\) |
default | \(\frac {\frac {b^{2} d^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {2 a b \,d^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3}+b^{2} c \,d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+\frac {\left (\tan ^{2}\left (f x +e \right )\right ) a^{2} d^{3}}{2}+3 a b c \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )+\frac {3 b^{2} c^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {b^{2} d^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{2}+3 \tan \left (f x +e \right ) a^{2} c \,d^{2}+6 a b \,c^{2} d \tan \left (f x +e \right )-2 a b \,d^{3} \tan \left (f x +e \right )+b^{2} c^{3} \tan \left (f x +e \right )-3 b^{2} c \,d^{2} \tan \left (f x +e \right )+\frac {\left (3 a^{2} c^{2} d -a^{2} d^{3}+2 a b \,c^{3}-6 a b c \,d^{2}-3 b^{2} c^{2} d +b^{2} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{2} c^{3}-3 a^{2} c \,d^{2}-6 a b \,c^{2} d +2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(307\) |
parallelrisch | \(\frac {12 b^{2} c^{3} \tan \left (f x +e \right )+3 b^{2} d^{3} \left (\tan ^{4}\left (f x +e \right )\right )-6 b^{2} d^{3} \left (\tan ^{2}\left (f x +e \right )\right )+12 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a b \,c^{3}-18 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) b^{2} c^{2} d -12 b^{2} c^{3} f x +24 a b \,d^{3} f x +36 b^{2} c \,d^{2} f x +8 a b \,d^{3} \left (\tan ^{3}\left (f x +e \right )\right )+12 b^{2} c \,d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+18 b^{2} c^{2} d \left (\tan ^{2}\left (f x +e \right )\right )-24 a b \,d^{3} \tan \left (f x +e \right )-36 b^{2} c \,d^{2} \tan \left (f x +e \right )+72 a b \,c^{2} d \tan \left (f x +e \right )+36 a b c \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )-36 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a b c \,d^{2}-36 x \,a^{2} c \,d^{2} f +12 x \,a^{2} c^{3} f +18 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{2} c^{2} d +36 \tan \left (f x +e \right ) a^{2} c \,d^{2}+6 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) b^{2} d^{3}+6 \left (\tan ^{2}\left (f x +e \right )\right ) a^{2} d^{3}-6 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{2} d^{3}-72 a b \,c^{2} d f x}{12 f}\) | \(367\) |
risch | \(a^{2} c^{3} x -b^{2} c^{3} x -6 a b \,c^{2} d x -3 a^{2} c \,d^{2} x +2 a b \,d^{3} x +3 b^{2} c \,d^{2} x -\frac {2 i a^{2} d^{3} e}{f}+\frac {2 i b^{2} d^{3} e}{f}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{2} c^{2} d}{f}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a b \,c^{3}}{f}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) b^{2} c^{2} d}{f}+3 i a^{2} c^{2} d x +2 i a b \,c^{3} x -3 i b^{2} c^{2} d x +\frac {2 i \left (-8 a b \,d^{3}-12 b^{2} c \,d^{2}-9 i b^{2} c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}+18 a b \,c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}-9 i b^{2} c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+54 a b \,c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}+54 a b \,c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+9 a^{2} c \,d^{2}+3 b^{2} c^{3}+18 a b \,c^{2} d +9 b^{2} c^{3} {\mathrm e}^{2 i \left (f x +e \right )}+9 b^{2} c^{3} {\mathrm e}^{4 i \left (f x +e \right )}+3 b^{2} c^{3} {\mathrm e}^{6 i \left (f x +e \right )}-18 i b^{2} c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}-3 i a^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-6 i a^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+6 i b^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-3 i a^{2} d^{3} {\mathrm e}^{6 i \left (f x +e \right )}+6 i b^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+6 i b^{2} d^{3} {\mathrm e}^{6 i \left (f x +e \right )}-24 a b \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-36 b^{2} c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-18 b^{2} c \,d^{2} {\mathrm e}^{6 i \left (f x +e \right )}+27 a^{2} c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+27 a^{2} c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-20 a b \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-30 b^{2} c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-12 a b \,d^{3} {\mathrm e}^{6 i \left (f x +e \right )}+9 a^{2} c \,d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-18 i a b c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-36 i a b c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-18 i a b c \,d^{2} {\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 {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a^{2} d^{3}}{f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) b^{2} d^{3}}{f}-i a^{2} d^{3} x +i b^{2} d^{3} x +\frac {6 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a b c \,d^{2}}{f}-6 i a b c \,d^{2} x +\frac {6 i a^{2} c^{2} d e}{f}+\frac {4 i a b \,c^{3} e}{f}-\frac {6 i b^{2} c^{2} d e}{f}-\frac {12 i a b c \,d^{2} e}{f}\) | \(872\) |
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Time = 0.24 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.12 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx=\frac {3 \, b^{2} d^{3} \tan \left (f x + e\right )^{4} + 4 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} \tan \left (f x + e\right )^{3} - 12 \, {\left (6 \, a b c^{2} d - 2 \, a b d^{3} - {\left (a^{2} - b^{2}\right )} c^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} f x + 6 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + {\left (a^{2} - b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} - 6 \, {\left (2 \, a b c^{3} - 6 \, a b c d^{2} + 3 \, {\left (a^{2} - b^{2}\right )} c^{2} d - {\left (a^{2} - b^{2}\right )} d^{3}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 12 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 2 \, a b d^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (194) = 388\).
Time = 0.18 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.03 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx=\begin {cases} a^{2} c^{3} x + \frac {3 a^{2} c^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 3 a^{2} c d^{2} x + \frac {3 a^{2} c d^{2} \tan {\left (e + f x \right )}}{f} - \frac {a^{2} d^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {a^{2} d^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} + \frac {a b c^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - 6 a b c^{2} d x + \frac {6 a b c^{2} d \tan {\left (e + f x \right )}}{f} - \frac {3 a b c d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {3 a b c d^{2} \tan ^{2}{\left (e + f x \right )}}{f} + 2 a b d^{3} x + \frac {2 a b d^{3} \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 a b d^{3} \tan {\left (e + f x \right )}}{f} - b^{2} c^{3} x + \frac {b^{2} c^{3} \tan {\left (e + f x \right )}}{f} - \frac {3 b^{2} c^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 b^{2} c^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} + 3 b^{2} c d^{2} x + \frac {b^{2} c d^{2} \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 b^{2} c d^{2} \tan {\left (e + f x \right )}}{f} + \frac {b^{2} d^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {b^{2} d^{3} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {b^{2} d^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \left (a + b \tan {\left (e \right )}\right )^{2} \left (c + d \tan {\left (e \right )}\right )^{3} & \text {otherwise} \end {cases} \]
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Time = 0.43 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.12 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx=\frac {3 \, b^{2} d^{3} \tan \left (f x + e\right )^{4} + 4 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} \tan \left (f x + e\right )^{3} + 6 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + {\left (a^{2} - b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} - 12 \, {\left (6 \, a b c^{2} d - 2 \, a b d^{3} - {\left (a^{2} - b^{2}\right )} c^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} {\left (f x + e\right )} + 6 \, {\left (2 \, a b c^{3} - 6 \, a b c d^{2} + 3 \, {\left (a^{2} - b^{2}\right )} c^{2} d - {\left (a^{2} - b^{2}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 12 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 2 \, a b d^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3977 vs. \(2 (213) = 426\).
Time = 3.06 (sec) , antiderivative size = 3977, normalized size of antiderivative = 18.16 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx=\text {Too large to display} \]
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Time = 6.92 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.18 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx=x\,\left (a^2\,c^3-3\,a^2\,c\,d^2-6\,a\,b\,c^2\,d+2\,a\,b\,d^3-b^2\,c^3+3\,b^2\,c\,d^2\right )+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (b^2\,c^3-b\,d^2\,\left (2\,a\,d+3\,b\,c\right )+3\,a^2\,c\,d^2+6\,a\,b\,c^2\,d\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (-\frac {3\,a^2\,c^2\,d}{2}+\frac {a^2\,d^3}{2}-a\,b\,c^3+3\,a\,b\,c\,d^2+\frac {3\,b^2\,c^2\,d}{2}-\frac {b^2\,d^3}{2}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {a^2\,d^3}{2}+3\,a\,b\,c\,d^2+\frac {3\,b^2\,c^2\,d}{2}-\frac {b^2\,d^3}{2}\right )}{f}+\frac {b^2\,d^3\,{\mathrm {tan}\left (e+f\,x\right )}^4}{4\,f}+\frac {b\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (2\,a\,d+3\,b\,c\right )}{3\,f} \]
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