Integrand size = 45, antiderivative size = 603 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\left (a^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x+\frac {\left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{f}-\frac {d \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{f}+\frac {\left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^2}{2 f}+\frac {\left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^3}{3 f}+\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{60 d^3 f}-\frac {b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f} \] Output:
(a^2*(A*c^3-3*A*c*d^2-3*B*c^2*d+B*d^3-C*c^3+3*C*c*d^2)+b^2*(c^3*C+3*B*c^2* d-3*C*c*d^2-B*d^3-A*(c^3-3*c*d^2))-2*a*b*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d ^2)))*x+(2*a*b*(c^3*C+3*B*c^2*d-3*C*c*d^2-B*d^3-A*(c^3-3*c*d^2))-a^2*((A-C )*d*(3*c^2-d^2)+B*(c^3-3*c*d^2))+b^2*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)) )*ln(cos(f*x+e))/f-d*(2*a*b*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-a^2*(2*c*(A- C)*d+B*(c^2-d^2))+b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*tan(f*x+e)/f+1/2*(2*a*b*( A*c-B*d-C*c)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*(c+d*tan(f*x+e))^2/f+1/3 *(B*a^2-B*b^2+2*a*b*(A-C))*(c+d*tan(f*x+e))^3/f+1/60*(5*a^2*C*d^2-6*a*b*d* (-5*B*d+C*c)+b^2*(c^2*C-3*B*c*d+15*(A-C)*d^2))*(c+d*tan(f*x+e))^4/d^3/f-1/ 15*b*(-3*B*b*d-C*a*d+C*b*c)*tan(f*x+e)*(c+d*tan(f*x+e))^4/d^2/f+1/6*C*(a+b *tan(f*x+e))^2*(c+d*tan(f*x+e))^4/d/f
Result contains complex when optimal does not.
Time = 6.40 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.69 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}+\frac {-\frac {2 b (b c C-3 b B d-a C d) \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac {-\frac {\left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (c^2 C-3 B c d+15 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^4}{2 d f}+\frac {5 \left (3 d \left (2 a b (A c-c C+B d)+a^2 (B c-(A-C) d)-b^2 (B c-(A-C) d)\right ) \left ((i c-d)^3 \log (i-\tan (e+f x))-(i c+d)^3 \log (i+\tan (e+f x))+6 c d^2 \tan (e+f x)+d^3 \tan ^2(e+f x)\right )+\left (a^2 B-b^2 B+2 a b (A-C)\right ) d \left (3 i (c+i d)^4 \log (i-\tan (e+f x))-3 i (c-i d)^4 \log (i+\tan (e+f x))-6 d^2 \left (6 c^2-d^2\right ) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-2 d^4 \tan ^3(e+f x)\right )\right )}{f}}{5 d}}{6 d} \] Input:
Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x ] + C*Tan[e + f*x]^2),x]
Output:
(C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f) + ((-2*b*(b*c*C - 3*b*B*d - a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^4)/(5*d*f) - (-1/2*(( 5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15*(A - C)*d^ 2))*(c + d*Tan[e + f*x])^4)/(d*f) + (5*(3*d*(2*a*b*(A*c - c*C + B*d) + a^2 *(B*c - (A - C)*d) - b^2*(B*c - (A - C)*d))*((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[ e + f*x]^2) + (a^2*B - b^2*B + 2*a*b*(A - C))*d*((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*(6*c^2 - d^2)*Tan[e + f*x] - 12*c*d^3*Tan[e + f*x]^2 - 2*d^4*Tan[e + f*x]^3)))/f)/( 5*d))/(6*d)
Time = 4.72 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.356, Rules used = {3042, 4130, 27, 3042, 4120, 25, 3042, 4113, 3042, 4011, 3042, 4011, 3042, 4008, 3042, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\int -2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \left ((b c C-a d C-3 b B d) \tan ^2(e+f x)-3 (A b-C b+a B) d \tan (e+f x)+b c C-3 a A d+2 a C d\right )dx}{6 d}+\frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \left ((b c C-a d C-3 b B d) \tan ^2(e+f x)-3 (A b-C b+a B) d \tan (e+f x)+b c C-a (3 A-2 C) d\right )dx}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \left ((b c C-a d C-3 b B d) \tan (e+f x)^2-3 (A b-C b+a B) d \tan (e+f x)+b c C-a (3 A-2 C) d\right )dx}{3 d}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}-\frac {\int -(c+d \tan (e+f x))^3 \left (-c (c C-3 B d) b^2+6 a c C d b-5 a^2 (3 A-2 C) d^2-\left (\left (C c^2-3 B d c+15 (A-C) d^2\right ) b^2-6 a d (c C-5 B d) b+5 a^2 C d^2\right ) \tan ^2(e+f x)-15 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{5 d}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {\int (c+d \tan (e+f x))^3 \left (-c (c C-3 B d) b^2+6 a c C d b-5 a^2 (3 A-2 C) d^2-\left (\left (C c^2-3 B d c+15 (A-C) d^2\right ) b^2-6 a d (c C-5 B d) b+5 a^2 C d^2\right ) \tan ^2(e+f x)-15 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {\int (c+d \tan (e+f x))^3 \left (-c (c C-3 B d) b^2+6 a c C d b-5 a^2 (3 A-2 C) d^2-\left (\left (C c^2-3 B d c+15 (A-C) d^2\right ) b^2-6 a d (c C-5 B d) b+5 a^2 C d^2\right ) \tan (e+f x)^2-15 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {\int (c+d \tan (e+f x))^3 \left (15 \left (-\left ((A-C) a^2\right )+2 b B a+b^2 (A-C)\right ) d^2-15 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {\int (c+d \tan (e+f x))^3 \left (15 \left (-\left ((A-C) a^2\right )+2 b B a+b^2 (A-C)\right ) d^2-15 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {\int (c+d \tan (e+f x))^2 \left (-15 \left ((A c-C c-B d) a^2-2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right ) d^2-15 \left ((B c+(A-C) d) a^2+2 b (A c-C c-B d) a-b^2 (B c+(A-C) d)\right ) \tan (e+f x) d^2\right )dx-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}-\frac {5 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {\int (c+d \tan (e+f x))^2 \left (-15 \left ((A c-C c-B d) a^2-2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right ) d^2-15 \left ((B c+(A-C) d) a^2+2 b (A c-C c-B d) a-b^2 (B c+(A-C) d)\right ) \tan (e+f x) d^2\right )dx-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}-\frac {5 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {\int (c+d \tan (e+f x)) \left (15 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) d^2+15 \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^2\right )dx-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}-\frac {5 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{f}-\frac {15 d^2 (c+d \tan (e+f x))^2 \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{2 f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {\int (c+d \tan (e+f x)) \left (15 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) d^2+15 \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^2\right )dx-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}-\frac {5 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{f}-\frac {15 d^2 (c+d \tan (e+f x))^2 \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{2 f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 4008 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {15 d^2 \left (-\left (a^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+2 a b \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )+b^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \int \tan (e+f x)dx-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}+\frac {15 d^3 \tan (e+f x) \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-15 d^2 x \left (a^2 \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )-2 a b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )-\frac {5 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{f}-\frac {15 d^2 (c+d \tan (e+f x))^2 \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{2 f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {15 d^2 \left (-\left (a^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+2 a b \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )+b^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \int \tan (e+f x)dx-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}+\frac {15 d^3 \tan (e+f x) \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-15 d^2 x \left (a^2 \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )-2 a b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )-\frac {5 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{f}-\frac {15 d^2 (c+d \tan (e+f x))^2 \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{2 f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}-\frac {\frac {-\frac {(c+d \tan (e+f x))^4 \left (5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left (15 d^2 (A-C)-3 B c d+c^2 C\right )\right )}{4 d f}+\frac {15 d^3 \tan (e+f x) \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {15 d^2 \log (\cos (e+f x)) \left (-\left (a^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+2 a b \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )+b^2 \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{f}-15 d^2 x \left (a^2 \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )-2 a b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )-\frac {5 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^3}{f}-\frac {15 d^2 (c+d \tan (e+f x))^2 \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{2 f}}{5 d}+\frac {b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}}{3 d}\) |
Input:
Int[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C* Tan[e + f*x]^2),x]
Output:
(C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f) - ((b*(b*c*C - 3 *b*B*d - a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^4)/(5*d*f) + (-15*d^2*(a ^2*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) + b^2*(c^3* C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) - 2*a*b*((A - C)*d* (3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x - (15*d^2*(2*a*b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) - a^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) + b^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*Lo g[Cos[e + f*x]])/f + (15*d^3*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^ 2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Tan[e + f*x])/f - (15*d^2*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*(c + d*Tan[e + f*x])^2)/(2*f) - (5*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2*(c + d*Tan[e + f*x])^3)/f - ((5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15*(A - C)*d^2))*(c + d*Tan [e + f*x])^4)/(4*d*f))/(5*d))/(3*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(a*c - b*d)*x, x] + (Simp[b*d*(Tan[e + f*x]/f), x] + Simp[(b*c + a*d) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2)) Int[(c + d*Tan[e + f*x])^n*Si mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && !LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Time = 0.41 (sec) , antiderivative size = 546, normalized size of antiderivative = 0.91
| method | result | size |
| parts | \(\frac {\left (3 A \,a^{2} c^{2} d +2 A a b \,c^{3}+B \,a^{2} c^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {\left (B \,b^{2} d^{3}+2 C a b \,d^{3}+3 C \,b^{2} c \,d^{2}\right ) \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (A \,b^{2} d^{3}+2 B a b \,d^{3}+3 B \,b^{2} c \,d^{2}+C \,a^{2} d^{3}+6 C a b c \,d^{2}+3 C \,b^{2} c^{2} d \right ) \left (\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (3 A \,a^{2} c \,d^{2}+6 A a b \,c^{2} d +A \,b^{2} c^{3}+3 B \,a^{2} c^{2} d +2 B a b \,c^{3}+C \,a^{2} c^{3}\right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (2 A a b \,d^{3}+3 A \,b^{2} c \,d^{2}+B \,a^{2} d^{3}+6 B a b c \,d^{2}+3 B \,b^{2} c^{2} d +3 C \,a^{2} c \,d^{2}+6 C a b \,c^{2} d +C \,b^{2} c^{3}\right ) \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (A \,a^{2} d^{3}+6 A a b c \,d^{2}+3 A \,b^{2} c^{2} d +3 B \,a^{2} c \,d^{2}+6 B a b \,c^{2} d +B \,b^{2} c^{3}+3 C \,a^{2} c^{2} d +2 C a b \,c^{3}\right ) \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+A \,a^{2} c^{3} x +\frac {C \,b^{2} d^{3} \left (\frac {\tan \left (f x +e \right )^{6}}{6}-\frac {\tan \left (f x +e \right )^{4}}{4}+\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}\) | \(546\) |
| norman | \(\left (A \,a^{2} c^{3}-3 A \,a^{2} c \,d^{2}-6 A a b \,c^{2} d +2 A a b \,d^{3}-A \,b^{2} c^{3}+3 A \,b^{2} c \,d^{2}-3 B \,a^{2} c^{2} d +B \,a^{2} d^{3}-2 B a b \,c^{3}+6 B a b c \,d^{2}+3 B \,b^{2} c^{2} d -B \,b^{2} d^{3}-C \,a^{2} c^{3}+3 C \,a^{2} c \,d^{2}+6 C a b \,c^{2} d -2 C a b \,d^{3}+C \,b^{2} c^{3}-3 C \,b^{2} c \,d^{2}\right ) x +\frac {\left (3 A \,a^{2} c \,d^{2}+6 A a b \,c^{2} d -2 A a b \,d^{3}+A \,b^{2} c^{3}-3 A \,b^{2} c \,d^{2}+3 B \,a^{2} c^{2} d -B \,a^{2} d^{3}+2 B a b \,c^{3}-6 B a b c \,d^{2}-3 B \,b^{2} c^{2} d +B \,b^{2} d^{3}+C \,a^{2} c^{3}-3 C \,a^{2} c \,d^{2}-6 C a b \,c^{2} d +2 C a b \,d^{3}-C \,b^{2} c^{3}+3 C \,b^{2} c \,d^{2}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (2 A a b \,d^{3}+3 A \,b^{2} c \,d^{2}+B \,a^{2} d^{3}+6 B a b c \,d^{2}+3 B \,b^{2} c^{2} d -B \,b^{2} d^{3}+3 C \,a^{2} c \,d^{2}+6 C a b \,c^{2} d -2 C a b \,d^{3}+C \,b^{2} c^{3}-3 C \,b^{2} c \,d^{2}\right ) \tan \left (f x +e \right )^{3}}{3 f}+\frac {\left (A \,a^{2} d^{3}+6 A a b c \,d^{2}+3 A \,b^{2} c^{2} d -A \,b^{2} d^{3}+3 B \,a^{2} c \,d^{2}+6 B a b \,c^{2} d -2 B a b \,d^{3}+B \,b^{2} c^{3}-3 B \,b^{2} c \,d^{2}+3 C \,a^{2} c^{2} d -C \,a^{2} d^{3}+2 C a b \,c^{3}-6 C a b c \,d^{2}-3 C \,b^{2} c^{2} d +C \,b^{2} d^{3}\right ) \tan \left (f x +e \right )^{2}}{2 f}+\frac {d \left (A \,b^{2} d^{2}+2 B a b \,d^{2}+3 B \,b^{2} c d +a^{2} C \,d^{2}+6 C a b c d +3 C \,b^{2} c^{2}-b^{2} d^{2} C \right ) \tan \left (f x +e \right )^{4}}{4 f}+\frac {C \,b^{2} d^{3} \tan \left (f x +e \right )^{6}}{6 f}+\frac {b \,d^{2} \left (B b d +2 C a d +3 C b c \right ) \tan \left (f x +e \right )^{5}}{5 f}+\frac {\left (3 A \,a^{2} c^{2} d -A \,a^{2} d^{3}+2 A a b \,c^{3}-6 A a b c \,d^{2}-3 A \,b^{2} c^{2} d +A \,b^{2} d^{3}+B \,a^{2} c^{3}-3 B \,a^{2} c \,d^{2}-6 B a b \,c^{2} d +2 B a b \,d^{3}-B \,b^{2} c^{3}+3 B \,b^{2} c \,d^{2}-3 C \,a^{2} c^{2} d +C \,a^{2} d^{3}-2 C a b \,c^{3}+6 C a b c \,d^{2}+3 C \,b^{2} c^{2} d -C \,b^{2} d^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}\) | \(896\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1239\) |
| default | \(\text {Expression too large to display}\) | \(1239\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1464\) |
| risch | \(\text {Expression too large to display}\) | \(4231\) |
Input:
int((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2), x,method=_RETURNVERBOSE)
Output:
1/2*(3*A*a^2*c^2*d+2*A*a*b*c^3+B*a^2*c^3)/f*ln(1+tan(f*x+e)^2)+(B*b^2*d^3+ 2*C*a*b*d^3+3*C*b^2*c*d^2)/f*(1/5*tan(f*x+e)^5-1/3*tan(f*x+e)^3+tan(f*x+e) -arctan(tan(f*x+e)))+(A*b^2*d^3+2*B*a*b*d^3+3*B*b^2*c*d^2+C*a^2*d^3+6*C*a* b*c*d^2+3*C*b^2*c^2*d)/f*(1/4*tan(f*x+e)^4-1/2*tan(f*x+e)^2+1/2*ln(1+tan(f *x+e)^2))+(3*A*a^2*c*d^2+6*A*a*b*c^2*d+A*b^2*c^3+3*B*a^2*c^2*d+2*B*a*b*c^3 +C*a^2*c^3)/f*(tan(f*x+e)-arctan(tan(f*x+e)))+(2*A*a*b*d^3+3*A*b^2*c*d^2+B *a^2*d^3+6*B*a*b*c*d^2+3*B*b^2*c^2*d+3*C*a^2*c*d^2+6*C*a*b*c^2*d+C*b^2*c^3 )/f*(1/3*tan(f*x+e)^3-tan(f*x+e)+arctan(tan(f*x+e)))+(A*a^2*d^3+6*A*a*b*c* d^2+3*A*b^2*c^2*d+3*B*a^2*c*d^2+6*B*a*b*c^2*d+B*b^2*c^3+3*C*a^2*c^2*d+2*C* a*b*c^3)/f*(1/2*tan(f*x+e)^2-1/2*ln(1+tan(f*x+e)^2))+A*a^2*c^3*x+C*b^2*d^3 /f*(1/6*tan(f*x+e)^6-1/4*tan(f*x+e)^4+1/2*tan(f*x+e)^2-1/2*ln(1+tan(f*x+e) ^2))
Time = 0.11 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.13 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx =\text {Too large to display} \] Input:
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+ e)^2),x, algorithm="fricas")
Output:
1/60*(10*C*b^2*d^3*tan(f*x + e)^6 + 12*(3*C*b^2*c*d^2 + (2*C*a*b + B*b^2)* d^3)*tan(f*x + e)^5 + 15*(3*C*b^2*c^2*d + 3*(2*C*a*b + B*b^2)*c*d^2 + (C*a ^2 + 2*B*a*b + (A - C)*b^2)*d^3)*tan(f*x + e)^4 + 20*(C*b^2*c^3 + 3*(2*C*a *b + B*b^2)*c^2*d + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c*d^2 + (B*a^2 + 2*( A - C)*a*b - B*b^2)*d^3)*tan(f*x + e)^3 + 60*(((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c^3 - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d - 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*f*x + 30*((2*C*a*b + B*b^2)*c^3 + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c^2*d + 3*( B*a^2 + 2*(A - C)*a*b - B*b^2)*c*d^2 + ((A - C)*a^2 - 2*B*a*b - (A - C)*b^ 2)*d^3)*tan(f*x + e)^2 - 30*((B*a^2 + 2*(A - C)*a*b - B*b^2)*c^3 + 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c^2*d - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2) *c*d^2 - ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*log(1/(tan(f*x + e)^2 + 1)) + 60*((C*a^2 + 2*B*a*b + (A - C)*b^2)*c^3 + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c^2*d + 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 - (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 1819 vs. \(2 (547) = 1094\).
Time = 0.41 (sec) , antiderivative size = 1819, normalized size of antiderivative = 3.02 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(f*x+e))**2*(c+d*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f* x+e)**2),x)
Output:
Piecewise((A*a**2*c**3*x + 3*A*a**2*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) - 3*A*a**2*c*d**2*x + 3*A*a**2*c*d**2*tan(e + f*x)/f - A*a**2*d**3*log(tan (e + f*x)**2 + 1)/(2*f) + A*a**2*d**3*tan(e + f*x)**2/(2*f) + A*a*b*c**3*l og(tan(e + f*x)**2 + 1)/f - 6*A*a*b*c**2*d*x + 6*A*a*b*c**2*d*tan(e + f*x) /f - 3*A*a*b*c*d**2*log(tan(e + f*x)**2 + 1)/f + 3*A*a*b*c*d**2*tan(e + f* x)**2/f + 2*A*a*b*d**3*x + 2*A*a*b*d**3*tan(e + f*x)**3/(3*f) - 2*A*a*b*d* *3*tan(e + f*x)/f - A*b**2*c**3*x + A*b**2*c**3*tan(e + f*x)/f - 3*A*b**2* c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) + 3*A*b**2*c**2*d*tan(e + f*x)**2/(2 *f) + 3*A*b**2*c*d**2*x + A*b**2*c*d**2*tan(e + f*x)**3/f - 3*A*b**2*c*d** 2*tan(e + f*x)/f + A*b**2*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + A*b**2*d** 3*tan(e + f*x)**4/(4*f) - A*b**2*d**3*tan(e + f*x)**2/(2*f) + B*a**2*c**3* log(tan(e + f*x)**2 + 1)/(2*f) - 3*B*a**2*c**2*d*x + 3*B*a**2*c**2*d*tan(e + f*x)/f - 3*B*a**2*c*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*a**2*c*d* *2*tan(e + f*x)**2/(2*f) + B*a**2*d**3*x + B*a**2*d**3*tan(e + f*x)**3/(3* f) - B*a**2*d**3*tan(e + f*x)/f - 2*B*a*b*c**3*x + 2*B*a*b*c**3*tan(e + f* x)/f - 3*B*a*b*c**2*d*log(tan(e + f*x)**2 + 1)/f + 3*B*a*b*c**2*d*tan(e + f*x)**2/f + 6*B*a*b*c*d**2*x + 2*B*a*b*c*d**2*tan(e + f*x)**3/f - 6*B*a*b* c*d**2*tan(e + f*x)/f + B*a*b*d**3*log(tan(e + f*x)**2 + 1)/f + B*a*b*d**3 *tan(e + f*x)**4/(2*f) - B*a*b*d**3*tan(e + f*x)**2/f - B*b**2*c**3*log(ta n(e + f*x)**2 + 1)/(2*f) + B*b**2*c**3*tan(e + f*x)**2/(2*f) + 3*B*b**2...
Time = 0.12 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.13 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx =\text {Too large to display} \] Input:
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+ e)^2),x, algorithm="maxima")
Output:
1/60*(10*C*b^2*d^3*tan(f*x + e)^6 + 12*(3*C*b^2*c*d^2 + (2*C*a*b + B*b^2)* d^3)*tan(f*x + e)^5 + 15*(3*C*b^2*c^2*d + 3*(2*C*a*b + B*b^2)*c*d^2 + (C*a ^2 + 2*B*a*b + (A - C)*b^2)*d^3)*tan(f*x + e)^4 + 20*(C*b^2*c^3 + 3*(2*C*a *b + B*b^2)*c^2*d + 3*(C*a^2 + 2*B*a*b + (A - C)*b^2)*c*d^2 + (B*a^2 + 2*( A - C)*a*b - B*b^2)*d^3)*tan(f*x + e)^3 + 30*((2*C*a*b + B*b^2)*c^3 + 3*(C *a^2 + 2*B*a*b + (A - C)*b^2)*c^2*d + 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c* d^2 + ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*tan(f*x + e)^2 + 60*(((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c^3 - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)* c^2*d - 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 + (B*a^2 + 2*(A - C) *a*b - B*b^2)*d^3)*(f*x + e) + 30*((B*a^2 + 2*(A - C)*a*b - B*b^2)*c^3 + 3 *((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c^2*d - 3*(B*a^2 + 2*(A - C)*a*b - B*b^2)*c*d^2 - ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d^3)*log(tan(f*x + e) ^2 + 1) + 60*((C*a^2 + 2*B*a*b + (A - C)*b^2)*c^3 + 3*(B*a^2 + 2*(A - C)*a *b - B*b^2)*c^2*d + 3*((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c*d^2 - (B*a^2 + 2*(A - C)*a*b - B*b^2)*d^3)*tan(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 1413 vs. \(2 (593) = 1186\).
Time = 1.18 (sec) , antiderivative size = 1413, normalized size of antiderivative = 2.34 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+ e)^2),x, algorithm="giac")
Output:
(A*a^2*c^3 - C*a^2*c^3 - 2*B*a*b*c^3 - A*b^2*c^3 + C*b^2*c^3 - 3*B*a^2*c^2 *d - 6*A*a*b*c^2*d + 6*C*a*b*c^2*d + 3*B*b^2*c^2*d - 3*A*a^2*c*d^2 + 3*C*a ^2*c*d^2 + 6*B*a*b*c*d^2 + 3*A*b^2*c*d^2 - 3*C*b^2*c*d^2 + B*a^2*d^3 + 2*A *a*b*d^3 - 2*C*a*b*d^3 - B*b^2*d^3)*(f*x + e)/f + 1/2*(B*a^2*c^3 + 2*A*a*b *c^3 - 2*C*a*b*c^3 - B*b^2*c^3 + 3*A*a^2*c^2*d - 3*C*a^2*c^2*d - 6*B*a*b*c ^2*d - 3*A*b^2*c^2*d + 3*C*b^2*c^2*d - 3*B*a^2*c*d^2 - 6*A*a*b*c*d^2 + 6*C *a*b*c*d^2 + 3*B*b^2*c*d^2 - A*a^2*d^3 + C*a^2*d^3 + 2*B*a*b*d^3 + A*b^2*d ^3 - C*b^2*d^3)*log(tan(f*x + e)^2 + 1)/f + 1/60*(10*C*b^2*d^3*f^5*tan(f*x + e)^6 + 36*C*b^2*c*d^2*f^5*tan(f*x + e)^5 + 24*C*a*b*d^3*f^5*tan(f*x + e )^5 + 12*B*b^2*d^3*f^5*tan(f*x + e)^5 + 45*C*b^2*c^2*d*f^5*tan(f*x + e)^4 + 90*C*a*b*c*d^2*f^5*tan(f*x + e)^4 + 45*B*b^2*c*d^2*f^5*tan(f*x + e)^4 + 15*C*a^2*d^3*f^5*tan(f*x + e)^4 + 30*B*a*b*d^3*f^5*tan(f*x + e)^4 + 15*A*b ^2*d^3*f^5*tan(f*x + e)^4 - 15*C*b^2*d^3*f^5*tan(f*x + e)^4 + 20*C*b^2*c^3 *f^5*tan(f*x + e)^3 + 120*C*a*b*c^2*d*f^5*tan(f*x + e)^3 + 60*B*b^2*c^2*d* f^5*tan(f*x + e)^3 + 60*C*a^2*c*d^2*f^5*tan(f*x + e)^3 + 120*B*a*b*c*d^2*f ^5*tan(f*x + e)^3 + 60*A*b^2*c*d^2*f^5*tan(f*x + e)^3 - 60*C*b^2*c*d^2*f^5 *tan(f*x + e)^3 + 20*B*a^2*d^3*f^5*tan(f*x + e)^3 + 40*A*a*b*d^3*f^5*tan(f *x + e)^3 - 40*C*a*b*d^3*f^5*tan(f*x + e)^3 - 20*B*b^2*d^3*f^5*tan(f*x + e )^3 + 60*C*a*b*c^3*f^5*tan(f*x + e)^2 + 30*B*b^2*c^3*f^5*tan(f*x + e)^2 + 90*C*a^2*c^2*d*f^5*tan(f*x + e)^2 + 180*B*a*b*c^2*d*f^5*tan(f*x + e)^2 ...
Time = 5.99 (sec) , antiderivative size = 891, normalized size of antiderivative = 1.48 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx =\text {Too large to display} \] Input:
int((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^3*(A + B*tan(e + f*x) + C* tan(e + f*x)^2),x)
Output:
x*(A*a^2*c^3 - A*b^2*c^3 + B*a^2*d^3 - C*a^2*c^3 - B*b^2*d^3 + C*b^2*c^3 + 2*A*a*b*d^3 - 2*B*a*b*c^3 - 2*C*a*b*d^3 - 3*A*a^2*c*d^2 + 3*A*b^2*c*d^2 - 3*B*a^2*c^2*d + 3*B*b^2*c^2*d + 3*C*a^2*c*d^2 - 3*C*b^2*c*d^2 - 6*A*a*b*c ^2*d + 6*B*a*b*c*d^2 + 6*C*a*b*c^2*d) - (tan(e + f*x)*(B*a^2*d^3 - A*b^2*c ^3 - b*d^2*(B*b*d + 2*C*a*d + 3*C*b*c) - C*a^2*c^3 + C*b^2*c^3 + 2*A*a*b*d ^3 - 2*B*a*b*c^3 - 3*A*a^2*c*d^2 + 3*A*b^2*c*d^2 - 3*B*a^2*c^2*d + 3*B*b^2 *c^2*d + 3*C*a^2*c*d^2 - 6*A*a*b*c^2*d + 6*B*a*b*c*d^2 + 6*C*a*b*c^2*d))/f - (log(tan(e + f*x)^2 + 1)*((A*a^2*d^3)/2 - (B*a^2*c^3)/2 - (A*b^2*d^3)/2 + (B*b^2*c^3)/2 - (C*a^2*d^3)/2 + (C*b^2*d^3)/2 - A*a*b*c^3 - B*a*b*d^3 + C*a*b*c^3 - (3*A*a^2*c^2*d)/2 + (3*A*b^2*c^2*d)/2 + (3*B*a^2*c*d^2)/2 - ( 3*B*b^2*c*d^2)/2 + (3*C*a^2*c^2*d)/2 - (3*C*b^2*c^2*d)/2 + 3*A*a*b*c*d^2 + 3*B*a*b*c^2*d - 3*C*a*b*c*d^2))/f + (tan(e + f*x)^4*((A*b^2*d^3)/4 + (C*a ^2*d^3)/4 - (C*b^2*d^3)/4 + (B*a*b*d^3)/2 + (3*B*b^2*c*d^2)/4 + (3*C*b^2*c ^2*d)/4 + (3*C*a*b*c*d^2)/2))/f + (tan(e + f*x)^3*((B*a^2*d^3)/3 - (b*d^2* (B*b*d + 2*C*a*d + 3*C*b*c))/3 + (C*b^2*c^3)/3 + (2*A*a*b*d^3)/3 + A*b^2*c *d^2 + B*b^2*c^2*d + C*a^2*c*d^2 + 2*B*a*b*c*d^2 + 2*C*a*b*c^2*d))/f + (ta n(e + f*x)^2*((A*a^2*d^3)/2 - (A*b^2*d^3)/2 + (B*b^2*c^3)/2 - (C*a^2*d^3)/ 2 + (C*b^2*d^3)/2 - B*a*b*d^3 + C*a*b*c^3 + (3*A*b^2*c^2*d)/2 + (3*B*a^2*c *d^2)/2 - (3*B*b^2*c*d^2)/2 + (3*C*a^2*c^2*d)/2 - (3*C*b^2*c^2*d)/2 + 3*A* a*b*c*d^2 + 3*B*a*b*c^2*d - 3*C*a*b*c*d^2))/f + (b*d^2*tan(e + f*x)^5*(...
Time = 0.16 (sec) , antiderivative size = 1163, normalized size of antiderivative = 1.93 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx =\text {Too large to display} \] Input:
int((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2), x)
Output:
(90*log(tan(e + f*x)**2 + 1)*a**3*c**2*d - 30*log(tan(e + f*x)**2 + 1)*a** 3*d**3 + 90*log(tan(e + f*x)**2 + 1)*a**2*b*c**3 - 270*log(tan(e + f*x)**2 + 1)*a**2*b*c*d**2 - 90*log(tan(e + f*x)**2 + 1)*a**2*c**3*d + 30*log(tan (e + f*x)**2 + 1)*a**2*c*d**3 - 270*log(tan(e + f*x)**2 + 1)*a*b**2*c**2*d + 90*log(tan(e + f*x)**2 + 1)*a*b**2*d**3 - 60*log(tan(e + f*x)**2 + 1)*a *b*c**4 + 180*log(tan(e + f*x)**2 + 1)*a*b*c**2*d**2 - 30*log(tan(e + f*x) **2 + 1)*b**3*c**3 + 90*log(tan(e + f*x)**2 + 1)*b**3*c*d**2 + 90*log(tan( e + f*x)**2 + 1)*b**2*c**3*d - 30*log(tan(e + f*x)**2 + 1)*b**2*c*d**3 + 1 0*tan(e + f*x)**6*b**2*c*d**3 + 24*tan(e + f*x)**5*a*b*c*d**3 + 12*tan(e + f*x)**5*b**3*d**3 + 36*tan(e + f*x)**5*b**2*c**2*d**2 + 15*tan(e + f*x)** 4*a**2*c*d**3 + 45*tan(e + f*x)**4*a*b**2*d**3 + 90*tan(e + f*x)**4*a*b*c* *2*d**2 + 45*tan(e + f*x)**4*b**3*c*d**2 + 45*tan(e + f*x)**4*b**2*c**3*d - 15*tan(e + f*x)**4*b**2*c*d**3 + 60*tan(e + f*x)**3*a**2*b*d**3 + 60*tan (e + f*x)**3*a**2*c**2*d**2 + 180*tan(e + f*x)**3*a*b**2*c*d**2 + 120*tan( e + f*x)**3*a*b*c**3*d - 40*tan(e + f*x)**3*a*b*c*d**3 + 60*tan(e + f*x)** 3*b**3*c**2*d - 20*tan(e + f*x)**3*b**3*d**3 + 20*tan(e + f*x)**3*b**2*c** 4 - 60*tan(e + f*x)**3*b**2*c**2*d**2 + 30*tan(e + f*x)**2*a**3*d**3 + 270 *tan(e + f*x)**2*a**2*b*c*d**2 + 90*tan(e + f*x)**2*a**2*c**3*d - 30*tan(e + f*x)**2*a**2*c*d**3 + 270*tan(e + f*x)**2*a*b**2*c**2*d - 90*tan(e + f* x)**2*a*b**2*d**3 + 60*tan(e + f*x)**2*a*b*c**4 - 180*tan(e + f*x)**2*a...