Integrand size = 47, antiderivative size = 464 \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {(a-i b)^3 (i A+B-i C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^3 (i A-B-i C) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (8 c^2 C-14 B c d+35 (A-C) d^2\right )-b^3 \left (16 c^3 C-24 B c^2 d+42 c (A-C) d^2+105 B d^3\right )\right ) (c+d \tan (e+f x))^{3/2}}{315 d^4 f}+\frac {2 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{105 d^3 f}-\frac {2 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f} \]
-(a-I*b)^3*(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))*(c-I* d)^(1/2)/f+(a+I*b)^3*(I*A-B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1 /2))*(c+I*d)^(1/2)/f+2*(B*a^3-3*B*a*b^2+3*a^2*b*(A-C)-b^3*(A-C))*(c+d*tan( f*x+e))^(1/2)/f+2/315*(40*a^3*C*d^3-6*a^2*b*d^2*(-45*B*d+16*C*c)+9*a*b^2*d *(8*c^2*C-14*B*c*d+35*(A-C)*d^2)-b^3*(16*c^3*C-24*B*c^2*d+42*c*(A-C)*d^2+1 05*B*d^3))*(c+d*tan(f*x+e))^(3/2)/d^4/f+2/105*b*(21*b*(A*b+B*a-C*b)*d^2+4* (-a*d+b*c)*(-3*B*b*d-2*C*a*d+2*C*b*c))*tan(f*x+e)*(c+d*tan(f*x+e))^(3/2)/d ^3/f-2/21*(-3*B*b*d-2*C*a*d+2*C*b*c)*(a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^( 3/2)/d^2/f+2/9*C*(a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(3/2)/d/f
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1232\) vs. \(2(464)=928\).
Time = 6.55 (sec) , antiderivative size = 1232, normalized size of antiderivative = 2.66 \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}+\frac {2 \left (-\frac {3 (2 b c C-3 b B d-2 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}+\frac {2 \left (\frac {3 b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{10 d f}-\frac {2 \left (\frac {2 \left (-\frac {15}{8} a d \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )+b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )\right ) (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {i \left (-\frac {15}{8} a d \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )+\frac {3}{4} b c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )+\frac {15}{8} a d \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )+\frac {5}{2} i d \left (\frac {63}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {3}{4} b \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )-\frac {3}{4} b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )-b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )\right ) \left (\frac {2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{-c+i d}+2 \sqrt {c+d \tan (e+f x)}\right )}{2 f}-\frac {i \left (-\frac {15}{8} a d \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )+\frac {3}{4} b c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )+\frac {15}{8} a d \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )-\frac {5}{2} i d \left (\frac {63}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {3}{4} b \left (a^2 (21 A-13 C) d^2+4 b^2 c (2 c C-3 B d)-a b d (16 c C+9 B d)\right )-\frac {3}{4} b \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )-b \left (-\frac {315}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {3}{4} c \left (21 b (A b+a B-b C) d^2+4 (b c-a d) (2 b c C-3 b B d-2 a C d)\right )\right )\right ) \left (\frac {2 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{-c-i d}+2 \sqrt {c+d \tan (e+f x)}\right )}{2 f}\right )}{5 d}\right )}{7 d}\right )}{9 d} \]
Integrate[(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f *x] + C*Tan[e + f*x]^2),x]
(2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2))/(9*d*f) + (2*((-3* (2*b*c*C - 3*b*B*d - 2*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^ (3/2))/(7*d*f) + (2*((3*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b *c*C - 3*b*B*d - 2*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(10*d* f) - (2*((2*((-15*a*d*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/8 + b*((-315*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/ 8 + (3*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4))*(c + d*Tan[e + f*x])^(3/2))/(3*d*f) + ((I/2)*((-15*a*d*(a^2 *(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/8 + (3*b*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4 + (15*a*d*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c* C - 3*b*B*d - 2*a*C*d)))/8 + ((5*I)/2)*d*((63*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 + (3*b*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b *d*(16*c*C + 9*B*d)))/4 - (3*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d) *(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4) - b*((-315*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (3*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4))*((2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f* x]]/Sqrt[c - I*d]])/(-c + I*d) + 2*Sqrt[c + d*Tan[e + f*x]]))/f - ((I/2)*( (-15*a*d*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/8 + (3*b*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*...
Time = 3.44 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.03, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.426, Rules used = {3042, 4130, 27, 3042, 4130, 27, 3042, 4120, 27, 3042, 4113, 3042, 4011, 3042, 4022, 3042, 4020, 25, 73, 221}
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.
| \(\displaystyle \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 \int -\frac {3}{2} (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)} \left ((2 b c C-2 a d C-3 b B d) \tan ^2(e+f x)-3 (A b-C b+a B) d \tan (e+f x)+2 b c C-a (3 A-C) d\right )dx}{9 d}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\int (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)} \left ((2 b c C-2 a d C-3 b B d) \tan ^2(e+f x)-3 (A b-C b+a B) d \tan (e+f x)+2 b c C-a (3 A-C) d\right )dx}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\int (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)} \left ((2 b c C-2 a d C-3 b B d) \tan (e+f x)^2-3 (A b-C b+a B) d \tan (e+f x)+2 b c C-a (3 A-C) d\right )dx}{3 d}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 \int -\frac {1}{2} (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)} \left (4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2+\left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right ) \tan ^2(e+f x)+21 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{7 d}+\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\int (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)} \left (4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2+\left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right ) \tan ^2(e+f x)+21 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\int (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)} \left (4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2+\left (21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right ) \tan (e+f x)^2+21 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}-\frac {2 \int -\frac {1}{2} \sqrt {c+d \tan (e+f x)} \left (-2 c \left (8 C c^2-12 B d c+21 (A-C) d^2\right ) b^3+18 a c d (4 c C-7 B d) b^2-3 a^2 d^2 (32 c C+15 B d) b+5 a^3 (21 A-13 C) d^3+\left (-\left (\left (16 C c^3-24 B d c^2+42 (A-C) d^2 c+105 B d^3\right ) b^3\right )+9 a d \left (8 C c^2-14 B d c+35 (A-C) d^2\right ) b^2-6 a^2 d^2 (16 c C-45 B d) b+40 a^3 C d^3\right ) \tan ^2(e+f x)+105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )dx}{5 d}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (-2 c \left (8 C c^2-12 B d c+21 (A-C) d^2\right ) b^3+18 a c d (4 c C-7 B d) b^2-3 a^2 d^2 (32 c C+15 B d) b+5 a^3 (21 A-13 C) d^3+\left (-\left (\left (16 C c^3-24 B d c^2+42 (A-C) d^2 c+105 B d^3\right ) b^3\right )+9 a d \left (8 C c^2-14 B d c+35 (A-C) d^2\right ) b^2-6 a^2 d^2 (16 c C-45 B d) b+40 a^3 C d^3\right ) \tan ^2(e+f x)+105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )dx}{5 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (-2 c \left (8 C c^2-12 B d c+21 (A-C) d^2\right ) b^3+18 a c d (4 c C-7 B d) b^2-3 a^2 d^2 (32 c C+15 B d) b+5 a^3 (21 A-13 C) d^3+\left (-\left (\left (16 C c^3-24 B d c^2+42 (A-C) d^2 c+105 B d^3\right ) b^3\right )+9 a d \left (8 C c^2-14 B d c+35 (A-C) d^2\right ) b^2-6 a^2 d^2 (16 c C-45 B d) b+40 a^3 C d^3\right ) \tan (e+f x)^2+105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )dx}{5 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)-105 \left (-\left ((A-C) a^3\right )+3 b B a^2+3 b^2 (A-C) a-b^3 B\right ) d^3\right )dx+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}}{5 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)-105 \left (-\left ((A-C) a^3\right )+3 b B a^2+3 b^2 (A-C) a-b^3 B\right ) d^3\right )dx+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}}{5 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {\int \frac {105 \left ((A c-C c-B d) a^3-3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a+b^3 (B c+(A-C) d)\right ) d^3+105 \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}+\frac {210 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}}{5 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {\int \frac {105 \left ((A c-C c-B d) a^3-3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a+b^3 (B c+(A-C) d)\right ) d^3+105 \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}+\frac {210 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}}{5 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}+\frac {\frac {105}{2} d^3 (a+i b)^3 (c+i d) (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {105}{2} d^3 (a-i b)^3 (c-i d) (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}+\frac {210 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}}{5 d}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}+\frac {\frac {105}{2} d^3 (a+i b)^3 (c+i d) (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {105}{2} d^3 (a-i b)^3 (c-i d) (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}+\frac {210 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}}{5 d}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}+\frac {\frac {105 i d^3 (a-i b)^3 (c-i d) (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {105 i d^3 (a+i b)^3 (c+i d) (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}+\frac {210 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}}{5 d}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}+\frac {-\frac {105 i d^3 (a-i b)^3 (c-i d) (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {105 i d^3 (a+i b)^3 (c+i d) (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}+\frac {210 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}}{5 d}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}+\frac {\frac {105 d^2 (a-i b)^3 (c-i d) (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {105 d^2 (a+i b)^3 (c+i d) (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}+\frac {210 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}}{5 d}}{7 d}}{3 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}-\frac {\frac {2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left (21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right )}{5 d f}+\frac {\frac {2 (c+d \tan (e+f x))^{3/2} \left (40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left (35 d^2 (A-C)-14 B c d+8 c^2 C\right )-\left (b^3 \left (42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right )\right )\right )}{3 d f}+\frac {210 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {105 d^3 (a-i b)^3 \sqrt {c-i d} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}+\frac {105 d^3 (a+i b)^3 \sqrt {c+i d} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}}{5 d}}{7 d}}{3 d}\) |
Int[(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]
(2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2))/(9*d*f) - ((2*(2*b *c*C - 3*b*B*d - 2*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2 ))/(7*d*f) - ((2*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(5*d*f) + ((1 05*(a - I*b)^3*(A - I*B - C)*Sqrt[c - I*d]*d^3*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/f + (105*(a + I*b)^3*(A + I*B - C)*Sqrt[c + I*d]*d^3*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/f + (210*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^ 3*(A - C))*d^3*Sqrt[c + d*Tan[e + f*x]])/f + (2*(40*a^3*C*d^3 - 6*a^2*b*d^ 2*(16*c*C - 45*B*d) + 9*a*b^2*d*(8*c^2*C - 14*B*c*d + 35*(A - C)*d^2) - b^ 3*(16*c^3*C - 24*B*c^2*d + 42*c*(A - C)*d^2 + 105*B*d^3))*(c + d*Tan[e + f *x])^(3/2))/(3*d*f))/(5*d))/(7*d))/(3*d)
3.1.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
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])))
Leaf count of result is larger than twice the leaf count of optimal. \(4472\) vs. \(2(424)=848\).
Time = 0.53 (sec) , antiderivative size = 4473, normalized size of antiderivative = 9.64
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(4473\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(6661\) |
| default | \(\text {Expression too large to display}\) | \(6661\) |
int((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e) ^2),x,method=_RETURNVERBOSE)
1/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c ^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*A*a^3+1/f*d/(2*(c ^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/ 2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*B*b^3-1/f*d/(2*(c^2+d^2)^(1/ 2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/ 2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*A*a^3-1/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/ 2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2 +d^2)^(1/2)-2*c)^(1/2))*B*b^3+1/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(( (2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2) -2*c)^(1/2))*C*a^3-1/f*d/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan( f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2) )*C*a^3+2/f/d*A*a*b^2*(c+d*tan(f*x+e))^(3/2)+2/f/d*B*a^2*b*(c+d*tan(f*x+e) )^(3/2)-12/5/f/d^3*C*a*b^2*c*(c+d*tan(f*x+e))^(5/2)-1/4/f/d*ln(d*tan(f*x+e )+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))* A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3+1/4/f/d*ln(d*tan(f*x+e )+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))* A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c-1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f *x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*B*(2*(c^2+d^2) ^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^3+1/4/f/d*ln(d*tan(f*x+e)+c+(c+d*tan(f *x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*B*(2*(c^2+d...
Leaf count of result is larger than twice the leaf count of optimal. 35153 vs. \(2 (414) = 828\).
Time = 10.87 (sec) , antiderivative size = 35153, normalized size of antiderivative = 75.76 \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="fricas")
\[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
Integral((a + b*tan(e + f*x))**3*sqrt(c + d*tan(e + f*x))*(A + B*tan(e + f *x) + C*tan(e + f*x)**2), x)
Timed out. \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="maxima")
Timed out. \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="giac")
Timed out. \[ \int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Hanged} \]