Integrand size = 47, antiderivative size = 550 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {(i a+b)^3 (A-i B-C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^3 (i A-B-i C) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )-b^3 \left (48 c^3 C-88 B c^2 d+198 c (A-C) d^2+693 B d^3\right )\right ) (c+d \tan (e+f x))^{5/2}}{3465 d^4 f}+\frac {2 b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{693 d^3 f}-\frac {2 (6 b c C-11 b B d-6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f} \]
(I*a+b)^3*(A-I*B-C)*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^( 1/2))/f+(a+I*b)^3*(I*A-B-I*C)*(c+I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2) /(c+I*d)^(1/2))/f+2*(3*a^2*b*(A*c-B*d-C*c)-b^3*(A*c-B*d-C*c)+a^3*(B*c+(A-C )*d)-3*a*b^2*(B*c+(A-C)*d))*(c+d*tan(f*x+e))^(1/2)/f+2/3*(B*a^3-3*B*a*b^2+ 3*a^2*b*(A-C)-b^3*(A-C))*(c+d*tan(f*x+e))^(3/2)/f+2/3465*(168*a^3*C*d^3-2* a^2*b*d^2*(-847*B*d+192*C*c)+33*a*b^2*d*(8*c^2*C-18*B*c*d+63*(A-C)*d^2)-b^ 3*(48*c^3*C-88*B*c^2*d+198*c*(A-C)*d^2+693*B*d^3))*(c+d*tan(f*x+e))^(5/2)/ d^4/f+2/693*b*(99*b*(A*b+B*a-C*b)*d^2+4*(-a*d+b*c)*(-11*B*b*d-6*C*a*d+6*C* b*c))*tan(f*x+e)*(c+d*tan(f*x+e))^(5/2)/d^3/f-2/99*(-11*B*b*d-6*C*a*d+6*C* b*c)*(a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)/d^2/f+2/11*C*(a+b*tan(f*x+e ))^3*(c+d*tan(f*x+e))^(5/2)/d/f
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1290\) vs. \(2(550)=1100\).
Time = 6.57 (sec) , antiderivative size = 1290, normalized size of antiderivative = 2.35 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \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))^{5/2}}{11 d f}+\frac {2 \left (\frac {(-6 b c C+11 b B d+6 a C d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}+\frac {2 \left (\frac {b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right ) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{14 d f}-\frac {2 \left (\frac {2 \left (-\frac {7}{8} a d \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )+b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )\right ) (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {i \left (-\frac {7}{8} a d \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )+\frac {1}{4} b c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )+\frac {7}{8} a d \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )+\frac {7}{2} i d \left (\frac {99}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {1}{4} b \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )-\frac {1}{4} b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )-b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )\right ) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \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 )\right )}{2 f}-\frac {i \left (-\frac {7}{8} a d \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )+\frac {1}{4} b c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )+\frac {7}{8} a d \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )-\frac {7}{2} i d \left (\frac {99}{4} a \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {1}{4} b \left (3 a^2 (33 A-25 C) d^2+4 b^2 c (6 c C-11 B d)-a b d (48 c C+55 B d)\right )-\frac {1}{4} b \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )-b \left (-\frac {693}{8} \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+\frac {1}{4} c \left (99 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-11 b B d-6 a C d)\right )\right )\right ) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \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 )\right )}{2 f}\right )}{7 d}\right )}{9 d}\right )}{11 d} \]
Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)*(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])^(5/2))/(11*d*f) + (2*(((- 6*b*c*C + 11*b*B*d + 6*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^ (5/2))/(9*d*f) + (2*((b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c *C - 11*b*B*d - 6*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(14*d*f ) - (2*((2*((-7*a*d*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/8 + b*((-693*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6* a*C*d)))/4))*(c + d*Tan[e + f*x])^(5/2))/(5*d*f) + ((I/2)*((-7*a*d*(3*a^2* (33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a*b*d*(48*c*C + 55*B*d)))/8 + (b*c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4 + (7*a*d*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/8 + ((7*I)/2)*d*((99*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 + (b*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a* b*d*(48*c*C + 55*B*d)))/4 - (b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d) *(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4) - b*((-693*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 1 1*b*B*d - 6*a*C*d)))/4))*((2*(c + d*Tan[e + f*x])^(3/2))/3 + (c - I*d)*((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)*((-7*a*d*(3*a^2*(33*A - 25*C )*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a*b*d*(48*c*C + 55*B*d)))/8 + (b*c*(...
Time = 4.50 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.03, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.468, Rules used = {3042, 4130, 27, 3042, 4130, 27, 3042, 4120, 27, 3042, 4113, 3042, 4011, 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.
| \(\displaystyle \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 \int -\frac {1}{2} (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left ((6 b c C-6 a d C-11 b B d) \tan ^2(e+f x)-11 (A b-C b+a B) d \tan (e+f x)+6 b c C-a (11 A-5 C) d\right )dx}{11 d}+\frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left ((6 b c C-6 a d C-11 b B d) \tan ^2(e+f x)-11 (A b-C b+a B) d \tan (e+f x)+6 b c C-a (11 A-5 C) d\right )dx}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left ((6 b c C-6 a d C-11 b B d) \tan (e+f x)^2-11 (A b-C b+a B) d \tan (e+f x)+6 b c C-a (11 A-5 C) d\right )dx}{11 d}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 \int -\frac {1}{2} (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left (4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2+\left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right ) \tan ^2(e+f x)+99 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{9 d}+\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left (4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2+\left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right ) \tan ^2(e+f x)+99 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left (4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2+\left (99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right ) \tan (e+f x)^2+99 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}-\frac {2 \int -\frac {1}{2} (c+d \tan (e+f x))^{3/2} \left (-2 c \left (24 C c^2-44 B d c+99 (A-C) d^2\right ) b^3+66 a c d (4 c C-9 B d) b^2-a^2 d^2 (384 c C+385 B d) b+21 a^3 (33 A-25 C) d^3+\left (-\left (\left (48 C c^3-88 B d c^2+198 (A-C) d^2 c+693 B d^3\right ) b^3\right )+33 a d \left (8 C c^2-18 B d c+63 (A-C) d^2\right ) b^2-2 a^2 d^2 (192 c C-847 B d) b+168 a^3 C d^3\right ) \tan ^2(e+f x)+693 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )dx}{7 d}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (-2 c \left (24 C c^2-44 B d c+99 (A-C) d^2\right ) b^3+66 a c d (4 c C-9 B d) b^2-a^2 d^2 (384 c C+385 B d) b+21 a^3 (33 A-25 C) d^3+\left (-\left (\left (48 C c^3-88 B d c^2+198 (A-C) d^2 c+693 B d^3\right ) b^3\right )+33 a d \left (8 C c^2-18 B d c+63 (A-C) d^2\right ) b^2-2 a^2 d^2 (192 c C-847 B d) b+168 a^3 C d^3\right ) \tan ^2(e+f x)+693 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )dx}{7 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (-2 c \left (24 C c^2-44 B d c+99 (A-C) d^2\right ) b^3+66 a c d (4 c C-9 B d) b^2-a^2 d^2 (384 c C+385 B d) b+21 a^3 (33 A-25 C) d^3+\left (-\left (\left (48 C c^3-88 B d c^2+198 (A-C) d^2 c+693 B d^3\right ) b^3\right )+33 a d \left (8 C c^2-18 B d c+63 (A-C) d^2\right ) b^2-2 a^2 d^2 (192 c C-847 B d) b+168 a^3 C d^3\right ) \tan (e+f x)^2+693 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )dx}{7 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (693 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)-693 \left (-\left ((A-C) a^3\right )+3 b B a^2+3 b^2 (A-C) a-b^3 B\right ) d^3\right )dx+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}}{7 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (693 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)-693 \left (-\left ((A-C) a^3\right )+3 b B a^2+3 b^2 (A-C) a-b^3 B\right ) d^3\right )dx+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}}{7 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (693 \left ((A c-C c-B d) a^3-3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a+b^3 (B c+(A-C) d)\right ) d^3+693 \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x) d^3\right )dx+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}}{7 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (693 \left ((A c-C c-B d) a^3-3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a+b^3 (B c+(A-C) d)\right ) d^3+693 \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x) d^3\right )dx+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}}{7 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int \frac {-693 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3+3 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) d^3-693 \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3\right )+3 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+3 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}+\frac {1386 d^3 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}}{7 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int \frac {-693 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3+3 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) d^3-693 \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3\right )+3 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+3 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}+\frac {1386 d^3 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}}{7 d}+\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}+\frac {\frac {693}{2} d^3 (a+i b)^3 (c+i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {693}{2} d^3 (a-i b)^3 (c-i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}+\frac {1386 d^3 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}}{7 d}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}+\frac {\frac {693}{2} d^3 (a+i b)^3 (c+i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {693}{2} d^3 (a-i b)^3 (c-i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}+\frac {1386 d^3 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}}{7 d}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}+\frac {\frac {693 i d^3 (a-i b)^3 (c-i d)^2 (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {693 i d^3 (a+i b)^3 (c+i d)^2 (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}+\frac {1386 d^3 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}}{7 d}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}+\frac {-\frac {693 i d^3 (a-i b)^3 (c-i d)^2 (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {693 i d^3 (a+i b)^3 (c+i d)^2 (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}+\frac {1386 d^3 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}}{7 d}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}+\frac {\frac {693 d^2 (a-i b)^3 (c-i d)^2 (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {693 d^2 (a+i b)^3 (c+i d)^2 (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}+\frac {1386 d^3 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}}{7 d}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}-\frac {\frac {2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left (99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right )}{7 d f}+\frac {\frac {2 (c+d \tan (e+f x))^{5/2} \left (168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )-\left (b^3 \left (198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right )\right )\right )}{5 d f}+\frac {462 d^3 \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{f}+\frac {1386 d^3 \sqrt {c+d \tan (e+f x)} \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+\frac {693 d^3 (a-i b)^3 (c-i d)^{3/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}+\frac {693 d^3 (a+i b)^3 (c+i d)^{3/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}}{7 d}}{9 d}}{11 d}\) |
Int[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)*(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])^(5/2))/(11*d*f) - ((2*(6* b*c*C - 11*b*B*d - 6*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5 /2))/(9*d*f) - ((2*b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(7*d*f) + ((693*(a - I*b)^3*(A - I*B - C)*(c - I*d)^(3/2)*d^3*ArcTan[Tan[e + f*x]/Sq rt[c - I*d]])/f + (693*(a + I*b)^3*(A + I*B - C)*(c + I*d)^(3/2)*d^3*ArcTa n[Tan[e + f*x]/Sqrt[c + I*d]])/f + (1386*d^3*(3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d)) *Sqrt[c + d*Tan[e + f*x]])/f + (462*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*d^3*(c + d*Tan[e + f*x])^(3/2))/f + (2*(168*a^3*C*d^3 - 2*a^ 2*b*d^2*(192*c*C - 847*B*d) + 33*a*b^2*d*(8*c^2*C - 18*B*c*d + 63*(A - C)* d^2) - b^3*(48*c^3*C - 88*B*c^2*d + 198*c*(A - C)*d^2 + 693*B*d^3))*(c + d *Tan[e + f*x])^(5/2))/(5*d*f))/(7*d))/(9*d))/(11*d)
3.1.97.3.1 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. \(10951\) vs. \(2(507)=1014\).
Time = 0.34 (sec) , antiderivative size = 10952, normalized size of antiderivative = 19.91
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(10952\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(11056\) |
| default | \(\text {Expression too large to display}\) | \(11056\) |
int((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e) ^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 84950 vs. \(2 (498) = 996\).
Time = 194.27 (sec) , antiderivative size = 84950, normalized size of antiderivative = 154.45 \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="fricas")
\[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
Integral((a + b*tan(e + f*x))**3*(c + d*tan(e + f*x))**(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)**2), x)
Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="maxima")
Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="giac")
Timed out. \[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Hanged} \]