Integrand size = 47, antiderivative size = 503 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {(a-i b)^2 (i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^2 (i A-B-i C) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}-\frac {2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{693 d^3 f}-\frac {2 b (4 b c C-11 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f} \]
-(a-I*b)^2*(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d )^(1/2))/f+(a+I*b)^2*(I*A-B-I*C)*(c+I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1 /2)/(c+I*d)^(1/2))/f-2*(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)))*(c+d*tan(f*x+e))^(1/2)/ f+2/3*(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))^(3/2)/f+2/5*(B*a^2-B*b^2+2*a*b*(A-C))*(c+d*tan(f*x+e))^(5/2)/f+2/69 3*(36*a^2*C*d^2-22*a*b*d*(-9*B*d+2*C*c)+b^2*(8*c^2*C-22*B*c*d+99*(A-C)*d^2 ))*(c+d*tan(f*x+e))^(7/2)/d^3/f-2/99*b*(-11*B*b*d-4*C*a*d+4*C*b*c)*tan(f*x +e)*(c+d*tan(f*x+e))^(7/2)/d^2/f+2/11*C*(a+b*tan(f*x+e))^2*(c+d*tan(f*x+e) )^(7/2)/d/f
Time = 6.58 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.12 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}+\frac {2 \left (\frac {b (-4 b c C+11 b B d+4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac {2 \left (\frac {\left (-36 a^2 C d^2+22 a b d (2 c C-9 B d)-b^2 \left (8 c^2 C-22 B c d+99 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{7/2}}{14 d f}+\frac {i \left (\frac {99}{4} i \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {99}{4} \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2\right ) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+(c-i d) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left (\frac {2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{-c+i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )\right )}{2 f}-\frac {i \left (-\frac {99}{4} i \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {99}{4} \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2\right ) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+(c+i d) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left (\frac {2 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{-c-i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )\right )}{2 f}\right )}{9 d}\right )}{11 d} \]
Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]
(2*C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(7/2))/(11*d*f) + (2*((b* (-4*b*c*C + 11*b*B*d + 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7/2))/( 9*d*f) - (2*(((-36*a^2*C*d^2 + 22*a*b*d*(2*c*C - 9*B*d) - b^2*(8*c^2*C - 2 2*B*c*d + 99*(A - C)*d^2))*(c + d*Tan[e + f*x])^(7/2))/(14*d*f) + ((I/2)*( ((99*I)/4)*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2 + (99*(2*a*b*B - a^2*(A - C ) + b^2*(A - C))*d^2)/4)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (c - I*d)*((2 *(c + d*Tan[e + f*x])^(3/2))/3 + (c - I*d)*((2*(c - I*d)^(3/2)*ArcTanh[Sqr t[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(-c + I*d) + 2*Sqrt[c + d*Tan[e + f* x]]))))/f - ((I/2)*(((-99*I)/4)*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2 + (99* (2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^2)/4)*((2*(c + d*Tan[e + f*x])^(5/ 2))/5 + (c + I*d)*((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))/(9*d)))/(11*d)
Time = 3.78 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.02, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.447, Rules used = {3042, 4130, 27, 3042, 4120, 27, 3042, 4113, 3042, 4011, 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.
| \(\displaystyle \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 \int -\frac {1}{2} (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \left ((4 b c C-4 a d C-11 b B d) \tan ^2(e+f x)-11 (A b-C b+a B) d \tan (e+f x)+4 b c C-a (11 A-7 C) d\right )dx}{11 d}+\frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \left ((4 b c C-4 a d C-11 b B d) \tan ^2(e+f x)-11 (A b-C b+a B) d \tan (e+f x)+4 b c C-a (11 A-7 C) d\right )dx}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \left ((4 b c C-4 a d C-11 b B d) \tan (e+f x)^2-11 (A b-C b+a B) d \tan (e+f x)+4 b c C-a (11 A-7 C) d\right )dx}{11 d}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac {2 \int -\frac {1}{2} (c+d \tan (e+f x))^{5/2} \left (-2 c (4 c C-11 B d) b^2+44 a c C d b-9 a^2 (11 A-7 C) d^2-\left (\left (8 C c^2-22 B d c+99 (A-C) d^2\right ) b^2-22 a d (2 c C-9 B d) b+36 a^2 C d^2\right ) \tan ^2(e+f x)-99 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{5/2} \left (-\left (\left (8 c^2 C-22 B c d\right ) b^2\right )+44 a c C d b-9 a^2 (11 A-7 C) d^2-\left (\left (8 C c^2-22 B d c+99 (A-C) d^2\right ) b^2-22 a d (2 c C-9 B d) b+36 a^2 C d^2\right ) \tan ^2(e+f x)-99 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{5/2} \left (-\left (\left (8 c^2 C-22 B c d\right ) b^2\right )+44 a c C d b-9 a^2 (11 A-7 C) d^2-\left (\left (8 C c^2-22 B d c+99 (A-C) d^2\right ) b^2-22 a d (2 c C-9 B d) b+36 a^2 C d^2\right ) \tan (e+f x)^2-99 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{5/2} \left (99 \left (-\left ((A-C) a^2\right )+2 b B a+b^2 (A-C)\right ) d^2-99 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{5/2} \left (99 \left (-\left ((A-C) a^2\right )+2 b B a+b^2 (A-C)\right ) d^2-99 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (-99 \left ((A c-C c-B d) a^2-2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right ) d^2-99 \left ((B c+(A-C) d) a^2+2 b (A c-C c-B d) a-b^2 (B c+(A-C) d)\right ) \tan (e+f x) d^2\right )dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (-99 \left ((A c-C c-B d) a^2-2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right ) d^2-99 \left ((B c+(A-C) d) a^2+2 b (A c-C c-B d) a-b^2 (B c+(A-C) d)\right ) \tan (e+f x) d^2\right )dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (99 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) d^2+99 \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^2\right )dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (99 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) d^2+99 \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^2\right )dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int \frac {99 d^2 \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)-99 d^2 \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}+\frac {198 d^2 \sqrt {c+d \tan (e+f x)} \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {\int \frac {99 d^2 \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)-99 d^2 \left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right )}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}+\frac {198 d^2 \sqrt {c+d \tan (e+f x)} \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}+\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}}{11 d}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {-\frac {99}{2} d^2 (a+i b)^2 (c+i d)^3 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {99}{2} d^2 (a-i b)^2 (c-i d)^3 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}+\frac {198 d^2 \sqrt {c+d \tan (e+f x)} \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {-\frac {99}{2} d^2 (a+i b)^2 (c+i d)^3 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {99}{2} d^2 (a-i b)^2 (c-i d)^3 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}+\frac {198 d^2 \sqrt {c+d \tan (e+f x)} \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {-\frac {99 i d^2 (a-i b)^2 (c-i d)^3 (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {99 i d^2 (a+i b)^2 (c+i d)^3 (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}+\frac {198 d^2 \sqrt {c+d \tan (e+f x)} \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {\frac {99 i d^2 (a-i b)^2 (c-i d)^3 (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {99 i d^2 (a+i b)^2 (c+i d)^3 (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}+\frac {198 d^2 \sqrt {c+d \tan (e+f x)} \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {-\frac {99 d (a+i b)^2 (c+i d)^3 (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}-\frac {99 d (a-i b)^2 (c-i d)^3 (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}+\frac {198 d^2 \sqrt {c+d \tan (e+f x)} \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{9 d}}{11 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {-\frac {2 (c+d \tan (e+f x))^{7/2} \left (36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left (99 d^2 (A-C)-22 B c d+8 c^2 C\right )\right )}{7 d f}+\frac {198 d^2 \sqrt {c+d \tan (e+f x)} \left (-\left (a^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )+2 a b \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b^2 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{f}-\frac {198 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {66 d^2 (c+d \tan (e+f x))^{3/2} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}-\frac {99 d^2 (a-i b)^2 (c-i d)^{5/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}-\frac {99 d^2 (a+i b)^2 (c+i d)^{5/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}}{9 d}}{11 d}\) |
Int[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]
(2*C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(7/2))/(11*d*f) - ((2*b*( 4*b*c*C - 11*b*B*d - 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7/2))/(9* d*f) + ((-99*(a - I*b)^2*(A - I*B - C)*(c - I*d)^(5/2)*d^2*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/f - (99*(a + I*b)^2*(A + I*B - C)*(c + I*d)^(5/2)*d^2 *ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/f + (198*d^2*(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)))*Sqrt[c + d*Tan[e + f*x]])/f - (66*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*T an[e + f*x])^(3/2))/f - (198*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2*(c + d*Ta n[e + f*x])^(5/2))/(5*f) - (2*(36*a^2*C*d^2 - 22*a*b*d*(2*c*C - 9*B*d) + b ^2*(8*c^2*C - 22*B*c*d + 99*(A - C)*d^2))*(c + d*Tan[e + f*x])^(7/2))/(7*d *f))/(9*d))/(11*d)
3.2.4.3.1 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. \(11279\) vs. \(2(459)=918\).
Time = 0.44 (sec) , antiderivative size = 11280, normalized size of antiderivative = 22.43
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(11280\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(11478\) |
| default | \(\text {Expression too large to display}\) | \(11478\) |
int((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)*(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. 91140 vs. \(2 (449) = 898\).
Time = 169.46 (sec) , antiderivative size = 91140, normalized size of antiderivative = 181.19 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="fricas")
\[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
Integral((a + b*tan(e + f*x))**2*(c + d*tan(e + f*x))**(5/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)**2), x)
Timed out. \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="maxima")
Timed out. \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="giac")
Timed out. \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Hanged} \]