Integrand size = 47, antiderivative size = 396 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {(a-i b)^2 (B+i (A-C)) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(a+i b)^2 (i A-B-i C) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{5/2}}{315 d^3 f}-\frac {2 b (4 b c C-9 b B d-4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{63 d^2 f}+\frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f} \]
-(a-I*b)^2*(B+I*(A-C))*(c-I*d)^(3/2)*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)^(3/2)*arctanh((c+d*tan(f*x+e))^(1 /2)/(c+I*d)^(1/2))/f+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))^(1/2)/f+2/3*(B*a^2-B*b^2+2*a*b*(A-C))*(c+d*tan(f*x +e))^(3/2)/f+2/315*(28*a^2*C*d^2-18*a*b*d*(-7*B*d+2*C*c)+b^2*(8*c^2*C-18*B *c*d+63*(A-C)*d^2))*(c+d*tan(f*x+e))^(5/2)/d^3/f-2/63*b*(-9*B*b*d-4*C*a*d+ 4*C*b*c)*tan(f*x+e)*(c+d*tan(f*x+e))^(5/2)/d^2/f+2/9*C*(a+b*tan(f*x+e))^2* (c+d*tan(f*x+e))^(5/2)/d/f
Time = 6.43 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.29 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}+\frac {2 \left (\frac {b (-4 b c C+9 b B d+4 a C d) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \left (\frac {\left (-28 a^2 C d^2+18 a b d (2 c C-7 B d)-b^2 \left (8 c^2 C-18 B c d+63 (A-C) d^2\right )\right ) (c+d \tan (e+f x))^{5/2}}{10 d f}+\frac {i \left (\frac {63}{4} i \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {63}{4} \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2\right ) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left (\frac {2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{-c+i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )}{2 f}-\frac {i \left (-\frac {63}{4} i \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+\frac {63}{4} \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2\right ) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left (\frac {2 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{-c-i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )}{2 f}\right )}{7 d}\right )}{9 d} \]
Integrate[(a + b*Tan[e + f*x])^2*(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])^2*(c + d*Tan[e + f*x])^(5/2))/(9*d*f) + (2*((b*( -4*b*c*C + 9*b*B*d + 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(7* d*f) - (2*(((-28*a^2*C*d^2 + 18*a*b*d*(2*c*C - 7*B*d) - b^2*(8*c^2*C - 18* B*c*d + 63*(A - C)*d^2))*(c + d*Tan[e + f*x])^(5/2))/(10*d*f) + ((I/2)*((( 63*I)/4)*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2 + (63*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^2)/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)*(((-63*I)/4)*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2 + (63*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^2)/4)*((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))/(7*d)))/(9*d)
Time = 2.92 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.02, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.404, Rules used = {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.
| \(\displaystyle \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 \int -\frac {1}{2} (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left ((4 b c C-4 a d C-9 b B d) \tan ^2(e+f x)-9 (A b-C b+a B) d \tan (e+f x)+4 b c C-a (9 A-5 C) d\right )dx}{9 d}+\frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left ((4 b c C-4 a d C-9 b B d) \tan ^2(e+f x)-9 (A b-C b+a B) d \tan (e+f x)+4 b c C-a (9 A-5 C) d\right )dx}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left ((4 b c C-4 a d C-9 b B d) \tan (e+f x)^2-9 (A b-C b+a B) d \tan (e+f x)+4 b c C-a (9 A-5 C) d\right )dx}{9 d}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int -\frac {1}{2} (c+d \tan (e+f x))^{3/2} \left (-2 c (4 c C-9 B d) b^2+36 a c C d b-7 a^2 (9 A-5 C) d^2-\left (\left (8 C c^2-18 B d c+63 (A-C) d^2\right ) b^2-18 a d (2 c C-7 B d) b+28 a^2 C d^2\right ) \tan ^2(e+f x)-63 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{7 d}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (-2 c (4 c C-9 B d) b^2+36 a c C d b-7 a^2 (9 A-5 C) d^2-\left (\left (8 C c^2-18 B d c+63 (A-C) d^2\right ) b^2-18 a d (2 c C-7 B d) b+28 a^2 C d^2\right ) \tan ^2(e+f x)-63 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{7 d}+\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (-2 c (4 c C-9 B d) b^2+36 a c C d b-7 a^2 (9 A-5 C) d^2-\left (\left (8 C c^2-18 B d c+63 (A-C) d^2\right ) b^2-18 a d (2 c C-7 B d) b+28 a^2 C d^2\right ) \tan (e+f x)^2-63 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{7 d}+\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}}{9 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (63 \left (-\left ((A-C) a^2\right )+2 b B a+b^2 (A-C)\right ) d^2-63 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}}{7 d}+\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int (c+d \tan (e+f x))^{3/2} \left (63 \left (-\left ((A-C) a^2\right )+2 b B a+b^2 (A-C)\right ) d^2-63 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}}{7 d}+\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}}{9 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (-63 \left ((A c-C c-B d) a^2-2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right ) d^2-63 \left ((B c+(A-C) d) a^2+2 b (A c-C c-B d) a-b^2 (B c+(A-C) d)\right ) \tan (e+f x) d^2\right )dx-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}}{7 d}+\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int \sqrt {c+d \tan (e+f x)} \left (-63 \left ((A c-C c-B d) a^2-2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right ) d^2-63 \left ((B c+(A-C) d) a^2+2 b (A c-C c-B d) a-b^2 (B c+(A-C) d)\right ) \tan (e+f x) d^2\right )dx-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}}{7 d}+\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}}{9 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int \frac {63 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) d^2+63 \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^2}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}-\frac {126 d^2 \sqrt {c+d \tan (e+f x)} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{7 d}+\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {\int \frac {63 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) d^2+63 \left (-\left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2\right )+2 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x) d^2}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}-\frac {126 d^2 \sqrt {c+d \tan (e+f x)} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{7 d}+\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}}{9 d}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {-\frac {63}{2} d^2 (a+i b)^2 (c+i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {63}{2} d^2 (a-i b)^2 (c-i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}-\frac {126 d^2 \sqrt {c+d \tan (e+f x)} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{7 d}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {-\frac {63}{2} d^2 (a+i b)^2 (c+i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {63}{2} d^2 (a-i b)^2 (c-i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}-\frac {126 d^2 \sqrt {c+d \tan (e+f x)} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{7 d}}{9 d}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {-\frac {63 i d^2 (a-i b)^2 (c-i d)^2 (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {63 i d^2 (a+i b)^2 (c+i d)^2 (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}-\frac {126 d^2 \sqrt {c+d \tan (e+f x)} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{7 d}}{9 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {\frac {63 i d^2 (a-i b)^2 (c-i d)^2 (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {63 i d^2 (a+i b)^2 (c+i d)^2 (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}-\frac {126 d^2 \sqrt {c+d \tan (e+f x)} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{7 d}}{9 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {-\frac {63 d (a+i b)^2 (c+i d)^2 (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}-\frac {63 d (a-i b)^2 (c-i d)^2 (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}-\frac {126 d^2 \sqrt {c+d \tan (e+f x)} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}}{7 d}}{9 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}-\frac {\frac {2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {-\frac {2 (c+d \tan (e+f x))^{5/2} \left (28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left (63 d^2 (A-C)-18 B c d+8 c^2 C\right )\right )}{5 d f}-\frac {42 d^2 \left (a^2 B+2 a b (A-C)-b^2 B\right ) (c+d \tan (e+f x))^{3/2}}{f}-\frac {126 d^2 \sqrt {c+d \tan (e+f x)} \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}-\frac {63 d^2 (a-i b)^2 (c-i d)^{3/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}-\frac {63 d^2 (a+i b)^2 (c+i d)^{3/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}}{7 d}}{9 d}\) |
Int[(a + b*Tan[e + f*x])^2*(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])^2*(c + d*Tan[e + f*x])^(5/2))/(9*d*f) - ((2*b*(4 *b*c*C - 9*b*B*d - 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(7*d* f) + ((-63*(a - I*b)^2*(A - I*B - C)*(c - I*d)^(3/2)*d^2*ArcTan[Tan[e + f* x]/Sqrt[c - I*d]])/f - (63*(a + I*b)^2*(A + I*B - C)*(c + I*d)^(3/2)*d^2*A rcTan[Tan[e + f*x]/Sqrt[c + I*d]])/f - (126*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))*Sqrt[c + d*Tan[e + f*x]])/ f - (42*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2*(c + d*Tan[e + f*x])^(3/2))/f - (2*(28*a^2*C*d^2 - 18*a*b*d*(2*c*C - 7*B*d) + b^2*(8*c^2*C - 18*B*c*d + 63*(A - C)*d^2))*(c + d*Tan[e + f*x])^(5/2))/(5*d*f))/(7*d))/(9*d)
3.1.98.3.1 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. \(7938\) vs. \(2(357)=714\).
Time = 0.21 (sec) , antiderivative size = 7939, normalized size of antiderivative = 20.05
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(7939\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(8031\) |
| default | \(\text {Expression too large to display}\) | \(8031\) |
int((a+b*tan(f*x+e))^2*(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. 58971 vs. \(2 (347) = 694\).
Time = 69.93 (sec) , antiderivative size = 58971, normalized size of antiderivative = 148.92 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
integrate((a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan( f*x+e)^2),x, algorithm="fricas")
\[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
Integral((a + b*tan(e + f*x))**2*(c + d*tan(e + f*x))**(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)**2), x)
Timed out. \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^2*(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. \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^2*(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. \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Hanged} \]