Integrand size = 47, antiderivative size = 336 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b) f}+\frac {(i A-B-i C) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{7/2} \left (a^2+b^2\right ) f}+\frac {2 \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \sqrt {c+d \tan (e+f x)}}{b^3 f}+\frac {2 (b c C+b B d-a C d) (c+d \tan (e+f x))^{3/2}}{3 b^2 f}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f} \]
-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/( a-I*b)/f+(I*A-B-I*C)*(c+I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^ (1/2))/(a+I*b)/f-2*(A*b^2-a*(B*b-C*a))*(-a*d+b*c)^(5/2)*arctanh(b^(1/2)*(c +d*tan(f*x+e))^(1/2)/(-a*d+b*c)^(1/2))/b^(7/2)/(a^2+b^2)/f+2*(b^2*d*(B*c+( A-C)*d)+(-a*d+b*c)*(B*b*d-C*a*d+C*b*c))*(c+d*tan(f*x+e))^(1/2)/b^3/f+2/3*( B*b*d-C*a*d+C*b*c)*(c+d*tan(f*x+e))^(3/2)/b^2/f+2/5*C*(c+d*tan(f*x+e))^(5/ 2)/b/f
Time = 5.76 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx=\frac {\frac {15 \left (b^{7/2} (-i a+b) (A-i B-C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+b^{7/2} (i a+b) (A+i B-C) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )-2 \left (A b^2+a (-b B+a C)\right ) (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )\right )}{b^{5/2} \left (a^2+b^2\right )}+\frac {30 \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \sqrt {c+d \tan (e+f x)}}{b^2}+\frac {10 (b c C+b B d-a C d) (c+d \tan (e+f x))^{3/2}}{b}+6 C (c+d \tan (e+f x))^{5/2}}{15 b f} \]
Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x] ^2))/(a + b*Tan[e + f*x]),x]
((15*(b^(7/2)*((-I)*a + b)*(A - I*B - C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + b^(7/2)*(I*a + b)*(A + I*B - C)*(c + I*d) ^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] - 2*(A*b^2 + a*(-(b *B) + a*C))*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/S qrt[b*c - a*d]]))/(b^(5/2)*(a^2 + b^2)) + (30*(b^2*d*(B*c + (A - C)*d) + ( b*c - a*d)*(b*c*C + b*B*d - a*C*d))*Sqrt[c + d*Tan[e + f*x]])/b^2 + (10*(b *c*C + b*B*d - a*C*d)*(c + d*Tan[e + f*x])^(3/2))/b + 6*C*(c + d*Tan[e + f *x])^(5/2))/(15*b*f)
Time = 3.71 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.01, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.489, Rules used = {3042, 4130, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4136, 25, 25, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 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.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{a+b \tan (e+f x)}dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 \int \frac {5 (c+d \tan (e+f x))^{3/2} \left ((b c C-a d C+b B d) \tan ^2(e+f x)+b (B c+(A-C) d) \tan (e+f x)+A b c-a C d\right )}{2 (a+b \tan (e+f x))}dx}{5 b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left ((b c C-a d C+b B d) \tan ^2(e+f x)+b (B c+(A-C) d) \tan (e+f x)+A b c-a C d\right )}{a+b \tan (e+f x)}dx}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left ((b c C-a d C+b B d) \tan (e+f x)^2+b (B c+(A-C) d) \tan (e+f x)+A b c-a C d\right )}{a+b \tan (e+f x)}dx}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {2 \int \frac {3 \sqrt {c+d \tan (e+f x)} \left (A c^2 b^2+\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x) b^2+\left (d (B c+(A-C) d) b^2+(b c-a d) (b c C-a d C+b B d)\right ) \tan ^2(e+f x)+a d (a C d-b (2 c C+B d))\right )}{2 (a+b \tan (e+f x))}dx}{3 b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (A c^2 b^2+\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x) b^2+\left (d (B c+(A-C) d) b^2+(b c-a d) (b c C-a d C+b B d)\right ) \tan ^2(e+f x)+a d (a C d-b (2 c C+B d))\right )}{a+b \tan (e+f x)}dx}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (A c^2 b^2+\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x) b^2+\left (d (B c+(A-C) d) b^2+(b c-a d) (b c C-a d C+b B d)\right ) \tan (e+f x)^2+a d (a C d-b (2 c C+B d))\right )}{a+b \tan (e+f x)}dx}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) \tan (e+f x) b^3+A \left (b c^3-a d^3\right ) b^2+\left (d \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) b^3+(b c-a d) \left (d (B c+(A-C) d) b^2+(b c-a d) (b c C-a d C+b B d)\right )\right ) \tan ^2(e+f x)-a d \left (\left (3 C c^2+3 B d c-C d^2\right ) b^2-a d (3 c C+B d) b+a^2 C d^2\right )}{2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) \tan (e+f x) b^3+A \left (b c^3-a d^3\right ) b^2+\left (d \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) b^3+(b c-a d) \left (d (B c+(A-C) d) b^2+(b c-a d) (b c C-a d C+b B d)\right )\right ) \tan ^2(e+f x)-a d \left (\left (3 C c^2+3 B d c-C d^2\right ) b^2-a d (3 c C+B d) b+a^2 C d^2\right )}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) \tan (e+f x) b^3+A \left (b c^3-a d^3\right ) b^2+\left (d \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) b^3+(b c-a d) \left (d (B c+(A-C) d) b^2+(b c-a d) (b c C-a d C+b B d)\right )\right ) \tan (e+f x)^2-a d \left (\left (3 C c^2+3 B d c-C d^2\right ) b^2-a d (3 c C+B d) b+a^2 C d^2\right )}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\int -\frac {b^3 \left (a \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}+\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {\int -\frac {\left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )+a \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}+\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\int \frac {\left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )-a \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}+\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\int \frac {\left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )-a \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}+\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}}{b}+\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}}{b}+\frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {1}{2} b^3 (a+i b) (c-i d)^3 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b^3 (a-i b) (c+i d)^3 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {1}{2} b^3 (a+i b) (c-i d)^3 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b^3 (a-i b) (c+i d)^3 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{b}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {i b^3 (a+i b) (c-i d)^3 (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i b^3 (a-i b) (c+i d)^3 (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}}{a^2+b^2}}{b}}{b}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {i b^3 (a-i b) (c+i d)^3 (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i b^3 (a+i b) (c-i d)^3 (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}}{a^2+b^2}}{b}}{b}}{b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {b^3 (a+i b) (c-i d)^3 (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {b^3 (a-i b) (c+i d)^3 (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}}{a^2+b^2}}{b}}{b}}{b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {b^3 (a+i b) (c-i d)^{5/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}+\frac {b^3 (a-i b) (c+i d)^{5/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}}{a^2+b^2}}{b}}{b}}{b}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {\frac {(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}+\frac {\frac {b^3 (a+i b) (c-i d)^{5/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}+\frac {b^3 (a-i b) (c+i d)^{5/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}}{a^2+b^2}}{b}}{b}}{b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {\frac {2 (b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}+\frac {\frac {b^3 (a+i b) (c-i d)^{5/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}+\frac {b^3 (a-i b) (c+i d)^{5/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}}{a^2+b^2}}{b}}{b}}{b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 C (c+d \tan (e+f x))^{5/2}}{5 b f}+\frac {\frac {2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b f}+\frac {-\frac {2 (b c-a d)^{5/2} \left (A b^2-a (b B-a C)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right )}+\frac {\frac {b^3 (a+i b) (c-i d)^{5/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}+\frac {b^3 (a-i b) (c+i d)^{5/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}}{a^2+b^2}}{b}}{b}}{b}\) |
Int[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/( a + b*Tan[e + f*x]),x]
(2*C*(c + d*Tan[e + f*x])^(5/2))/(5*b*f) + ((2*(b*c*C + b*B*d - a*C*d)*(c + d*Tan[e + f*x])^(3/2))/(3*b*f) + (((((a + I*b)*b^3*(A - I*B - C)*(c - I* d)^(5/2)*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/f + ((a - I*b)*b^3*(A + I*B - C)*(c + I*d)^(5/2)*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/f)/(a^2 + b^2) - ( 2*(A*b^2 - a*(b*B - a*C))*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Ta n[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*f))/b + (2*(b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(b*c*C + b*B*d - a*C*d))*Sqrt[c + d*Tan[e + f* x]])/(b*f))/b)/b
3.2.7.3.1 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[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_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
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])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*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[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(8697\) vs. \(2(294)=588\).
Time = 0.18 (sec) , antiderivative size = 8698, normalized size of antiderivative = 25.89
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(8698\) |
| default | \(\text {Expression too large to display}\) | \(8698\) |
int((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e) ),x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e)),x, algorithm="fricas")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx=\text {Exception raised: ValueError} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e)),x, algorithm="maxima")
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e)),x, algorithm="giac")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx=\text {Hanged} \]