Integrand size = 47, antiderivative size = 473 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}+\frac {(b c-a d)^{3/2} \left (3 a^3 b B d-5 a^4 C d-b^4 (2 B c+5 A d)-a b^3 (4 A c-4 c C-7 B d)+a^2 b^2 (2 B c-(A+9 C) d)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{7/2} \left (a^2+b^2\right )^2 f}-\frac {d \left (5 a^3 C d-A b^2 (b c-a d)-2 b^3 (2 c C+B d)-a^2 b (5 c C+3 B d)+a b^2 (B c+4 C d)\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right ) f}+\frac {\left (3 A b^2-3 a b B+5 a^2 C+2 b^2 C\right ) d (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right ) f}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]
-(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)^2/f-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d )^(1/2))/(a+I*b)^2/f+(-a*d+b*c)^(3/2)*(3*a^3*b*B*d-5*a^4*C*d-b^4*(5*A*d+2* B*c)-a*b^3*(4*A*c-7*B*d-4*C*c)+a^2*b^2*(2*B*c-(A+9*C)*d))*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)^2/f-d*(5*a^3*C* d-A*b^2*(-a*d+b*c)-2*b^3*(B*d+2*C*c)-a^2*b*(3*B*d+5*C*c)+a*b^2*(B*c+4*C*d) )*(c+d*tan(f*x+e))^(1/2)/b^3/(a^2+b^2)/f+1/3*(3*A*b^2-3*B*a*b+5*C*a^2+2*C* b^2)*d*(c+d*tan(f*x+e))^(3/2)/b^2/(a^2+b^2)/f-(A*b^2-a*(B*b-C*a))*(c+d*tan (f*x+e))^(5/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(6112\) vs. \(2(473)=946\).
Time = 7.02 (sec) , antiderivative size = 6112, normalized size of antiderivative = 12.92 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Result too large to show} \]
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])^2,x]
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.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^2}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\left (5 C a^2-3 b B a+3 A b^2+2 b^2 C\right ) d \tan ^2(e+f x)-2 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+2 (b B-a C) \left (b c-\frac {5 a d}{2}\right )+2 A b \left (a c+\frac {5 b d}{2}\right )\right )}{2 (a+b \tan (e+f x))}dx}{b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\left (5 C a^2-3 b B a+3 A b^2+2 b^2 C\right ) d \tan ^2(e+f x)-2 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (2 b c-5 a d)+A b (2 a c+5 b d)\right )}{a+b \tan (e+f x)}dx}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\left (5 C a^2-3 b B a+3 A b^2+2 b^2 C\right ) d \tan (e+f x)^2-2 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (2 b c-5 a d)+A b (2 a c+5 b d)\right )}{a+b \tan (e+f x)}dx}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {2 \int -\frac {3 \sqrt {c+d \tan (e+f x)} \left (-2 \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-c ((b B-a C) (2 b c-5 a d)+A b (2 a c+5 b d)) b+a \left (5 C a^2-3 b B a+3 A b^2+2 b^2 C\right ) d^2+d \left (5 C d a^3-b (5 c C+3 B d) a^2+b^2 (B c+4 C d) a-A b^2 (b c-a d)-2 b^3 (2 c C+B d)\right ) \tan ^2(e+f x)\right )}{2 (a+b \tan (e+f x))}dx}{3 b}+\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-2 \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-c ((b B-a C) (2 b c-5 a d)+A b (2 a c+5 b d)) b+a \left (5 C a^2-3 b B a+3 A b^2+2 b^2 C\right ) d^2+d \left (5 C d a^3-b (5 c C+3 B d) a^2+b^2 (B c+4 C d) a-A b^2 (b c-a d)-2 b^3 (2 c C+B d)\right ) \tan ^2(e+f x)\right )}{a+b \tan (e+f x)}dx}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-2 \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-c ((b B-a C) (2 b c-5 a d)+A b (2 a c+5 b d)) b+a \left (5 C a^2-3 b B a+3 A b^2+2 b^2 C\right ) d^2+d \left (5 C d a^3-b (5 c C+3 B d) a^2+b^2 (B c+4 C d) a-A b^2 (b c-a d)-2 b^3 (2 c C+B d)\right ) \tan (e+f x)^2\right )}{a+b \tan (e+f x)}dx}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 \int -\frac {5 C d^3 a^4-b d^2 (10 c C+3 B d) a^3+b^2 d \left (5 C c^2+4 B d c+(A+4 C) d^2\right ) a^2+b^3 \left (2 A c^3-2 C c^3-5 B d c^2-4 A d^2 c-6 C d^2 c-2 B d^3\right ) a+d \left (5 C d^2 a^4-b d (10 c C+3 B d) a^3+b^2 \left (5 C c^2+4 B d c+4 C d^2\right ) a^2+b^3 \left (B c^2-12 C d c-4 B d^2\right ) a+2 b^4 \left (3 C c^2+3 B d c-C d^2\right )+A b^2 \left (-\left (\left (c^2-2 d^2\right ) b^2\right )+2 a c d b+a^2 d^2\right )\right ) \tan ^2(e+f x)+b^4 c^2 (2 B c+5 A d)+2 b^3 \left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\int \frac {5 C d^3 a^4-b d^2 (10 c C+3 B d) a^3+b^2 d \left (5 C c^2+4 B d c+(A+4 C) d^2\right ) a^2+b^3 \left (2 A c^3-2 C c^3-5 B d c^2-4 A d^2 c-6 C d^2 c-2 B d^3\right ) a+d \left (5 C d^2 a^4-b d (10 c C+3 B d) a^3+b^2 \left (5 C c^2+4 B d c+4 C d^2\right ) a^2+b^3 \left (B c^2-12 C d c-4 B d^2\right ) a+2 b^4 \left (3 C c^2+3 B d c-C d^2\right )+A b^2 \left (-\left (\left (c^2-2 d^2\right ) b^2\right )+2 a c d b+a^2 d^2\right )\right ) \tan ^2(e+f x)+b^4 c^2 (2 B c+5 A d)+2 b^3 \left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\int \frac {5 C d^3 a^4-b d^2 (10 c C+3 B d) a^3+b^2 d \left (5 C c^2+4 B d c+(A+4 C) d^2\right ) a^2+b^3 \left (2 A c^3-2 C c^3-5 B d c^2-4 A d^2 c-6 C d^2 c-2 B d^3\right ) a+d \left (5 C d^2 a^4-b d (10 c C+3 B d) a^3+b^2 \left (5 C c^2+4 B d c+4 C d^2\right ) a^2+b^3 \left (B c^2-12 C d c-4 B d^2\right ) a+2 b^4 \left (3 C c^2+3 B d c-C d^2\right )+A b^2 \left (-\left (\left (c^2-2 d^2\right ) b^2\right )+2 a c d b+a^2 d^2\right )\right ) \tan (e+f x)^2+b^4 c^2 (2 B c+5 A d)+2 b^3 \left (b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {\int \frac {2 \left (\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {-\frac {2 \int -\frac {\left (\left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) b^3+\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2+2 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 d \left (5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right ) (c+d \tan (e+f x))^{3/2}}{3 b f}-\frac {\frac {2 d \sqrt {c+d \tan (e+f x)} \left (5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right )}{b f}-\frac {\frac {2 \int -\frac {\left (\left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^2 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) b^3+\left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^2\right )+2 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x) b^3}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{b}}{b}}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\) |
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])^2,x]
3.2.8.3.1 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_.)*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[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 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])))
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. \(14118\) vs. \(2(434)=868\).
Time = 0.20 (sec) , antiderivative size = 14119, normalized size of antiderivative = 29.85
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(14119\) |
| default | \(\text {Expression too large to display}\) | \(14119\) |
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) )^2,x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
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))^2,x, algorithm="fricas")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]
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))^2,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. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]
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))^2,x, algorithm="giac")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Hanged} \]