Integrand size = 47, antiderivative size = 543 \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=-\frac {(A-i B-C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(A+i B-C) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\left (3 a^5 b B d^2+a^6 C d^2-3 a^4 b^2 d (4 B c+5 A d-6 C d)-3 a^2 b^4 \left (8 A c^2-8 c^2 C-16 B c d-6 A d^2+5 C d^2\right )+2 a^3 b^3 \left (20 c (A-C) d+B \left (4 c^2-13 d^2\right )\right )-3 a b^5 \left (8 c (A-C) d+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (2 c C+B d)-A \left (8 c^2+d^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 (b c-a d)^{3/2} f}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (3 a^3 b B d+a^4 C d+b^4 (4 B c+A d)+a b^3 (8 A c-8 c C-5 B d)-a^2 b^2 (4 B c+7 A d-9 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))} \]
1/4*(3*a^5*b*B*d^2+a^6*C*d^2-3*a^4*b^2*d*(5*A*d+4*B*c-6*C*d)-3*a^2*b^4*(8* A*c^2-6*A*d^2-16*B*c*d-8*C*c^2+5*C*d^2)+2*a^3*b^3*(20*c*(A-C)*d+B*(4*c^2-1 3*d^2))-3*a*b^5*(8*c*(A-C)*d+B*(8*c^2-d^2))-b^6*(4*c*(B*d+2*C*c)-A*(8*c^2+ d^2)))*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(a ^2+b^2)^3/(-a*d+b*c)^(3/2)/f-(A-I*B-C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I *d)^(1/2))*(c-I*d)^(1/2)/(I*a+b)^3/f+(A+I*B-C)*arctanh((c+d*tan(f*x+e))^(1 /2)/(c+I*d)^(1/2))*(c+I*d)^(1/2)/(I*a-b)^3/f-1/2*(A*b^2-a*(B*b-C*a))*(c+d* tan(f*x+e))^(1/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^2-1/4*(3*a^3*b*B*d+a^4*C* d+b^4*(A*d+4*B*c)+a*b^3*(8*A*c-5*B*d-8*C*c)-a^2*b^2*(7*A*d+4*B*c-9*C*d))*( c+d*tan(f*x+e))^(1/2)/b/(a^2+b^2)^2/(-a*d+b*c)/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. \(2819\) vs. \(2(543)=1086\).
Time = 6.72 (sec) , antiderivative size = 2819, normalized size of antiderivative = 5.19 \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\text {Result too large to show} \]
Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2 ))/(a + b*Tan[e + f*x])^3,x]
(-2*C*Sqrt[c + d*Tan[e + f*x]])/(3*b*f*(a + b*Tan[e + f*x])^2) - (2*(-1/2* (((b^2*(-3*A*b*c + 4*b*c*C - a*C*d))/2 - a*((-3*b^2*(B*c + (A - C)*d))/2 - (a*(b*c*C - 3*b*B*d - a*C*d))/2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)* (b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((((I*Sqrt[c - I*d]*(b*(b*c - a* d)*((3*b*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 + 3*a*b*(b *c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + (3*b*(b*c - a* d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4) + a*((3*( b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(a^2*C*d + b^2*(4*B*c + A*d) + a*b* (4*A*c - 4*c*C - B*d)))/4 + (-(b*c) + (a*d)/2)*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c *C - a*A*d - b*B*d + a*C*d)) - (d*((3*b^2*(b*c - a*d)*(a^2*C*d + b^2*(4*B* c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 - a*((3*a*(3*A*b^2 - 3*a*b*B - a^ 2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))))/2) - I*(a*(b*c - a*d)*((3*b*(3*A*b^2 - 3*a*b* B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 + 3*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + (3*b*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4) - b*((3*(b*c - a*d)*((b^2*d)/2 - a* (b*c - a*d))*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 + (-(b*c) + (a*d)/2)*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*...
Time = 4.28 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.08, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.404, Rules used = {3042, 4128, 27, 3042, 4132, 27, 3042, 4136, 27, 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.
| \(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^3}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\int \frac {-\left (\left (-C a^2-3 b B a+3 A b^2-4 b^2 C\right ) d \tan ^2(e+f x)\right )-4 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+2 (b B-a C) \left (2 b c-\frac {a d}{2}\right )+A b (4 a c+b d)}{2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}dx}{2 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-\left (\left (-C a^2-3 b B a+3 A b^2-4 b^2 C\right ) d \tan ^2(e+f x)\right )-4 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (4 b c-a d)+A b (4 a c+b d)}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}dx}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-\left (\left (-C a^2-3 b B a+3 A b^2-4 b^2 C\right ) d \tan (e+f x)^2\right )-4 b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (4 b c-a d)+A b (4 a c+b d)}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}dx}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {\int -\frac {-d \left (C d a^4+3 b B d a^3-b^2 (4 B c+7 A d-9 C d) a^2+b^3 (8 A c-8 C c-5 B d) a+b^4 (4 B c+A d)\right ) \tan ^2(e+f x)-8 b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)+2 \left (-d a^2+b c a-\frac {b^2 d}{2}\right ) ((b B-a C) (4 b c-a d)+A b (4 a c+b d))-(2 b c-a d) \left (C d a^3+3 b B d a^2-4 b^2 (B c-2 C d) a+A b^2 (4 b c-7 a d)-4 b^3 (c C+B d)\right )}{2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-d \left (C d a^4+3 b B d a^3-b^2 (4 B c+7 A d-9 C d) a^2+b^3 (8 A c-8 C c-5 B d) a+b^4 (4 B c+A d)\right ) \tan ^2(e+f x)-8 b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)+2 \left (-d a^2+b c a-\frac {b^2 d}{2}\right ) ((b B-a C) (4 b c-a d)+A b (4 a c+b d))-(2 b c-a d) \left (C d a^3+3 b B d a^2-4 b^2 (B c-2 C d) a+A b^2 (4 b c-7 a d)-4 b^3 (c C+B d)\right )}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-d \left (C d a^4+3 b B d a^3-b^2 (4 B c+7 A d-9 C d) a^2+b^3 (8 A c-8 C c-5 B d) a+b^4 (4 B c+A d)\right ) \tan (e+f x)^2-8 b (b c-a d) \left (-\left ((B c+(A-C) d) a^2\right )+2 b (A c-C c-B d) a+b^2 (B c+(A-C) d)\right ) \tan (e+f x)+2 \left (-d a^2+b c a-\frac {b^2 d}{2}\right ) ((b B-a C) (4 b c-a d)+A b (4 a c+b d))-(2 b c-a d) \left (C d a^3+3 b B d a^2-4 b^2 (B c-2 C d) a+A b^2 (4 b c-7 a d)-4 b^3 (c C+B d)\right )}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {8 \left (b (b c-a d) \left ((A c-C c-B d) a^3+3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a-b^3 (B c+(A-C) d)\right )-b (b c-a d) \left (-\left ((B c+(A-C) d) a^3\right )+3 b (A c-C c-B d) a^2+3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {8 \int \frac {b (b c-a d) \left ((A c-C c-B d) a^3+3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a-b^3 (B c+(A-C) d)\right )-b (b c-a d) \left (-\left ((B c+(A-C) d) a^3\right )+3 b (A c-C c-B d) a^2+3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {8 \int \frac {b (b c-a d) \left ((A c-C c-B d) a^3+3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a-b^3 (B c+(A-C) d)\right )-b (b c-a d) \left (-\left ((B c+(A-C) d) a^3\right )+3 b (A c-C c-B d) a^2+3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {1}{2} b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {1}{2} b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {i b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {i b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {b (a-i b)^3 (c+i d) (A+i B-C) (b c-a d) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {b (a+i b)^3 (c-i d) (A-i B-C) (b c-a d) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {b (a-i b)^3 \sqrt {c+i d} (A+i B-C) (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {b (a+i b)^3 \sqrt {c-i d} (A-i B-C) (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {-\frac {\left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}+\frac {8 \left (\frac {b (a-i b)^3 \sqrt {c+i d} (A+i B-C) (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {b (a+i b)^3 \sqrt {c-i d} (A-i B-C) (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {-\frac {2 \left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}+\frac {8 \left (\frac {b (a-i b)^3 \sqrt {c+i d} (A+i B-C) (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {b (a+i b)^3 \sqrt {c-i d} (A-i B-C) (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {\frac {2 \left (a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left (20 c d (A-C)+B \left (4 c^2-13 d^2\right )\right )-3 a^2 b^4 \left (8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right )-3 a b^5 \left (8 c d (A-C)+B \left (8 c^2-d^2\right )\right )-b^6 \left (4 c (B d+2 c C)-A \left (8 c^2+d^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right ) \sqrt {b c-a d}}+\frac {8 \left (\frac {b (a-i b)^3 \sqrt {c+i d} (A+i B-C) (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {b (a+i b)^3 \sqrt {c-i d} (A-i B-C) (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
Int[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]
-1/2*((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*f*( a + b*Tan[e + f*x])^2) + (((8*(((a + I*b)^3*b*(A - I*B - C)*Sqrt[c - I*d]* (b*c - a*d)*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/f + ((a - I*b)^3*b*(A + I* B - C)*Sqrt[c + I*d]*(b*c - a*d)*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/f))/( a^2 + b^2) + (2*(3*a^5*b*B*d^2 + a^6*C*d^2 - 3*a^4*b^2*d*(4*B*c + 5*A*d - 6*C*d) - 3*a^2*b^4*(8*A*c^2 - 8*c^2*C - 16*B*c*d - 6*A*d^2 + 5*C*d^2) + 2* a^3*b^3*(20*c*(A - C)*d + B*(4*c^2 - 13*d^2)) - 3*a*b^5*(8*c*(A - C)*d + B *(8*c^2 - d^2)) - b^6*(4*c*(2*c*C + B*d) - A*(8*c^2 + d^2)))*ArcTanh[(Sqrt [b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*Sqrt[ b*c - a*d]*f))/(2*(a^2 + b^2)*(b*c - a*d)) - ((3*a^3*b*B*d + a^4*C*d + b^4 *(4*B*c + A*d) + a*b^3*(8*A*c - 8*c*C - 5*B*d) - a^2*b^2*(4*B*c + 7*A*d - 9*C*d))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])))/(4*b*(a^2 + b^2))
3.1.96.3.1 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[(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[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !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. \(9796\) vs. \(2(503)=1006\).
Time = 0.15 (sec) , antiderivative size = 9797, normalized size of antiderivative = 18.04
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(9797\) |
| default | \(\text {Expression too large to display}\) | \(9797\) |
int((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e) )^3,x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^3,x, algorithm="fricas")
\[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\int \frac {\sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \]
Integral(sqrt(c + d*tan(e + f*x))*(A + B*tan(e + f*x) + C*tan(e + f*x)**2) /(a + b*tan(e + f*x))**3, x)
Exception generated. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^3,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. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^3,x, algorithm="giac")
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\text {Hanged} \]