3.2.21 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx\) [121]

3.2.21.1 Optimal result

Integrand size = 47, antiderivative size = 447 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=-\frac {(i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 (c-i d)^{3/2} f}-\frac {(B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 (c+i d)^{3/2} f}-\frac {\sqrt {b} \left (5 a^3 b B d-3 a^4 C d+b^4 (2 B c-3 A d)+a b^3 (4 A c-4 c C+B d)-a^2 b^2 (2 B c+(7 A-C) d)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{5/2} f}-\frac {d \left (2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )+A \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \]

output

-(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(a-I*b)^2/(c-I* 
d)^(3/2)/f-(B-I*(A-C))*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(a+I* 
b)^2/(c+I*d)^(3/2)/f-(5*a^3*b*B*d-3*a^4*C*d+b^4*(-3*A*d+2*B*c)+a*b^3*(4*A* 
c+B*d-4*C*c)-a^2*b^2*(2*B*c+(7*A-C)*d))*arctanh(b^(1/2)*(c+d*tan(f*x+e))^( 
1/2)/(-a*d+b*c)^(1/2))*b^(1/2)/(a^2+b^2)^2/(-a*d+b*c)^(5/2)/f-d*(2*b^2*c*( 
-B*d+C*c)-a*b*B*(c^2+d^2)+a^2*(-2*B*c*d+3*C*c^2+C*d^2)+A*(2*a^2*d^2+b^2*(c 
^2+3*d^2)))/(a^2+b^2)/(-a*d+b*c)^2/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)+(-A* 
b^2+a*(B*b-C*a))/(a^2+b^2)/(-a*d+b*c)/f/(c+d*tan(f*x+e))^(1/2)/(a+b*tan(f* 
x+e))
 

3.2.21.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2078\) vs. \(2(447)=894\).

Time = 6.43 (sec) , antiderivative size = 2078, normalized size of antiderivative = 4.65 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\text {Result too large to show} \]

input

Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2* 
(c + d*Tan[e + f*x])^(3/2)),x]
 

output

-((A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])* 
Sqrt[c + d*Tan[e + f*x]])) - ((-2*(((I*Sqrt[c - I*d]*((b*(-(b*c) + a*d)*(( 
-3*(A*b^2 - a*(b*B - a*C))*d^2)/2 - c*(A*b - a*B - b*C)*(b*c - a*d) + (d*( 
3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2 + a*(-1/ 
2*(a*d*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a* 
d))) + (((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(3*A*b^2*d - 2*a*A*(b*c - a*d) 
- (b*B - a*C)*(2*b*c + a*d)))/2 - (b*(-(c*((-3*c*(A*b^2 - a*(b*B - a*C))*d 
)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (d^2*(3*A*b^2*d - 2*a*A*(b*c - a 
*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2) - I*((a*(-(b*c) + a*d)*((-3*(A*b^ 
2 - a*(b*B - a*C))*d^2)/2 - c*(A*b - a*B - b*C)*(b*c - a*d) + (d*(3*A*b^2* 
d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2 - b*(-1/2*(a*d*( 
(-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + ( 
((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - 
 a*C)*(2*b*c + a*d)))/2 - (b*(-(c*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A 
*b - a*B - b*C)*d*(b*c - a*d))) + (d^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b 
*B - a*C)*(2*b*c + a*d)))/2))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c 
 - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*((b*(-(b*c) + a*d)*((-3*(A*b^2 
 - a*(b*B - a*C))*d^2)/2 - c*(A*b - a*B - b*C)*(b*c - a*d) + (d*(3*A*b^2*d 
 - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2 + a*(-1/2*(a*d*(( 
-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) +...
 

3.2.21.3 Rubi [A] (warning: unable to verify)

Time = 4.08 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.15, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.404, Rules used = {3042, 4132, 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.

\(\displaystyle \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \tan (e+f x)+C \tan (e+f x)^2}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 4132

\(\displaystyle -\frac {\int \frac {3 A d b^2+3 \left (A b^2-a (b B-a C)\right ) d \tan ^2(e+f x)-2 a A (b c-a d)-(b B-a C) (2 b c+a d)+2 (A b-C b-a B) (b c-a d) \tan (e+f x)}{2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {3 A d b^2+3 \left (A b^2-a (b B-a C)\right ) d \tan ^2(e+f x)-2 a A (b c-a d)-(b B-a C) (2 b c+a d)+2 (A b-C b-a B) (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {3 A d b^2+3 \left (A b^2-a (b B-a C)\right ) d \tan (e+f x)^2-2 a A (b c-a d)-(b B-a C) (2 b c+a d)+2 (A b-C b-a B) (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 4132

\(\displaystyle -\frac {\frac {2 \int -\frac {2 d^2 (A c-C c+B d) a^3-b (4 A-C) d \left (c^2+d^2\right ) a^2-b^2 \left (2 C c^3+B d c^2+4 C d^2 c-B d^3-2 A \left (c^3+2 d^2 c\right )\right ) a-b d \left (2 A d^2 a^2+\left (3 C c^2-2 B d c+C d^2\right ) a^2-b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (c^2+3 d^2\right )\right ) \tan ^2(e+f x)+b^3 (2 B c-3 A d) \left (c^2+d^2\right )+2 (b c-a d)^2 (b c C-b B d-A (b c+a d)+a (B c+C d)) \tan (e+f x)}{2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {2 d^2 (A c-C c+B d) a^3-b (4 A-C) d \left (c^2+d^2\right ) a^2-b^2 \left (2 C c^3+B d c^2+4 C d^2 c-B d^3-2 A \left (c^3+2 d^2 c\right )\right ) a-b d \left (2 A d^2 a^2+\left (3 C c^2-2 B d c+C d^2\right ) a^2-b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (c^2+3 d^2\right )\right ) \tan ^2(e+f x)+b^3 (2 B c-3 A d) \left (c^2+d^2\right )+2 (b c-a d)^2 (a B c+b C c-b B d+a C d-A (b c+a d)) \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {2 d^2 (A c-C c+B d) a^3-b (4 A-C) d \left (c^2+d^2\right ) a^2-b^2 \left (2 C c^3+B d c^2+4 C d^2 c-B d^3-2 A \left (c^3+2 d^2 c\right )\right ) a-b d \left (2 A d^2 a^2+\left (3 C c^2-2 B d c+C d^2\right ) a^2-b B \left (c^2+d^2\right ) a+2 b^2 c (c C-B d)+A b^2 \left (c^2+3 d^2\right )\right ) \tan (e+f x)^2+b^3 (2 B c-3 A d) \left (c^2+d^2\right )+2 (b c-a d)^2 (a B c+b C c-b B d+a C d-A (b c+a d)) \tan (e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 4136

\(\displaystyle -\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {\int \frac {2 \left ((b c-a d)^2 \left ((A c-C c+B d) a^2+2 b (B c-(A-C) d) a-b^2 (A c-C c+B d)\right )-(b c-a d)^2 \left (-\left ((B c-(A-C) d) a^2\right )+2 b (A c-C c+B d) a+b^2 (B c-(A-C) d)\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 \int \frac {(b c-a d)^2 \left ((A c-C c+B d) a^2+2 b (B c-(A-C) d) a-b^2 (A c-C c+B d)\right )-(b c-a d)^2 \left (-\left ((B c-(A-C) d) a^2\right )+2 b (A c-C c+B d) a+b^2 (B c-(A-C) d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 \int \frac {(b c-a d)^2 \left ((A c-C c+B d) a^2+2 b (B c-(A-C) d) a-b^2 (A c-C c+B d)\right )-(b c-a d)^2 \left (-\left ((B c-(A-C) d) a^2\right )+2 b (A c-C c+B d) a+b^2 (B c-(A-C) d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {1}{2} (a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {1}{2} (a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {i (a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i (a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {i (a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i (a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {(a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {(a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {(a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}+\frac {2 \left (\frac {(a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 b \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}+\frac {2 \left (\frac {(a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\frac {2 d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {-\frac {2 \sqrt {b} \left (c^2+d^2\right ) \left (-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right ) \sqrt {b c-a d}}+\frac {2 \left (\frac {(a+i b)^2 (c+i d) (A-i B-C) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b)^2 (c-i d) (A+i B-C) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

input

Int[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d 
*Tan[e + f*x])^(3/2)),x]
 

output

-((A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])* 
Sqrt[c + d*Tan[e + f*x]])) - (-(((2*(((a + I*b)^2*(A - I*B - C)*(c + I*d)* 
(b*c - a*d)^2*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + ((a 
- I*b)^2*(A + I*B - C)*(c - I*d)*(b*c - a*d)^2*ArcTan[Tan[e + f*x]/Sqrt[c 
+ I*d]])/(Sqrt[c + I*d]*f)))/(a^2 + b^2) - (2*Sqrt[b]*(c^2 + d^2)*(5*a^3*b 
*B*d - 3*a^4*C*d + b^4*(2*B*c - 3*A*d) + a*b^3*(4*A*c - 4*c*C + B*d) - a^2 
*b^2*(2*B*c + (7*A - C)*d))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqr 
t[b*c - a*d]])/((a^2 + b^2)*Sqrt[b*c - a*d]*f))/((b*c - a*d)*(c^2 + d^2))) 
 + (2*d*(2*a^2*A*d^2 + 2*b^2*c*(c*C - B*d) - a*b*B*(c^2 + d^2) + A*b^2*(c^ 
2 + 3*d^2) + a^2*(3*c^2*C - 2*B*c*d + C*d^2)))/((b*c - a*d)*(c^2 + d^2)*f* 
Sqrt[c + d*Tan[e + f*x]]))/(2*(a^2 + b^2)*(b*c - a*d))
 

3.2.21.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 221

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4020

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]
 

rule 4022

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]
 

rule 4117

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]
 

rule 4132

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])))
 

rule 4136

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]
 

3.2.21.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(40618\) vs. \(2(411)=822\).

Time = 0.24 (sec) , antiderivative size = 40619, normalized size of antiderivative = 90.87

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(40619\)
default	\(\text {Expression too large to display}\)	\(40619\)

input

int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3 
/2),x,method=_RETURNVERBOSE)
 

output

result too large to display
 

3.2.21.5 Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

input

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+ 
e))^(3/2),x, algorithm="fricas")
 

output

Timed out
 

3.2.21.6 Sympy [F]

\[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))**2/(c+d*tan(f* 
x+e))**(3/2),x)
 

output

Integral((A + B*tan(e + f*x) + C*tan(e + f*x)**2)/((a + b*tan(e + f*x))**2 
*(c + d*tan(e + f*x))**(3/2)), x)
 

3.2.21.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]

input

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+ 
e))^(3/2),x, algorithm="maxima")
 

output

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
 

3.2.21.8 Giac [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

input

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+ 
e))^(3/2),x, algorithm="giac")
 

output

Timed out
 

3.2.21.9 Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\text {Hanged} \]

input

int((A + B*tan(e + f*x) + C*tan(e + f*x)^2)/((a + b*tan(e + f*x))^2*(c + d 
*tan(e + f*x))^(3/2)),x)
 

output

\text{Hanged}