\(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx\) [126]

Optimal result

Integrand size = 47, antiderivative size = 365 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\frac {(A-i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b) (c-i d)^{5/2} f}+\frac {(i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b) (c+i d)^{5/2} f}-\frac {2 b^{3/2} \left (A b^2-a (b B-a C)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) (b c-a d)^{5/2} f}+\frac {2 \left (c^2 C-B c d+A d^2\right )}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \left (b \left (c^4 C-2 B c^3 d+c^2 (3 A-C) d^2+A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1948\) vs. \(2(365)=730\).

Time = 6.25 (sec) , antiderivative size = 1948, normalized size of antiderivative = 5.34 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c 
 + d*Tan[e + f*x])^(5/2)),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 5.92 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.22, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.383, Rules used = {3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 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.

\(\displaystyle \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \tan (e+f x)+C \tan (e+f x)^2}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*T 
an[e + f*x])^(5/2)),x]
 

Output:

(2*(c^2*C - B*c*d + A*d^2))/(3*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f* 
x])^(3/2)) - (((-(((a + I*b)*(A - I*B - C)*(c + I*d)^2*(b*c - a*d)^2*ArcTa 
n[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) - ((a - I*b)*(A + I*B - 
C)*(c - I*d)^2*(b*c - a*d)^2*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + 
 I*d]*f))/(a^2 + b^2) + (2*b^(3/2)*(A*b^2 - a*(b*B - a*C))*(c^2 + d^2)^2*A 
rcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*S 
qrt[b*c - a*d]*f))/((b*c - a*d)*(c^2 + d^2)) - (2*(b*(c^4*C - 2*B*c^3*d + 
c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2))))/((b*c 
 - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]))/((b*c - a*d)*(c^2 + d^2))
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(45118\) vs. \(2(328)=656\).

Time = 0.28 (sec) , antiderivative size = 45119, normalized size of antiderivative = 123.61

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(45119\)
default	\(\text {Expression too large to display}\)	\(45119\)

Input:

int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(5/2 
),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))/(c+d*tan(f*x+e) 
)^(5/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\left (a + b \tan {\left (e + f x \right )}\right ) \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))/(c+d*tan(f*x+e 
))**(5/2),x)
 

Output:

Integral((A + B*tan(e + f*x) + C*tan(e + f*x)**2)/((a + b*tan(e + f*x))*(c 
 + d*tan(e + f*x))**(5/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))/(c+d*tan(f*x+e) 
)^(5/2),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
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))/(c+d*tan(f*x+e) 
)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \] Input:

int((A + B*tan(e + f*x) + C*tan(e + f*x)^2)/((a + b*tan(e + f*x))*(c + d*t 
an(e + f*x))^(5/2)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {too large to display} \] Input:

int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(5/2 
),x)
 

Output:

( - 4*sqrt(tan(e + f*x)*d + c)*a + 2*sqrt(tan(e + f*x)*d + c)*c - 3*int(sq 
rt(tan(e + f*x)*d + c)/(tan(e + f*x)**4*b*d**3 + tan(e + f*x)**3*a*d**3 + 
3*tan(e + f*x)**3*b*c*d**2 + 3*tan(e + f*x)**2*a*c*d**2 + 3*tan(e + f*x)** 
2*b*c**2*d + 3*tan(e + f*x)*a*c**2*d + tan(e + f*x)*b*c**3 + a*c**3),x)*ta 
n(e + f*x)**2*a**2*d**3*f + 3*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)** 
4*b*d**3 + tan(e + f*x)**3*a*d**3 + 3*tan(e + f*x)**3*b*c*d**2 + 3*tan(e + 
 f*x)**2*a*c*d**2 + 3*tan(e + f*x)**2*b*c**2*d + 3*tan(e + f*x)*a*c**2*d + 
 tan(e + f*x)*b*c**3 + a*c**3),x)*tan(e + f*x)**2*a*c*d**3*f - 6*int(sqrt( 
tan(e + f*x)*d + c)/(tan(e + f*x)**4*b*d**3 + tan(e + f*x)**3*a*d**3 + 3*t 
an(e + f*x)**3*b*c*d**2 + 3*tan(e + f*x)**2*a*c*d**2 + 3*tan(e + f*x)**2*b 
*c**2*d + 3*tan(e + f*x)*a*c**2*d + tan(e + f*x)*b*c**3 + a*c**3),x)*tan(e 
 + f*x)*a**2*c*d**2*f + 6*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**4*b* 
d**3 + tan(e + f*x)**3*a*d**3 + 3*tan(e + f*x)**3*b*c*d**2 + 3*tan(e + f*x 
)**2*a*c*d**2 + 3*tan(e + f*x)**2*b*c**2*d + 3*tan(e + f*x)*a*c**2*d + tan 
(e + f*x)*b*c**3 + a*c**3),x)*tan(e + f*x)*a*c**2*d**2*f - 3*int(sqrt(tan( 
e + f*x)*d + c)/(tan(e + f*x)**4*b*d**3 + tan(e + f*x)**3*a*d**3 + 3*tan(e 
 + f*x)**3*b*c*d**2 + 3*tan(e + f*x)**2*a*c*d**2 + 3*tan(e + f*x)**2*b*c** 
2*d + 3*tan(e + f*x)*a*c**2*d + tan(e + f*x)*b*c**3 + a*c**3),x)*a**2*c**2 
*d*f + 3*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**4*b*d**3 + tan(e + f* 
x)**3*a*d**3 + 3*tan(e + f*x)**3*b*c*d**2 + 3*tan(e + f*x)**2*a*c*d**2 ...