3.2.3 \(\int \frac {(c+d \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^3} \, dx\) [103]

3.2.3.1 Optimal result

Integrand size = 47, antiderivative size = 532 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=-\frac {(A-i B-C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(A+i B-C) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}-\frac {\left (a^5 b B d^2+3 a^6 C d^2+a^4 b^2 d (4 B c+3 (A+2 C) d)-b^6 \left (8 A c^2-8 c^2 C-12 B c d-3 A d^2\right )+a^2 b^4 \left (24 A c^2-24 c^2 C-48 B c d-26 A d^2+35 C d^2\right )-2 a^3 b^3 \left (12 c (A-C) d+B \left (4 c^2-9 d^2\right )\right )+a b^5 \left (40 c (A-C) d+3 B \left (8 c^2-5 d^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {\left (a^3 b B d+3 a^4 C d+b^4 (4 B c+3 A d)+a b^3 (8 A c-8 c C-7 B d)-a^2 b^2 (4 B c+5 A d-11 C d)\right ) \sqrt {c+d \tan (e+f x)}}{4 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2} \]

output

-(A-I*B-C)*(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(I* 
a+b)^3/f+(A+I*B-C)*(c+I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1 
/2))/(I*a-b)^3/f-1/4*(a^5*b*B*d^2+3*a^6*C*d^2+a^4*b^2*d*(4*B*c+3*(A+2*C)*d 
)-b^6*(8*A*c^2-3*A*d^2-12*B*c*d-8*C*c^2)+a^2*b^4*(24*A*c^2-26*A*d^2-48*B*c 
*d-24*C*c^2+35*C*d^2)-2*a^3*b^3*(12*c*(A-C)*d+B*(4*c^2-9*d^2))+a*b^5*(40*c 
*(A-C)*d+3*B*(8*c^2-5*d^2)))*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+ 
b*c)^(1/2))/b^(5/2)/(a^2+b^2)^3/f/(-a*d+b*c)^(1/2)-1/4*(a^3*b*B*d+3*a^4*C* 
d+b^4*(3*A*d+4*B*c)+a*b^3*(8*A*c-7*B*d-8*C*c)-a^2*b^2*(5*A*d+4*B*c-11*C*d) 
)*(c+d*tan(f*x+e))^(1/2)/b^2/(a^2+b^2)^2/f/(a+b*tan(f*x+e))-1/2*(A*b^2-a*( 
B*b-C*a))*(c+d*tan(f*x+e))^(3/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^2
 

3.2.3.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(7678\) vs. \(2(532)=1064\).

Time = 7.18 (sec) , antiderivative size = 7678, normalized size of antiderivative = 14.43 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

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

Time = 4.40 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.06, 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, 4128, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4128

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4128

\(\displaystyle \frac {\frac {\int \frac {-8 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+2 \left (a c+\frac {b d}{2}\right ) ((b B-a C) (4 b c-3 a d)+A b (4 a c+3 b d)) b+d \left (3 C d a^4+b B d a^3+b^2 (4 B c+3 (A+C) d) a^2-b^3 (8 A c-8 C c-9 B d) a-b^4 (4 B c+5 A d-8 C d)\right ) \tan ^2(e+f x)-(2 b c-a d) \left (3 C d a^3+b B d a^2-4 b^2 (B c-2 C d) a+A b^2 (4 b c-5 a d)-4 b^3 (c C+B d)\right )}{2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{b \left (a^2+b^2\right )}-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-8 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+2 \left (a c+\frac {b d}{2}\right ) ((b B-a C) (4 b c-3 a d)+A b (4 a c+3 b d)) b+d \left (3 C d a^4+b B d a^3+b^2 (4 B c+3 (A+C) d) a^2-b^3 (8 A c-8 C c-9 B d) a-b^4 (4 B c+5 A d-8 C d)\right ) \tan ^2(e+f x)-(2 b c-a d) \left (3 C d a^3+b B d a^2-4 b^2 (B c-2 C d) a+A b^2 (4 b c-5 a d)-4 b^3 (c C+B d)\right )}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{2 b \left (a^2+b^2\right )}-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {-8 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2+2 \left (a c+\frac {b d}{2}\right ) ((b B-a C) (4 b c-3 a d)+A b (4 a c+3 b d)) b+d \left (3 C d a^4+b B d a^3+b^2 (4 B c+3 (A+C) d) a^2-b^3 (8 A c-8 C c-9 B d) a-b^4 (4 B c+5 A d-8 C d)\right ) \tan (e+f x)^2-(2 b c-a d) \left (3 C d a^3+b B d a^2-4 b^2 (B c-2 C d) a+A b^2 (4 b c-5 a d)-4 b^3 (c C+B d)\right )}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{2 b \left (a^2+b^2\right )}-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {\frac {\frac {\int -\frac {8 \left (b^2 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3-3 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 \left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {8 \int \frac {b^2 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3-3 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 \left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {8 \int \frac {b^2 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^3-3 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 \left (\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3+3 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2-3 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^3 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {8 \left (-\frac {1}{2} b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {1}{2} b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {8 \left (-\frac {1}{2} b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx-\frac {1}{2} b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {8 \left (\frac {i b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {8 \left (\frac {i b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {8 \left (-\frac {b^2 (a-i b)^3 (c+i d)^2 (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}-\frac {b^2 (a+i b)^3 (c-i d)^2 (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {8 \left (-\frac {b^2 (a-i b)^3 (c+i d)^{3/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}-\frac {b^2 (a+i b)^3 (c-i d)^{3/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {\left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}-\frac {8 \left (-\frac {b^2 (a-i b)^3 (c+i d)^{3/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}-\frac {b^2 (a+i b)^3 (c-i d)^{3/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {2 \left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}-\frac {8 \left (-\frac {b^2 (a-i b)^3 (c+i d)^{3/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}-\frac {b^2 (a+i b)^3 (c-i d)^{3/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\sqrt {c+d \tan (e+f x)} \left (3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right )}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {2 \left (3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left (12 c d (A-C)+B \left (4 c^2-9 d^2\right )\right )+a^2 b^4 \left (24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right )+a b^5 \left (40 c d (A-C)+3 B \left (8 c^2-5 d^2\right )\right )-b^6 \left (8 A c^2-3 A d^2-12 B c d-8 c^2 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right ) \sqrt {b c-a d}}-\frac {8 \left (-\frac {b^2 (a-i b)^3 (c+i d)^{3/2} (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}-\frac {b^2 (a+i b)^3 (c-i d)^{3/2} (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\)

input

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

output

-1/2*((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(b*(a^2 + b^2)*f 
*(a + b*Tan[e + f*x])^2) + (((-8*(-(((a + I*b)^3*b^2*(A - I*B - C)*(c - I* 
d)^(3/2)*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/f) - ((a - I*b)^3*b^2*(A + I* 
B - C)*(c + I*d)^(3/2)*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/f))/(a^2 + b^2) 
 - (2*(a^5*b*B*d^2 + 3*a^6*C*d^2 + a^4*b^2*d*(4*B*c + 3*(A + 2*C)*d) - b^6 
*(8*A*c^2 - 8*c^2*C - 12*B*c*d - 3*A*d^2) + a^2*b^4*(24*A*c^2 - 24*c^2*C - 
 48*B*c*d - 26*A*d^2 + 35*C*d^2) - 2*a^3*b^3*(12*c*(A - C)*d + B*(4*c^2 - 
9*d^2)) + a*b^5*(40*c*(A - C)*d + 3*B*(8*c^2 - 5*d^2)))*ArcTanh[(Sqrt[b]*S 
qrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*Sqrt[b*c - 
 a*d]*f))/(2*b*(a^2 + b^2)) - ((a^3*b*B*d + 3*a^4*C*d + b^4*(4*B*c + 3*A*d 
) + a*b^3*(8*A*c - 8*c*C - 7*B*d) - a^2*b^2*(4*B*c + 5*A*d - 11*C*d))*Sqrt 
[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])))/(4*b*(a^2 + 
b^2))
 

3.2.3.3.1 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 4128

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]
 

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]
 

3.2.3.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(14440\) vs. \(2(492)=984\).

Time = 0.16 (sec) , antiderivative size = 14441, normalized size of antiderivative = 27.14

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(14441\)
default	\(\text {Expression too large to display}\)	\(14441\)

input

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

output

result too large to display
 

3.2.3.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.2.3.6 Sympy [F]

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

input

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

output

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

3.2.3.7 Maxima [F(-2)]

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

input

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

3.2.3.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.2.3.9 Mupad [F(-1)]

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

input

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

output

\text{Hanged}