3.2.2 \(\int \frac {(c+d \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^2} \, dx\) [102]

3.2.2.1 Optimal result

Integrand size = 47, antiderivative size = 372 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=-\frac {(i A+B-i C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}-\frac {(B-i (A-C)) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}+\frac {\sqrt {b c-a d} \left (a^3 b B d-3 a^4 C d-b^4 (2 B c+3 A d)-a b^3 (4 A c-4 c C-5 B d)+a^2 b^2 (2 B c+(A-7 C) d)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 f}+\frac {\left (A b^2-a b B+3 a^2 C+2 b^2 C\right ) d \sqrt {c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

output

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

3.2.2.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2738\) vs. \(2(372)=744\).

Time = 6.61 (sec) , antiderivative size = 2738, normalized size of antiderivative = 7.36 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx=\text {Result too large to show} \]

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])^2,x]
 

output

(2*C*(c + d*Tan[e + f*x])^(3/2))/(b*f*(a + b*Tan[e + f*x])) + (2*(-(((3*b* 
c*C + b*B*d - 3*a*C*d)*Sqrt[c + d*Tan[e + f*x]])/(b*f*(a + b*Tan[e + f*x]) 
)) - (2*(-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((b*(-(A*b^2*c^2) + 3*a^2*C*d 
^2 + 2*b^2*c*(2*c*C + B*d) - a*b*d*(6*c*C + B*d)))/4 + (a*b^2*(2*c*(A - C) 
*d + B*(c^2 - d^2)))/4 - (b*(3*a^2*C*d^2 - a*b*d*(6*c*C + B*d) + b^2*(3*c^ 
2*C - (A - C)*d^2)))/4) + a*((((b^2*d)/2 - a*(b*c - a*d))*(-(A*b^2*c^2) + 
3*a^2*C*d^2 + 2*b^2*c*(2*c*C + B*d) - a*b*d*(6*c*C + B*d)))/4 + (-(b*c) + 
(a*d)/2)*(-1/4*(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))) - (a*(3*a^2*C*d^2 - a 
*b*d*(6*c*C + B*d) + b^2*(3*c^2*C - (A - C)*d^2)))/4) - (d*((b^2*(-(A*b^2* 
c^2) + 3*a^2*C*d^2 + 2*b^2*c*(2*c*C + B*d) - a*b*d*(6*c*C + B*d)))/4 - a*( 
-1/4*(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))) - (a*(3*a^2*C*d^2 - a*b*d*(6*c* 
C + B*d) + b^2*(3*c^2*C - (A - C)*d^2)))/4)))/2) - I*(a*(b*c - a*d)*((b*(- 
(A*b^2*c^2) + 3*a^2*C*d^2 + 2*b^2*c*(2*c*C + B*d) - a*b*d*(6*c*C + B*d)))/ 
4 + (a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))/4 - (b*(3*a^2*C*d^2 - a*b*d*(6 
*c*C + B*d) + b^2*(3*c^2*C - (A - C)*d^2)))/4) - b*((((b^2*d)/2 - a*(b*c - 
 a*d))*(-(A*b^2*c^2) + 3*a^2*C*d^2 + 2*b^2*c*(2*c*C + B*d) - a*b*d*(6*c*C 
+ B*d)))/4 + (-(b*c) + (a*d)/2)*(-1/4*(b^3*(2*c*(A - C)*d + B*(c^2 - d^2)) 
) - (a*(3*a^2*C*d^2 - a*b*d*(6*c*C + B*d) + b^2*(3*c^2*C - (A - C)*d^2)))/ 
4) - (d*((b^2*(-(A*b^2*c^2) + 3*a^2*C*d^2 + 2*b^2*c*(2*c*C + B*d) - a*b*d* 
(6*c*C + B*d)))/4 - a*(-1/4*(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))) - (a*...
 

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

Time = 3.69 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.04, 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, 4130, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4128

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4130

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

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])^2,x]
 

output

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

3.2.2.3.1 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 4130

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. 
) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ 
e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a 
+ b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C 
*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* 
m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, 
b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && 
 NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ 
c, 0] && NeQ[a, 0])))
 

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]
 

3.2.2.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(9864\) vs. \(2(337)=674\).

Time = 0.17 (sec) , antiderivative size = 9865, normalized size of antiderivative = 26.52

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(9865\)
default	\(\text {Expression too large to display}\)	\(9865\)

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) 
)^2,x,method=_RETURNVERBOSE)
 

output

result too large to display
 

3.2.2.5 Fricas [F(-1)]

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

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))^2,x, algorithm="fricas")
 

output

Timed out
 

3.2.2.6 Sympy [F]

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

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))**2,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))**2, x)
 

3.2.2.7 Maxima [F(-2)]

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

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))^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
 

3.2.2.8 Giac [F(-1)]

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

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))^2,x, algorithm="giac")
 

output

Timed out
 

3.2.2.9 Mupad [F(-1)]

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

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))^2,x)
 

output

\text{Hanged}