\(\int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx\) [1228]

Optimal result

Integrand size = 25, antiderivative size = 457 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=-\frac {\left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )+a b \left (6 c^2 d-2 d^3\right )\right ) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^3}+\frac {b^4 \left (2 a b c-5 a^2 d-3 b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^2 (b c-a d)^4 f}+\frac {d^3 \left (a^2 d^2 \left (3 c^2-d^2\right )-2 a b c d \left (5 c^2+d^2\right )+b^2 \left (10 c^4+9 c^2 d^2+3 d^4\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^4 \left (c^2+d^2\right )^3 f}-\frac {d \left (a^2 d^2+b^2 \left (2 c^2+3 d^2\right )\right )}{2 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}+\frac {d \left (2 a^3 c d^3+2 a b^2 c d^3-2 a^2 b d^2 \left (2 c^2+d^2\right )-b^3 \left (c^4+6 c^2 d^2+3 d^4\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^3 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \] Output:

-(b^2*c*(c^2-3*d^2)-a^2*(c^3-3*c*d^2)+a*b*(6*c^2*d-2*d^3))*x/(a^2+b^2)^2/( 
c^2+d^2)^3+b^4*(-5*a^2*d+2*a*b*c-3*b^2*d)*ln(a*cos(f*x+e)+b*sin(f*x+e))/(a 
^2+b^2)^2/(-a*d+b*c)^4/f+d^3*(a^2*d^2*(3*c^2-d^2)-2*a*b*c*d*(5*c^2+d^2)+b^ 
2*(10*c^4+9*c^2*d^2+3*d^4))*ln(c*cos(f*x+e)+d*sin(f*x+e))/(-a*d+b*c)^4/(c^ 
2+d^2)^3/f-1/2*d*(a^2*d^2+b^2*(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))^2-b^2/(a^2+b^2)/(-a*d+b*c)/f/(a+b*tan(f*x+e))/(c+d* 
tan(f*x+e))^2+d*(2*a^3*c*d^3+2*a*b^2*c*d^3-2*a^2*b*d^2*(2*c^2+d^2)-b^3*(c^ 
4+6*c^2*d^2+3*d^4))/(a^2+b^2)/(-a*d+b*c)^3/(c^2+d^2)^2/f/(c+d*tan(f*x+e))
 

Mathematica [A] (verified)

Time = 6.84 (sec) , antiderivative size = 827, normalized size of antiderivative = 1.81 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}-\frac {-\frac {d^2 \left (-a b c+a^2 d+3 b^2 d\right )-c \left (-3 b^2 c d+b d (b c-a d)\right )}{2 (-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {-\frac {\frac {b (b c-a d)^3 \left (2 a b c^3+3 a^2 c^2 d-3 b^2 c^2 d-6 a b c d^2-a^2 d^3+b^2 d^3+\frac {b \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )+a b \left (6 c^2 d-2 d^3\right )\right )}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 b^5 \left (2 a b c-5 a^2 d-3 b^2 d\right ) \left (c^2+d^2\right )^2 \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}+\frac {b (b c-a d)^3 \left (2 a b c^3+3 a^2 c^2 d-3 b^2 c^2 d-6 a b c d^2-a^2 d^3+b^2 d^3+\frac {\sqrt {-b^2} \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )+a b \left (6 c^2 d-2 d^3\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 b \left (a^2+b^2\right ) d^3 \left (a^2 d^2 \left (3 c^2-d^2\right )-2 a b c d \left (5 c^2+d^2\right )+b^2 \left (10 c^4+9 c^2 d^2+3 d^4\right )\right ) \log (c+d \tan (e+f x))}{(b c-a d) \left (c^2+d^2\right )}}{b (-b c+a d) \left (c^2+d^2\right ) f}-\frac {2 d^2 \left (a b d^2 (2 b c+a d)+\frac {1}{2} \left (a b c-a^2 d-3 b^2 d\right ) \left (2 a c d-2 b \left (c^2+d^2\right )\right )\right )-c \left (2 d (b c-a d)^2 (b c+a d)-2 b c d \left (a^2 d^2+b^2 \left (2 c^2+3 d^2\right )\right )\right )}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}}{2 (-b c+a d) \left (c^2+d^2\right )}}{\left (a^2+b^2\right ) (b c-a d)} \] Input:

Integrate[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3),x]
 

Output:

-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]) 
^2)) - (-1/2*(d^2*(-(a*b*c) + a^2*d + 3*b^2*d) - c*(-3*b^2*c*d + b*d*(b*c 
- a*d)))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-(((b*(b 
*c - a*d)^3*(2*a*b*c^3 + 3*a^2*c^2*d - 3*b^2*c^2*d - 6*a*b*c*d^2 - a^2*d^3 
 + b^2*d^3 + (b*(b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d 
- 2*d^3)))/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 
 + d^2)) - (2*b^5*(2*a*b*c - 5*a^2*d - 3*b^2*d)*(c^2 + d^2)^2*Log[a + b*Ta 
n[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) + (b*(b*c - a*d)^3*(2*a*b*c^3 + 3*a 
^2*c^2*d - 3*b^2*c^2*d - 6*a*b*c*d^2 - a^2*d^3 + b^2*d^3 + (Sqrt[-b^2]*(b^ 
2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3)))/b)*Log[S 
qrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) - (2*b*(a^2 + b^2)* 
d^3*(a^2*d^2*(3*c^2 - d^2) - 2*a*b*c*d*(5*c^2 + d^2) + b^2*(10*c^4 + 9*c^2 
*d^2 + 3*d^4))*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)))/(b*(-(b 
*c) + a*d)*(c^2 + d^2)*f)) - (2*d^2*(a*b*d^2*(2*b*c + a*d) + ((a*b*c - a^2 
*d - 3*b^2*d)*(2*a*c*d - 2*b*(c^2 + d^2)))/2) - c*(2*d*(b*c - a*d)^2*(b*c 
+ a*d) - 2*b*c*d*(a^2*d^2 + b^2*(2*c^2 + 3*d^2))))/((-(b*c) + a*d)*(c^2 + 
d^2)*f*(c + d*Tan[e + f*x])))/(2*(-(b*c) + a*d)*(c^2 + d^2)))/((a^2 + b^2) 
*(b*c - a*d))
 

Rubi [A] (verified)

Time = 3.05 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4052, 25, 3042, 4132, 27, 3042, 4132, 3042, 4134, 3042, 4013}

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.

\(\displaystyle \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}dx\)

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle \frac {-\frac {\frac {\int \frac {d^3 \left (c^2-d^2\right ) a^4-b c d^2 \left (3 c^2+d^2\right ) a^3+b^2 d \left (3 c^4+7 d^2 c^2+2 d^4\right ) a^2-b^3 c \left (c^4+5 d^2 c^2+2 d^4\right ) a+3 b^4 d \left (c^2+d^2\right )^2-b d \left (-\left (\left (c^4+6 d^2 c^2+3 d^4\right ) b^3\right )+2 a c d^3 b^2-2 a^2 d^2 \left (2 c^2+d^2\right ) b+2 a^3 c d^3\right ) \tan ^2(e+f x)+(b c-a d)^3 \left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (2 a^3 c d^3-2 a^2 b d^2 \left (2 c^2+d^2\right )+2 a b^2 c d^3-\left (b^3 \left (c^4+6 c^2 d^2+3 d^4\right )\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (2 c^2+3 d^2\right )\right )}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\int \frac {d^3 \left (c^2-d^2\right ) a^4-b c d^2 \left (3 c^2+d^2\right ) a^3+b^2 d \left (3 c^4+7 d^2 c^2+2 d^4\right ) a^2-b^3 c \left (c^4+5 d^2 c^2+2 d^4\right ) a+3 b^4 d \left (c^2+d^2\right )^2-b d \left (-\left (\left (c^4+6 d^2 c^2+3 d^4\right ) b^3\right )+2 a c d^3 b^2-2 a^2 d^2 \left (2 c^2+d^2\right ) b+2 a^3 c d^3\right ) \tan (e+f x)^2+(b c-a d)^3 \left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (2 a^3 c d^3-2 a^2 b d^2 \left (2 c^2+d^2\right )+2 a b^2 c d^3-\left (b^3 \left (c^4+6 c^2 d^2+3 d^4\right )\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (2 c^2+3 d^2\right )\right )}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 4134

\(\displaystyle \frac {-\frac {\frac {-\frac {d^3 \left (a^2+b^2\right ) \left (a^2 d^2 \left (3 c^2-d^2\right )-2 a b c d \left (5 c^2+d^2\right )+b^2 \left (10 c^4+9 c^2 d^2+3 d^4\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {b^4 \left (c^2+d^2\right )^2 \left (-5 a^2 d+2 a b c-3 b^2 d\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {x (b c-a d)^3 \left (-\left (a^2 \left (c^3-3 c d^2\right )\right )+a b \left (6 c^2 d-2 d^3\right )+b^2 c \left (c^2-3 d^2\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (2 a^3 c d^3-2 a^2 b d^2 \left (2 c^2+d^2\right )+2 a b^2 c d^3-\left (b^3 \left (c^4+6 c^2 d^2+3 d^4\right )\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (2 c^2+3 d^2\right )\right )}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {-\frac {d^3 \left (a^2+b^2\right ) \left (a^2 d^2 \left (3 c^2-d^2\right )-2 a b c d \left (5 c^2+d^2\right )+b^2 \left (10 c^4+9 c^2 d^2+3 d^4\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {b^4 \left (c^2+d^2\right )^2 \left (-5 a^2 d+2 a b c-3 b^2 d\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {x (b c-a d)^3 \left (-\left (a^2 \left (c^3-3 c d^2\right )\right )+a b \left (6 c^2 d-2 d^3\right )+b^2 c \left (c^2-3 d^2\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (2 a^3 c d^3-2 a^2 b d^2 \left (2 c^2+d^2\right )+2 a b^2 c d^3-\left (b^3 \left (c^4+6 c^2 d^2+3 d^4\right )\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^2 d^2+b^2 \left (2 c^2+3 d^2\right )\right )}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 4013

\(\displaystyle \frac {-\frac {d \left (a^2 d^2+b^2 \left (2 c^2+3 d^2\right )\right )}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}-\frac {\frac {-\frac {d^3 \left (a^2+b^2\right ) \left (a^2 d^2 \left (3 c^2-d^2\right )-2 a b c d \left (5 c^2+d^2\right )+b^2 \left (10 c^4+9 c^2 d^2+3 d^4\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)}+\frac {x (b c-a d)^3 \left (-\left (a^2 \left (c^3-3 c d^2\right )\right )+a b \left (6 c^2 d-2 d^3\right )+b^2 c \left (c^2-3 d^2\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {b^4 \left (c^2+d^2\right )^2 \left (-5 a^2 d+2 a b c-3 b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (2 a^3 c d^3-2 a^2 b d^2 \left (2 c^2+d^2\right )+2 a b^2 c d^3-\left (b^3 \left (c^4+6 c^2 d^2+3 d^4\right )\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}}{\left (c^2+d^2\right ) (b c-a d)}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}\)

Input:

Int[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3),x]
 

Output:

-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]) 
^2)) + (-1/2*(d*(a^2*d^2 + b^2*(2*c^2 + 3*d^2)))/((b*c - a*d)*(c^2 + d^2)* 
f*(c + d*Tan[e + f*x])^2) - ((((b*c - a*d)^3*(b^2*c*(c^2 - 3*d^2) - a^2*(c 
^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3))*x)/((a^2 + b^2)*(c^2 + d^2)) - (b^4 
*(2*a*b*c - 5*a^2*d - 3*b^2*d)*(c^2 + d^2)^2*Log[a*Cos[e + f*x] + b*Sin[e 
+ f*x]])/((a^2 + b^2)*(b*c - a*d)*f) - ((a^2 + b^2)*d^3*(a^2*d^2*(3*c^2 - 
d^2) - 2*a*b*c*d*(5*c^2 + d^2) + b^2*(10*c^4 + 9*c^2*d^2 + 3*d^4))*Log[c*C 
os[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)*(c^2 + d^2)*f))/((b*c - a*d)*( 
c^2 + d^2)) - (d*(2*a^3*c*d^3 + 2*a*b^2*c*d^3 - 2*a^2*b*d^2*(2*c^2 + d^2) 
- b^3*(c^4 + 6*c^2*d^2 + 3*d^4)))/((b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e 
+ f*x])))/((b*c - a*d)*(c^2 + d^2)))/((a^2 + b^2)*(b*c - a*d))
 

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 3042

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

rule 4013

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* 
(x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si 
n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && 
 NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
 

rule 4052

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

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 4134

Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^ 
2)/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) 
*(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))*(x/ 
((a^2 + b^2)*(c^2 + d^2))), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d) 
*(a^2 + b^2))   Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Sim 
p[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2))   Int[(d - c*Tan[e + f* 
x])/(c + d*Tan[e + f*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]
 

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-3 a^{2} c^{2} d +a^{2} d^{3}-2 a b \,c^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d -b^{2} d^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (a^{2} c^{3}-3 a^{2} c \,d^{2}-6 a b \,c^{2} d +2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{3}}+\frac {b^{4}}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}-\frac {b^{4} \left (5 a^{2} d -2 a b c +3 b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (a^{2}+b^{2}\right )^{2}}-\frac {d^{3}}{2 \left (c^{2}+d^{2}\right ) \left (a d -b c \right )^{2} \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {d^{3} \left (3 a^{2} c^{2} d^{2}-a^{2} d^{4}-10 a b \,c^{3} d -2 a b c \,d^{3}+10 b^{2} c^{4}+9 b^{2} c^{2} d^{2}+3 b^{2} d^{4}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3} \left (a d -b c \right )^{4}}-\frac {2 d^{3} \left (a c d -2 b \,c^{2}-b \,d^{2}\right )}{\left (c^{2}+d^{2}\right )^{2} \left (a d -b c \right )^{3} \left (c +d \tan \left (f x +e \right )\right )}}{f}\)	\(419\)
default	\(\frac {\frac {\frac {\left (-3 a^{2} c^{2} d +a^{2} d^{3}-2 a b \,c^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d -b^{2} d^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (a^{2} c^{3}-3 a^{2} c \,d^{2}-6 a b \,c^{2} d +2 a b \,d^{3}-b^{2} c^{3}+3 b^{2} c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{3}}+\frac {b^{4}}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}-\frac {b^{4} \left (5 a^{2} d -2 a b c +3 b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (a^{2}+b^{2}\right )^{2}}-\frac {d^{3}}{2 \left (c^{2}+d^{2}\right ) \left (a d -b c \right )^{2} \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {d^{3} \left (3 a^{2} c^{2} d^{2}-a^{2} d^{4}-10 a b \,c^{3} d -2 a b c \,d^{3}+10 b^{2} c^{4}+9 b^{2} c^{2} d^{2}+3 b^{2} d^{4}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3} \left (a d -b c \right )^{4}}-\frac {2 d^{3} \left (a c d -2 b \,c^{2}-b \,d^{2}\right )}{\left (c^{2}+d^{2}\right )^{2} \left (a d -b c \right )^{3} \left (c +d \tan \left (f x +e \right )\right )}}{f}\)	\(419\)
norman	\(\text {Expression too large to display}\)	\(1840\)
risch	\(\text {Expression too large to display}\)	\(8456\)
parallelrisch	\(\text {Expression too large to display}\)	\(9321\)

Input:

int(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^3,x,method=_RETURNVERBOSE)
 

Output:

1/f*(1/(a^2+b^2)^2/(c^2+d^2)^3*(1/2*(-3*a^2*c^2*d+a^2*d^3-2*a*b*c^3+6*a*b* 
c*d^2+3*b^2*c^2*d-b^2*d^3)*ln(1+tan(f*x+e)^2)+(a^2*c^3-3*a^2*c*d^2-6*a*b*c 
^2*d+2*a*b*d^3-b^2*c^3+3*b^2*c*d^2)*arctan(tan(f*x+e)))+b^4/(a*d-b*c)^3/(a 
^2+b^2)/(a+b*tan(f*x+e))-b^4*(5*a^2*d-2*a*b*c+3*b^2*d)/(a*d-b*c)^4/(a^2+b^ 
2)^2*ln(a+b*tan(f*x+e))-1/2*d^3/(c^2+d^2)/(a*d-b*c)^2/(c+d*tan(f*x+e))^2+d 
^3*(3*a^2*c^2*d^2-a^2*d^4-10*a*b*c^3*d-2*a*b*c*d^3+10*b^2*c^4+9*b^2*c^2*d^ 
2+3*b^2*d^4)/(c^2+d^2)^3/(a*d-b*c)^4*ln(c+d*tan(f*x+e))-2*d^3*(a*c*d-2*b*c 
^2-b*d^2)/(c^2+d^2)^2/(a*d-b*c)^3/(c+d*tan(f*x+e)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4799 vs. \(2 (459) = 918\).

Time = 2.81 (sec) , antiderivative size = 4799, normalized size of antiderivative = 10.50 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=\text {Exception raised: NotImplementedError} \] Input:

integrate(1/(a+b*tan(f*x+e))**2/(c+d*tan(f*x+e))**3,x)
 

Output:

Exception raised: NotImplementedError >> no valid subset found
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1849 vs. \(2 (459) = 918\).

Time = 0.19 (sec) , antiderivative size = 1849, normalized size of antiderivative = 4.05 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^3,x, algorithm="maxima")
 

Output:

-1/2*(2*(6*a*b*c^2*d - 2*a*b*d^3 - (a^2 - b^2)*c^3 + 3*(a^2 - b^2)*c*d^2)* 
(f*x + e)/((a^4 + 2*a^2*b^2 + b^4)*c^6 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*d^2 
 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d^4 + (a^4 + 2*a^2*b^2 + b^4)*d^6) - 2*(2 
*a*b^5*c - (5*a^2*b^4 + 3*b^6)*d)*log(b*tan(f*x + e) + a)/((a^4*b^4 + 2*a^ 
2*b^6 + b^8)*c^4 - 4*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*c^3*d + 6*(a^6*b^2 + 2* 
a^4*b^4 + a^2*b^6)*c^2*d^2 - 4*(a^7*b + 2*a^5*b^3 + a^3*b^5)*c*d^3 + (a^8 
+ 2*a^6*b^2 + a^4*b^4)*d^4) - 2*(10*b^2*c^4*d^3 - 10*a*b*c^3*d^4 - 2*a*b*c 
*d^6 + 3*(a^2 + 3*b^2)*c^2*d^5 - (a^2 - 3*b^2)*d^7)*log(d*tan(f*x + e) + c 
)/(b^4*c^10 - 4*a*b^3*c^9*d - 4*a^3*b*c*d^9 + a^4*d^10 + 3*(2*a^2*b^2 + b^ 
4)*c^8*d^2 - 4*(a^3*b + 3*a*b^3)*c^7*d^3 + (a^4 + 18*a^2*b^2 + 3*b^4)*c^6* 
d^4 - 12*(a^3*b + a*b^3)*c^5*d^5 + (3*a^4 + 18*a^2*b^2 + b^4)*c^4*d^6 - 4* 
(3*a^3*b + a*b^3)*c^3*d^7 + 3*(a^4 + 2*a^2*b^2)*c^2*d^8) + (2*a*b*c^3 - 6* 
a*b*c*d^2 + 3*(a^2 - b^2)*c^2*d - (a^2 - b^2)*d^3)*log(tan(f*x + e)^2 + 1) 
/((a^4 + 2*a^2*b^2 + b^4)*c^6 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*d^2 + 3*(a^4 
 + 2*a^2*b^2 + b^4)*c^2*d^4 + (a^4 + 2*a^2*b^2 + b^4)*d^6) + (2*b^4*c^6 + 
4*b^4*c^4*d^2 + 9*(a^3*b + a*b^3)*c^3*d^3 - (5*a^4 + 5*a^2*b^2 - 2*b^4)*c^ 
2*d^4 + 5*(a^3*b + a*b^3)*c*d^5 - (a^4 + a^2*b^2)*d^6 + 2*(b^4*c^4*d^2 + 2 
*(2*a^2*b^2 + 3*b^4)*c^2*d^4 - 2*(a^3*b + a*b^3)*c*d^5 + (2*a^2*b^2 + 3*b^ 
4)*d^6)*tan(f*x + e)^2 + (4*b^4*c^5*d + (9*a^2*b^2 + 17*b^4)*c^3*d^3 + 3*( 
a^3*b + a*b^3)*c^2*d^4 - (4*a^4 - a^2*b^2 - 9*b^4)*c*d^5 + 3*(a^3*b + a...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2015 vs. \(2 (459) = 918\).

Time = 0.48 (sec) , antiderivative size = 2015, normalized size of antiderivative = 4.41 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^3,x, algorithm="giac")
 

Output:

(a^2*c^3 - b^2*c^3 - 6*a*b*c^2*d - 3*a^2*c*d^2 + 3*b^2*c*d^2 + 2*a*b*d^3)* 
(f*x + e)/(a^4*c^6*f + 2*a^2*b^2*c^6*f + b^4*c^6*f + 3*a^4*c^4*d^2*f + 6*a 
^2*b^2*c^4*d^2*f + 3*b^4*c^4*d^2*f + 3*a^4*c^2*d^4*f + 6*a^2*b^2*c^2*d^4*f 
 + 3*b^4*c^2*d^4*f + a^4*d^6*f + 2*a^2*b^2*d^6*f + b^4*d^6*f) - 1/2*(2*a*b 
*c^3 + 3*a^2*c^2*d - 3*b^2*c^2*d - 6*a*b*c*d^2 - a^2*d^3 + b^2*d^3)*log(ta 
n(f*x + e)^2 + 1)/(a^4*c^6*f + 2*a^2*b^2*c^6*f + b^4*c^6*f + 3*a^4*c^4*d^2 
*f + 6*a^2*b^2*c^4*d^2*f + 3*b^4*c^4*d^2*f + 3*a^4*c^2*d^4*f + 6*a^2*b^2*c 
^2*d^4*f + 3*b^4*c^2*d^4*f + a^4*d^6*f + 2*a^2*b^2*d^6*f + b^4*d^6*f) + (2 
*a*b^6*c - 5*a^2*b^5*d - 3*b^7*d)*log(abs(b*tan(f*x + e) + a))/(a^4*b^5*c^ 
4*f + 2*a^2*b^7*c^4*f + b^9*c^4*f - 4*a^5*b^4*c^3*d*f - 8*a^3*b^6*c^3*d*f 
- 4*a*b^8*c^3*d*f + 6*a^6*b^3*c^2*d^2*f + 12*a^4*b^5*c^2*d^2*f + 6*a^2*b^7 
*c^2*d^2*f - 4*a^7*b^2*c*d^3*f - 8*a^5*b^4*c*d^3*f - 4*a^3*b^6*c*d^3*f + a 
^8*b*d^4*f + 2*a^6*b^3*d^4*f + a^4*b^5*d^4*f) + (10*b^2*c^4*d^4 - 10*a*b*c 
^3*d^5 + 3*a^2*c^2*d^6 + 9*b^2*c^2*d^6 - 2*a*b*c*d^7 - a^2*d^8 + 3*b^2*d^8 
)*log(abs(d*tan(f*x + e) + c))/(b^4*c^10*d*f - 4*a*b^3*c^9*d^2*f + 6*a^2*b 
^2*c^8*d^3*f + 3*b^4*c^8*d^3*f - 4*a^3*b*c^7*d^4*f - 12*a*b^3*c^7*d^4*f + 
a^4*c^6*d^5*f + 18*a^2*b^2*c^6*d^5*f + 3*b^4*c^6*d^5*f - 12*a^3*b*c^5*d^6* 
f - 12*a*b^3*c^5*d^6*f + 3*a^4*c^4*d^7*f + 18*a^2*b^2*c^4*d^7*f + b^4*c^4* 
d^7*f - 12*a^3*b*c^3*d^8*f - 4*a*b^3*c^3*d^8*f + 3*a^4*c^2*d^9*f + 6*a^2*b 
^2*c^2*d^9*f - 4*a^3*b*c*d^10*f + a^4*d^11*f) - 1/2*(2*a^2*b^5*c^9 + 2*...
 

Mupad [B] (verification not implemented)

Time = 16.01 (sec) , antiderivative size = 1417, normalized size of antiderivative = 3.10 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx =\text {Too large to display} \] Input:

int(1/((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))^3),x)
 

Output:

((2*b^4*c^6 - a^4*d^6 - a^2*b^2*d^6 - 5*a^4*c^2*d^4 + 2*b^4*c^2*d^4 + 4*b^ 
4*c^4*d^2 + 9*a*b^3*c^3*d^3 + 9*a^3*b*c^3*d^3 - 5*a^2*b^2*c^2*d^4 + 5*a*b^ 
3*c*d^5 + 5*a^3*b*c*d^5)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c 
*d^2)*(a^2*c^4 + a^2*d^4 + b^2*c^4 + b^2*d^4 + 2*a^2*c^2*d^2 + 2*b^2*c^2*d 
^2)) + (tan(e + f*x)*(3*a*b^3*d^6 + 3*a^3*b*d^6 - 4*a^4*c*d^5 + 9*b^4*c*d^ 
5 + 4*b^4*c^5*d + 17*b^4*c^3*d^3 + 3*a*b^3*c^2*d^4 + a^2*b^2*c*d^5 + 3*a^3 
*b*c^2*d^4 + 9*a^2*b^2*c^3*d^3))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3 
*a^2*b*c*d^2)*(a^2*c^4 + a^2*d^4 + b^2*c^4 + b^2*d^4 + 2*a^2*c^2*d^2 + 2*b 
^2*c^2*d^2)) + (tan(e + f*x)^2*(3*b^4*d^6 + 2*a^2*b^2*d^6 + 6*b^4*c^2*d^4 
+ b^4*c^4*d^2 + 4*a^2*b^2*c^2*d^4 - 2*a*b^3*c*d^5 - 2*a^3*b*c*d^5))/((a^3* 
d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^2*c^4 + a^2*d^4 + b^2*c^ 
4 + b^2*d^4 + 2*a^2*c^2*d^2 + 2*b^2*c^2*d^2)))/(f*(tan(e + f*x)*(b*c^2 + 2 
*a*c*d) + a*c^2 + tan(e + f*x)^2*(a*d^2 + 2*b*c*d) + b*d^2*tan(e + f*x)^3) 
) - (log(tan(e + f*x) - 1i)*1i)/(2*f*(a^2*c^3 - a^2*d^3*1i - b^2*c^3 + b^2 
*d^3*1i - 3*a^2*c*d^2 + a^2*c^2*d*3i + 3*b^2*c*d^2 - b^2*c^2*d*3i + a*b*c^ 
3*2i + 2*a*b*d^3 - a*b*c*d^2*6i - 6*a*b*c^2*d)) + (log(tan(e + f*x) + 1i)* 
1i)/(2*f*(a^2*c^3 + a^2*d^3*1i - b^2*c^3 - b^2*d^3*1i - 3*a^2*c*d^2 - a^2* 
c^2*d*3i + 3*b^2*c*d^2 + b^2*c^2*d*3i - a*b*c^3*2i + 2*a*b*d^3 + a*b*c*d^2 
*6i - 6*a*b*c^2*d)) - (log(a + b*tan(e + f*x))*(d*(3*b^6 + 5*a^2*b^4) - 2* 
a*b^5*c))/(f*(a^8*d^4 + b^8*c^4 + 2*a^2*b^6*c^4 + a^4*b^4*c^4 + a^4*b^4...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 14325, normalized size of antiderivative = 31.35 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx =\text {Too large to display} \] Input:

int(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^3,x)
 

Output:

( - 3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**7*b*c**2*d**8 + log(tan( 
e + f*x)**2 + 1)*tan(e + f*x)**3*a**7*b*d**10 + 4*log(tan(e + f*x)**2 + 1) 
*tan(e + f*x)**3*a**6*b**2*c**3*d**7 + 4*log(tan(e + f*x)**2 + 1)*tan(e + 
f*x)**3*a**6*b**2*c*d**9 + 10*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a** 
5*b**3*c**4*d**6 - 11*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**5*b**3*c 
**2*d**8 - log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**5*b**3*d**10 - 20*l 
og(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**4*b**4*c**5*d**5 - 10*log(tan(e 
 + f*x)**2 + 1)*tan(e + f*x)**3*a**4*b**4*c**3*d**7 + 2*log(tan(e + f*x)** 
2 + 1)*tan(e + f*x)**3*a**4*b**4*c*d**9 + 5*log(tan(e + f*x)**2 + 1)*tan(e 
 + f*x)**3*a**3*b**5*c**6*d**4 + 35*log(tan(e + f*x)**2 + 1)*tan(e + f*x)* 
*3*a**3*b**5*c**4*d**6 + 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**3*b 
**5*c**2*d**8 + 8*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**2*b**6*c**7* 
d**3 - 16*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**2*b**6*c**5*d**5 - 8 
*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**2*b**6*c**3*d**7 - 4*log(tan( 
e + f*x)**2 + 1)*tan(e + f*x)**3*a*b**7*c**8*d**2 - 9*log(tan(e + f*x)**2 
+ 1)*tan(e + f*x)**3*a*b**7*c**6*d**4 + 7*log(tan(e + f*x)**2 + 1)*tan(e + 
 f*x)**3*a*b**7*c**4*d**6 + 6*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*b** 
8*c**7*d**3 - 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*b**8*c**5*d**5 - 
3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2*a**8*c**2*d**8 + log(tan(e + f* 
x)**2 + 1)*tan(e + f*x)**2*a**8*d**10 - 2*log(tan(e + f*x)**2 + 1)*tan(...