\(\int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx\) [1222]

Optimal result

Integrand size = 25, antiderivative size = 457 \[ \int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=-\frac {\left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )^2}-\frac {b^3 \left (10 a^3 b c d+2 a b^3 c d-10 a^4 d^2+b^4 \left (c^2-3 d^2\right )-3 a^2 b^2 \left (c^2+3 d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 (b c-a d)^4 f}-\frac {d^4 \left (5 b c^2-2 a c d+3 b d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^4 \left (c^2+d^2\right )^2 f}+\frac {d \left (a^4 d^3-2 a b^3 c \left (c^2+d^2\right )+2 a^2 b^2 d \left (2 c^2+3 d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{\left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {b^2 \left (4 a b c-7 a^2 d-3 b^2 d\right )}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))} \] Output:

-(6*a^2*b*c*d-2*b^3*c*d-a^3*(c^2-d^2)+3*a*b^2*(c^2-d^2))*x/(a^2+b^2)^3/(c^ 
2+d^2)^2-b^3*(10*a^3*b*c*d+2*a*b^3*c*d-10*a^4*d^2+b^4*(c^2-3*d^2)-3*a^2*b^ 
2*(c^2+3*d^2))*ln(a*cos(f*x+e)+b*sin(f*x+e))/(a^2+b^2)^3/(-a*d+b*c)^4/f-d^ 
4*(-2*a*c*d+5*b*c^2+3*b*d^2)*ln(c*cos(f*x+e)+d*sin(f*x+e))/(-a*d+b*c)^4/(c 
^2+d^2)^2/f+d*(a^4*d^3-2*a*b^3*c*(c^2+d^2)+2*a^2*b^2*d*(2*c^2+3*d^2)+b^4*d 
*(2*c^2+3*d^2))/(a^2+b^2)^2/(-a*d+b*c)^3/(c^2+d^2)/f/(c+d*tan(f*x+e))-1/2* 
b^2/(a^2+b^2)/(-a*d+b*c)/f/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))-1/2*b^2*(-7 
*a^2*d+4*a*b*c-3*b^2*d)/(a^2+b^2)^2/(-a*d+b*c)^2/f/(a+b*tan(f*x+e))/(c+d*t 
an(f*x+e))
 

Mathematica [A] (verified)

Time = 6.86 (sec) , antiderivative size = 840, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=-\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {-\frac {b^2 \left (-2 a b c+2 a^2 d+3 b^2 d\right )-a \left (-3 a b^2 d+2 b^2 (b c-a d)\right )}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {-\frac {-\frac {b (b c-a d)^3 \left (3 a^2 b c^2-b^3 c^2+2 a^3 c d-6 a b^2 c d-3 a^2 b d^2+b^3 d^2-\frac {\sqrt {-b^2} \left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 b^4 \left (c^2+d^2\right ) \left (10 a^3 b c d+2 a b^3 c d-10 a^4 d^2+b^4 \left (c^2-3 d^2\right )-3 a^2 b^2 \left (c^2+3 d^2\right )\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b (b c-a d)^3 \left (3 a^2 b c^2-b^3 c^2+2 a^3 c d-6 a b^2 c d-3 a^2 b d^2+b^3 d^2+\frac {\sqrt {-b^2} \left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 b \left (a^2+b^2\right )^2 d^4 \left (5 b c^2-2 a c d+3 b d^2\right ) \log (c+d \tan (e+f x))}{(b c-a d) \left (c^2+d^2\right )}}{b (-b c+a d) \left (c^2+d^2\right ) f}-\frac {-c \left (-4 a b d (b c-a d)^2+2 b^2 c d \left (4 a b c-7 a^2 d-3 b^2 d\right )\right )-2 d^2 \left (2 a^3 b c d+2 a b^3 c d-a^4 d^2+b^4 \left (c^2-3 d^2\right )-a^2 b^2 \left (c^2+6 d^2\right )\right )}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)} \] Input:

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

Output:

-1/2*b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + 
f*x])) - (-((b^2*(-2*a*b*c + 2*a^2*d + 3*b^2*d) - a*(-3*a*b^2*d + 2*b^2*(b 
*c - a*d)))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + 
 f*x]))) - (-((-((b*(b*c - a*d)^3*(3*a^2*b*c^2 - b^3*c^2 + 2*a^3*c*d - 6*a 
*b^2*c*d - 3*a^2*b*d^2 + b^3*d^2 - (Sqrt[-b^2]*(6*a^2*b*c*d - 2*b^3*c*d - 
a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2)))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x] 
])/((a^2 + b^2)*(c^2 + d^2))) - (2*b^4*(c^2 + d^2)*(10*a^3*b*c*d + 2*a*b^3 
*c*d - 10*a^4*d^2 + b^4*(c^2 - 3*d^2) - 3*a^2*b^2*(c^2 + 3*d^2))*Log[a + b 
*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) - (b*(b*c - a*d)^3*(3*a^2*b*c^2 
- b^3*c^2 + 2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*d^2 + b^3*d^2 + (Sqrt[-b^2]* 
(6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2)))/b)*Log[ 
Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) - (2*b*(a^2 + b^2) 
^2*d^4*(5*b*c^2 - 2*a*c*d + 3*b*d^2)*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)) - (-(c*(-4*a*b*d*(b*c - a 
*d)^2 + 2*b^2*c*d*(4*a*b*c - 7*a^2*d - 3*b^2*d))) - 2*d^2*(2*a^3*b*c*d + 2 
*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 3*d^2) - a^2*b^2*(c^2 + 6*d^2)))/((-(b*c 
) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/((a^2 + b^2)*(b*c - a*d)))/( 
2*(a^2 + b^2)*(b*c - a*d))
 

Rubi [A] (verified)

Time = 3.17 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.18, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4134

\(\displaystyle \frac {-\frac {2 \left (\frac {\frac {d^4 \left (a^2+b^2\right )^2 \left (-2 a c d+5 b c^2+3 b d^2\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {b^3 \left (c^2+d^2\right ) \left (-10 a^4 d^2+10 a^3 b c d-3 a^2 b^2 \left (c^2+3 d^2\right )+2 a b^3 c d+b^4 \left (c^2-3 d^2\right )\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {x (b c-a d)^3 \left (-\left (a^3 \left (c^2-d^2\right )\right )+6 a^2 b c d+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (2 c^2+3 d^2\right )-2 a b^3 c \left (c^2+d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\right )}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2 \left (-7 a^2 d+4 a b c-3 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 \left (\frac {\frac {d^4 \left (a^2+b^2\right )^2 \left (-2 a c d+5 b c^2+3 b d^2\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{\left (c^2+d^2\right ) (b c-a d)}+\frac {b^3 \left (c^2+d^2\right ) \left (-10 a^4 d^2+10 a^3 b c d-3 a^2 b^2 \left (c^2+3 d^2\right )+2 a b^3 c d+b^4 \left (c^2-3 d^2\right )\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {x (b c-a d)^3 \left (-\left (a^3 \left (c^2-d^2\right )\right )+6 a^2 b c d+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (2 c^2+3 d^2\right )-2 a b^3 c \left (c^2+d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\right )}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b^2 \left (-7 a^2 d+4 a b c-3 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4013

\(\displaystyle \frac {-\frac {b^2 \left (-7 a^2 d+4 a b c-3 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {2 \left (\frac {\frac {d^4 \left (a^2+b^2\right )^2 \left (-2 a c d+5 b c^2+3 b d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)}+\frac {x (b c-a d)^3 \left (-\left (a^3 \left (c^2-d^2\right )\right )+6 a^2 b c d+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac {b^3 \left (c^2+d^2\right ) \left (-10 a^4 d^2+10 a^3 b c d-3 a^2 b^2 \left (c^2+3 d^2\right )+2 a b^3 c d+b^4 \left (c^2-3 d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (2 c^2+3 d^2\right )-2 a b^3 c \left (c^2+d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}\)

Input:

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

Output:

-1/2*b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + 
f*x])) + (-((b^2*(4*a*b*c - 7*a^2*d - 3*b^2*d))/((a^2 + b^2)*(b*c - a*d)*f 
*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))) - (2*((((b*c - a*d)^3*(6*a^2* 
b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2))*x)/((a^2 + b^2) 
*(c^2 + d^2)) + (b^3*(c^2 + d^2)*(10*a^3*b*c*d + 2*a*b^3*c*d - 10*a^4*d^2 
+ b^4*(c^2 - 3*d^2) - 3*a^2*b^2*(c^2 + 3*d^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)^2*d^4*(5*b*c^2 - 2*a 
*c*d + 3*b*d^2)*Log[c*Cos[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*(a^4*d^3 - 2*a*b^3*c*(c^2 + d^2) + 
 2*a^2*b^2*d*(2*c^2 + 3*d^2) + b^4*d*(2*c^2 + 3*d^2)))/((b*c - a*d)*(c^2 + 
 d^2)*f*(c + d*Tan[e + f*x]))))/((a^2 + b^2)*(b*c - a*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 = 1.88 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-2 a^{3} c d -3 a^{2} b \,c^{2}+3 a^{2} b \,d^{2}+6 a \,b^{2} c d +b^{3} c^{2}-b^{3} d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {d^{4}}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{4} \left (2 a c d -5 b \,c^{2}-3 b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (c^{2}+d^{2}\right )^{2}}-\frac {b^{3}}{2 \left (a^{2}+b^{2}\right ) \left (a d -b c \right )^{2} \left (a +b \tan \left (f x +e \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4} d^{2}-10 a^{3} b c d +3 a^{2} b^{2} c^{2}+9 a^{2} b^{2} d^{2}-2 a \,b^{3} c d -b^{4} c^{2}+3 b^{4} d^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a d -b c \right )^{4}}-\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right )}{\left (a^{2}+b^{2}\right )^{2} \left (a d -b c \right )^{3} \left (a +b \tan \left (f x +e \right )\right )}}{f}\)	\(419\)
default	\(\frac {\frac {\frac {\left (-2 a^{3} c d -3 a^{2} b \,c^{2}+3 a^{2} b \,d^{2}+6 a \,b^{2} c d +b^{3} c^{2}-b^{3} d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {d^{4}}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{4} \left (2 a c d -5 b \,c^{2}-3 b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (c^{2}+d^{2}\right )^{2}}-\frac {b^{3}}{2 \left (a^{2}+b^{2}\right ) \left (a d -b c \right )^{2} \left (a +b \tan \left (f x +e \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4} d^{2}-10 a^{3} b c d +3 a^{2} b^{2} c^{2}+9 a^{2} b^{2} d^{2}-2 a \,b^{3} c d -b^{4} c^{2}+3 b^{4} d^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a d -b c \right )^{4}}-\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right )}{\left (a^{2}+b^{2}\right )^{2} \left (a d -b c \right )^{3} \left (a +b \tan \left (f x +e \right )\right )}}{f}\)	\(419\)
norman	\(\text {Expression too large to display}\)	\(1676\)
risch	\(\text {Expression too large to display}\)	\(8456\)
parallelrisch	\(\text {Expression too large to display}\)	\(9339\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4749 vs. \(2 (453) = 906\).

Time = 2.12 (sec) , antiderivative size = 4749, normalized size of antiderivative = 10.39 \[ \int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-2)]

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

integrate(1/(a+b*tan(f*x+e))**3/(c+d*tan(f*x+e))**2,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. 1833 vs. \(2 (453) = 906\).

Time = 0.19 (sec) , antiderivative size = 1833, normalized size of antiderivative = 4.01 \[ \int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2016 vs. \(2 (453) = 906\).

Time = 0.46 (sec) , antiderivative size = 2016, normalized size of antiderivative = 4.41 \[ \int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 31.47 (sec) , antiderivative size = 1421, normalized size of antiderivative = 3.11 \[ \int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 14325, normalized size of antiderivative = 31.35 \[ \int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**8*b**2*c*d**7 + 8*log(ta 
n(e + f*x)**2 + 1)*tan(e + f*x)**3*a**7*b**3*c**2*d**6 + 6*log(tan(e + f*x 
)**2 + 1)*tan(e + f*x)**3*a**7*b**3*d**8 + 5*log(tan(e + f*x)**2 + 1)*tan( 
e + f*x)**3*a**6*b**4*c**3*d**5 - 9*log(tan(e + f*x)**2 + 1)*tan(e + f*x)* 
*3*a**6*b**4*c*d**7 - 20*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**5*b** 
5*c**4*d**4 - 16*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**5*b**5*c**2*d 
**6 - 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**5*b**5*d**8 + 10*log(t 
an(e + f*x)**2 + 1)*tan(e + f*x)**3*a**4*b**6*c**5*d**3 + 35*log(tan(e + f 
*x)**2 + 1)*tan(e + f*x)**3*a**4*b**6*c**3*d**5 + 7*log(tan(e + f*x)**2 + 
1)*tan(e + f*x)**3*a**4*b**6*c*d**7 + 4*log(tan(e + f*x)**2 + 1)*tan(e + f 
*x)**3*a**3*b**7*c**6*d**2 - 10*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a 
**3*b**7*c**4*d**4 - 8*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**3*b**7* 
c**2*d**6 - 3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**2*b**8*c**7*d - 
11*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*a**2*b**8*c**5*d**3 + 2*log(ta 
n(e + f*x)**2 + 1)*tan(e + f*x)**3*a**2*b**8*c**3*d**5 + 4*log(tan(e + f*x 
)**2 + 1)*tan(e + f*x)**3*a*b**9*c**6*d**2 + 2*log(tan(e + f*x)**2 + 1)*ta 
n(e + f*x)**3*a*b**9*c**4*d**4 + log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3* 
b**10*c**7*d - log(tan(e + f*x)**2 + 1)*tan(e + f*x)**3*b**10*c**5*d**3 - 
8*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2*a**9*b*c*d**7 + 12*log(tan(e + 
f*x)**2 + 1)*tan(e + f*x)**2*a**8*b**2*c**2*d**6 + 12*log(tan(e + f*x)*...