\(\int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx\) [1221]

Optimal result

Integrand size = 25, antiderivative size = 290 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\frac {(b (c-d)+a (c+d)) (a (c-d)-b (c+d)) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}+\frac {2 b^3 \left (a b c-2 a^2 d-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^2 (b c-a d)^3 f}-\frac {2 d^3 \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right )^2 f}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))} \] Output:

(b*(c-d)+a*(c+d))*(a*(c-d)-b*(c+d))*x/(a^2+b^2)^2/(c^2+d^2)^2+2*b^3*(-2*a^ 
2*d+a*b*c-b^2*d)*ln(a*cos(f*x+e)+b*sin(f*x+e))/(a^2+b^2)^2/(-a*d+b*c)^3/f- 
2*d^3*(a*c*d-b*(2*c^2+d^2))*ln(c*cos(f*x+e)+d*sin(f*x+e))/(-a*d+b*c)^3/(c^ 
2+d^2)^2/f-d*(a^2*d^2+b^2*(c^2+2*d^2))/(a^2+b^2)/(-a*d+b*c)^2/(c^2+d^2)/f/ 
(c+d*tan(f*x+e))-b^2/(a^2+b^2)/(-a*d+b*c)/f/(a+b*tan(f*x+e))/(c+d*tan(f*x+ 
e))
 

Mathematica [A] (verified)

Time = 6.18 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\frac {-\frac {(b c-a d)^3 \left (2 a b c^2+2 a^2 c d-2 b^2 c d-2 a b d^2+\frac {b \left (4 a b c d+b^2 \left (c^2-d^2\right )+a^2 \left (-c^2+d^2\right )\right )}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )+4 b^3 \left (-a b c+2 a^2 d+b^2 d\right ) \left (c^2+d^2\right )^2 \log (a+b \tan (e+f x))+(b c-a d)^3 \left (2 a b c^2+2 a^2 c d-2 b^2 c d-2 a b d^2+\frac {\sqrt {-b^2} \left (4 a b c d+b^2 \left (c^2-d^2\right )+a^2 \left (-c^2+d^2\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )+4 \left (a^2+b^2\right )^2 d^3 \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c+d \tan (e+f x))}{2 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right )^2}+\frac {a^2 d^3+b^2 d \left (c^2+2 d^2\right )}{(-b c+a d) \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac {b^2}{(a+b \tan (e+f x)) (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d) f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.80 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4052, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4013

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

Input:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 = 0.62 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {-\frac {b^{3}}{\left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}+\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right )^{2}}-\frac {d^{3}}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {2 d^{3} \left (a c d -2 b \,c^{2}-b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right )^{2}}+\frac {\frac {\left (-2 a^{2} c d -2 a b \,c^{2}+2 a b \,d^{2}+2 b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}}{f}\)	\(289\)
default	\(\frac {-\frac {b^{3}}{\left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}+\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right )^{2}}-\frac {d^{3}}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {2 d^{3} \left (a c d -2 b \,c^{2}-b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right )^{2}}+\frac {\frac {\left (-2 a^{2} c d -2 a b \,c^{2}+2 a b \,d^{2}+2 b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}}{f}\)	\(289\)
norman	\(\frac {\frac {\left (a d +b c \right ) \left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) x \tan \left (f x +e \right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) a c x}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b d \left (a^{2} c^{2}-a^{2} d^{2}-4 a b c d -b^{2} c^{2}+b^{2} d^{2}\right ) x \tan \left (f x +e \right )^{2}}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (a^{4} d^{4}+a^{2} b^{2} d^{4}+b^{4} c^{4}+b^{4} c^{2} d^{2}\right ) \tan \left (f x +e \right )}{f c a \left (c^{2}+d^{2}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a^{2}+b^{2}\right )}+\frac {b d \left (a^{3} d^{3}+a \,b^{2} d^{3}+b^{3} c^{3}+b^{3} c \,d^{2}\right ) \tan \left (f x +e \right )^{2}}{f c a \left (c^{2}+d^{2}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}-\frac {\left (a^{2} c d +a b \,c^{2}-a b \,d^{2}-b^{2} c d \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{f \left (a^{4} c^{4}+2 a^{4} c^{2} d^{2}+a^{4} d^{4}+2 a^{2} b^{2} c^{4}+4 a^{2} b^{2} c^{2} d^{2}+2 a^{2} b^{2} d^{4}+b^{4} c^{4}+2 b^{4} c^{2} d^{2}+b^{4} d^{4}\right )}+\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) f}+\frac {2 d^{3} \left (a c d -2 b \,c^{2}-b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}\)	\(770\)
risch	\(\text {Expression too large to display}\)	\(3046\)
parallelrisch	\(\text {Expression too large to display}\)	\(4238\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2217 vs. \(2 (291) = 582\).

Time = 0.87 (sec) , antiderivative size = 2217, normalized size of antiderivative = 7.64 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-2)]

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

integrate(1/(a+b*tan(f*x+e))**2/(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. 878 vs. \(2 (291) = 582\).

Time = 0.16 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.03 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (291) = 582\).

Time = 0.31 (sec) , antiderivative size = 910, normalized size of antiderivative = 3.14 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 7.37 (sec) , antiderivative size = 725, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d\,\left (4\,a^2\,b^3+2\,b^5\right )-2\,a\,b^4\,c\right )}{f\,\left (a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d+2\,a^5\,b^2\,d^3-a^4\,b^3\,c^3-6\,a^4\,b^3\,c\,d^2+6\,a^3\,b^4\,c^2\,d+a^3\,b^4\,d^3-2\,a^2\,b^5\,c^3-3\,a^2\,b^5\,c\,d^2+3\,a\,b^6\,c^2\,d-b^7\,c^3\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}{2\,f\,\left (-a^2\,c^2\,1{}\mathrm {i}+2\,a^2\,c\,d+a^2\,d^2\,1{}\mathrm {i}+2\,a\,b\,c^2+a\,b\,c\,d\,4{}\mathrm {i}-2\,a\,b\,d^2+b^2\,c^2\,1{}\mathrm {i}-2\,b^2\,c\,d-b^2\,d^2\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{2\,f\,\left (-a^2\,c^2\,1{}\mathrm {i}-2\,a^2\,c\,d+a^2\,d^2\,1{}\mathrm {i}-2\,a\,b\,c^2+a\,b\,c\,d\,4{}\mathrm {i}+2\,a\,b\,d^2+b^2\,c^2\,1{}\mathrm {i}+2\,b^2\,c\,d-b^2\,d^2\,1{}\mathrm {i}\right )}-\frac {\frac {a^3\,d^3+a\,b^2\,d^3+b^3\,c^3+b^3\,c\,d^2}{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,c^2+a^2\,d^2+b^2\,c^2+b^2\,d^2\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b\,d^3+b^3\,c^2\,d+2\,b^3\,d^3\right )}{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,c^2+a^2\,d^2+b^2\,c^2+b^2\,d^2\right )}}{f\,\left (b\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2+\left (a\,d+b\,c\right )\,\mathrm {tan}\left (e+f\,x\right )+a\,c\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (b\,\left (4\,c^2\,d^3+2\,d^5\right )-2\,a\,c\,d^4\right )}{f\,\left (a^3\,c^4\,d^3+2\,a^3\,c^2\,d^5+a^3\,d^7-3\,a^2\,b\,c^5\,d^2-6\,a^2\,b\,c^3\,d^4-3\,a^2\,b\,c\,d^6+3\,a\,b^2\,c^6\,d+6\,a\,b^2\,c^4\,d^3+3\,a\,b^2\,c^2\,d^5-b^3\,c^7-2\,b^3\,c^5\,d^2-b^3\,c^3\,d^4\right )} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 6247, normalized size of antiderivative = 21.54 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2*a**6*b*c*d**6 + log(tan(e + f 
*x)**2 + 1)*tan(e + f*x)**2*a**5*b**2*c**2*d**5 + log(tan(e + f*x)**2 + 1) 
*tan(e + f*x)**2*a**5*b**2*d**7 + 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)* 
*2*a**4*b**3*c**3*d**4 - log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2*a**4*b** 
3*c*d**6 - 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2*a**3*b**4*c**4*d**3 
- 2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2*a**3*b**4*c**2*d**5 - log(tan 
(e + f*x)**2 + 1)*tan(e + f*x)**2*a**2*b**5*c**5*d**2 + 2*log(tan(e + f*x) 
**2 + 1)*tan(e + f*x)**2*a**2*b**5*c**3*d**4 + log(tan(e + f*x)**2 + 1)*ta 
n(e + f*x)**2*a*b**6*c**6*d + log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2*a*b 
**6*c**4*d**3 - log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2*b**7*c**5*d**2 - 
log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**7*c*d**6 + log(tan(e + f*x)**2 + 
1)*tan(e + f*x)*a**6*b*d**7 + 3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**5 
*b**2*c**3*d**4 - 3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**4*b**3*c**2*d 
**5 - 3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)*a**3*b**4*c**5*d**2 + 3*log( 
tan(e + f*x)**2 + 1)*tan(e + f*x)*a**2*b**5*c**4*d**3 + log(tan(e + f*x)** 
2 + 1)*tan(e + f*x)*a*b**6*c**7 - log(tan(e + f*x)**2 + 1)*tan(e + f*x)*b* 
*7*c**6*d - log(tan(e + f*x)**2 + 1)*a**7*c**2*d**5 + log(tan(e + f*x)**2 
+ 1)*a**6*b*c**3*d**4 + log(tan(e + f*x)**2 + 1)*a**6*b*c*d**6 + 2*log(tan 
(e + f*x)**2 + 1)*a**5*b**2*c**4*d**3 - log(tan(e + f*x)**2 + 1)*a**5*b**2 
*c**2*d**5 - 2*log(tan(e + f*x)**2 + 1)*a**4*b**3*c**5*d**2 - 2*log(tan...