\(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [485]

Optimal result

Integrand size = 21, antiderivative size = 211 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {3 b \log (\sin (c+d x))}{a^4 d}+\frac {b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.49 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.84 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \cot (c+d x)}{a^3}+\frac {b^5}{a^4 \left (a^2+b^2\right ) (b+a \cot (c+d x))^2}-\frac {2 b^4 \left (5 a^2+3 b^2\right )}{a^4 \left (a^2+b^2\right )^2 (b+a \cot (c+d x))}+\frac {\log (i-\cot (c+d x))}{(i a+b)^3}+\frac {\log (i+\cot (c+d x))}{(-i a+b)^3}-\frac {2 b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (b+a \cot (c+d x))}{a^4 \left (a^2+b^2\right )^3}}{2 d} \] Input:

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

Output:

-1/2*((2*Cot[c + d*x])/a^3 + b^5/(a^4*(a^2 + b^2)*(b + a*Cot[c + d*x])^2) 
- (2*b^4*(5*a^2 + 3*b^2))/(a^4*(a^2 + b^2)^2*(b + a*Cot[c + d*x])) + Log[I 
 - Cot[c + d*x]]/(I*a + b)^3 + Log[I + Cot[c + d*x]]/((-I)*a + b)^3 - (2*b 
^3*(10*a^4 + 9*a^2*b^2 + 3*b^4)*Log[b + a*Cot[c + d*x]])/(a^4*(a^2 + b^2)^ 
3))/d
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.22, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 4052, 3042, 4132, 27, 3042, 4132, 3042, 4134, 3042, 25, 3956, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {-\frac {3 b \left (a^2+b^2\right )^2 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^4 x \left (a^2-3 b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^2+3 b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 4013

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

Input:

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

Output:

-(Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^2)) - ((b*(2*a^2 + 3*b^2))/(2*a*( 
a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((a^4*(a^2 - 3*b^2)*x)/(a^2 + b^2) 
 + (3*b*(a^2 + b^2)^2*Log[-Sin[c + d*x]])/(a*d) - (b^3*(10*a^4 + 9*a^2*b^2 
 + 3*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a*(a^2 
 + b^2)) + (b*(a^4 + 6*a^2*b^2 + 3*b^4))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d 
*x])))/(a*(a^2 + b^2)))/a
 

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 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, 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 = 1.98 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {-\frac {b^{3}}{2 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 b^{2} a^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 b^{3} \left (2 a^{2}+b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {1}{a^{3} \tan \left (d x +c \right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}}{d}\)	\(201\)
default	\(\frac {-\frac {b^{3}}{2 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 b^{2} a^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 b^{3} \left (2 a^{2}+b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {1}{a^{3} \tan \left (d x +c \right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}}{d}\)	\(201\)
parallelrisch	\(\frac {20 b^{3} \left (a^{4}+\frac {9}{10} b^{2} a^{2}+\frac {3}{10} b^{4}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (a +b \tan \left (d x +c \right )\right )+3 b \,a^{4} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (a^{2}-\frac {b^{2}}{3}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )-6 b \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\tan \left (d x +c \right )\right )+\left (-2 x \,a^{7} b^{2} d +6 x \,a^{5} b^{4} d +4 a^{6} b^{3}+21 a^{4} b^{5}+26 a^{2} b^{7}+9 b^{9}\right ) \tan \left (d x +c \right )^{2}+\left (-4 x \,a^{8} b d +12 x \,a^{6} b^{3} d +6 a^{7} b^{2}+28 a^{5} b^{4}+34 a^{3} b^{6}+12 b^{8} a \right ) \tan \left (d x +c \right )-2 \left (\left (a^{2}+b^{2}\right )^{3} \cot \left (d x +c \right )+a^{4} d x \left (a^{2}-3 b^{2}\right )\right ) a^{3}}{2 \left (a^{2}+b^{2}\right )^{3} d \,a^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}\)	\(295\)
norman	\(\frac {\frac {b \left (3 a^{4} b +11 a^{2} b^{3}+6 b^{5}\right ) \tan \left (d x +c \right )^{2}}{d \,a^{3} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {1}{a d}+\frac {b^{2} \left (4 a^{4} b +17 a^{2} b^{3}+9 b^{5}\right ) \tan \left (d x +c \right )^{3}}{2 d \,a^{4} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {\left (a^{2}-3 b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 b \left (a^{2}-3 b^{2}\right ) a^{2} x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (a^{2}-3 b^{2}\right ) a x \tan \left (d x +c \right )^{3}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 b^{2} a^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(422\)
risch	\(\frac {x}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {6 i b x}{a^{4}}+\frac {6 i b c}{a^{4} d}-\frac {20 i b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {20 i b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {18 i b^{5} x}{a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {18 i b^{5} c}{a^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i b^{7} x}{a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i b^{7} c}{a^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i \left (a^{7} {\mathrm e}^{4 i \left (d x +c \right )}+2 a^{7} {\mathrm e}^{2 i \left (d x +c \right )}+a^{7}-15 i a^{2} b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+3 a \,b^{6}+3 i a^{4} b^{3}+i a^{6} b -6 i b^{7} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{3} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-5 i a^{4} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+8 a^{3} b^{4}+3 i b^{7}-3 i a^{6} b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i b^{7} {\mathrm e}^{4 i \left (d x +c \right )}+8 i a^{2} b^{5}+3 a^{5} b^{2}-6 i a^{4} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a^{2} b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-a^{5} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-10 a^{3} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-6 a \,b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{5} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 i a^{6} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 a \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} \left (-i b +a \right )^{3} a^{3} d}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}+\frac {10 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {9 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{4} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(862\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (209) = 418\).

Time = 0.13 (sec) , antiderivative size = 585, normalized size of antiderivative = 2.77 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {2 \, a^{9} + 6 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 2 \, a^{3} b^{6} - {\left (9 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - 2 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{7} b^{2} - 2 \, a^{5} b^{4} + 6 \, a^{3} b^{6} + 3 \, a b^{8} + 2 \, {\left (a^{8} b - 3 \, a^{6} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left ({\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left ({\left (10 \, a^{4} b^{5} + 9 \, a^{2} b^{7} + 3 \, b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (10 \, a^{5} b^{4} + 9 \, a^{3} b^{6} + 3 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (10 \, a^{6} b^{3} + 9 \, a^{4} b^{5} + 3 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (4 \, a^{8} b + 12 \, a^{6} b^{3} + 23 \, a^{4} b^{5} + 9 \, a^{2} b^{7} + 2 \, {\left (a^{9} - 3 \, a^{7} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} d \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6}\right )} d \tan \left (d x + c\right )\right )}} \] Input:

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

Output:

-1/2*(2*a^9 + 6*a^7*b^2 + 6*a^5*b^4 + 2*a^3*b^6 - (9*a^4*b^5 + 3*a^2*b^7 - 
 2*(a^7*b^2 - 3*a^5*b^4)*d*x)*tan(d*x + c)^3 + 2*(a^7*b^2 - 2*a^5*b^4 + 6* 
a^3*b^6 + 3*a*b^8 + 2*(a^8*b - 3*a^6*b^3)*d*x)*tan(d*x + c)^2 + 3*((a^6*b^ 
3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*tan(d*x + c)^3 + 2*(a^7*b^2 + 3*a^5*b^4 + 
 3*a^3*b^6 + a*b^8)*tan(d*x + c)^2 + (a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2* 
b^7)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - ((10*a^4*b^5 
 + 9*a^2*b^7 + 3*b^9)*tan(d*x + c)^3 + 2*(10*a^5*b^4 + 9*a^3*b^6 + 3*a*b^8 
)*tan(d*x + c)^2 + (10*a^6*b^3 + 9*a^4*b^5 + 3*a^2*b^7)*tan(d*x + c))*log( 
(b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) + (4 
*a^8*b + 12*a^6*b^3 + 23*a^4*b^5 + 9*a^2*b^7 + 2*(a^9 - 3*a^7*b^2)*d*x)*ta 
n(d*x + c))/((a^10*b^2 + 3*a^8*b^4 + 3*a^6*b^6 + a^4*b^8)*d*tan(d*x + c)^3 
 + 2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d*tan(d*x + c)^2 + (a^12 + 
 3*a^10*b^2 + 3*a^8*b^4 + a^6*b^6)*d*tan(d*x + c))
 

Sympy [F(-2)]

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

integrate(cot(d*x+c)**2/(a+b*tan(d*x+c))**3,x)
 

Output:

Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr 
imitive'
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.65 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (10 \, a^{4} b^{3} + 9 \, a^{2} b^{5} + 3 \, b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, a^{6} + 4 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + 2 \, {\left (a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, a^{5} b + 17 \, a^{3} b^{3} + 9 \, a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \tan \left (d x + c\right )} + \frac {6 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.60 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {{\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} + \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} + \frac {{\left (10 \, a^{4} b^{4} + 9 \, a^{2} b^{6} + 3 \, b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b d + 3 \, a^{8} b^{3} d + 3 \, a^{6} b^{5} d + a^{4} b^{7} d} - \frac {3 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4} d} - \frac {2 \, a^{9} + 6 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 2 \, a^{3} b^{6} + 2 \, {\left (a^{7} b^{2} + 7 \, a^{5} b^{4} + 9 \, a^{3} b^{6} + 3 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, a^{8} b + 21 \, a^{6} b^{3} + 26 \, a^{4} b^{5} + 9 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{4} d \tan \left (d x + c\right )} \] Input:

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

Output:

-(a^3 - 3*a*b^2)*(d*x + c)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4*d + b^6*d) + 1 
/2*(3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^ 
4*d + b^6*d) + (10*a^4*b^4 + 9*a^2*b^6 + 3*b^8)*log(abs(b*tan(d*x + c) + a 
))/(a^10*b*d + 3*a^8*b^3*d + 3*a^6*b^5*d + a^4*b^7*d) - 3*b*log(abs(tan(d* 
x + c)))/(a^4*d) - 1/2*(2*a^9 + 6*a^7*b^2 + 6*a^5*b^4 + 2*a^3*b^6 + 2*(a^7 
*b^2 + 7*a^5*b^4 + 9*a^3*b^6 + 3*a*b^8)*tan(d*x + c)^2 + (4*a^8*b + 21*a^6 
*b^3 + 26*a^4*b^5 + 9*a^2*b^7)*tan(d*x + c))/((a^2 + b^2)^3*(b*tan(d*x + c 
) + a)^2*a^4*d*tan(d*x + c))
 

Mupad [B] (verification not implemented)

Time = 1.72 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.39 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (10\,a^4+9\,a^2\,b^2+3\,b^4\right )}{a^4\,d\,{\left (a^2+b^2\right )}^3}-\frac {\frac {1}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^4\,b^2+6\,a^2\,b^4+3\,b^6\right )}{a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^4\,b+17\,a^2\,b^3+9\,b^5\right )}{2\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {3\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1617, normalized size of antiderivative = 7.66 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(48*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**7*b**3 - 
16*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**5*b**5 + 1 
60*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin( 
c + d*x)**2*a**5*b**5 + 144*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan 
((c + d*x)/2)*b - a)*sin(c + d*x)**2*a**3*b**7 + 48*cos(c + d*x)*log(tan(( 
c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**2*a*b**9 - 48*c 
os(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**7*b**3 - 144*cos(c + 
d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**5*b**5 - 144*cos(c + d*x)*lo 
g(tan((c + d*x)/2))*sin(c + d*x)**2*a**3*b**7 - 48*cos(c + d*x)*log(tan((c 
 + d*x)/2))*sin(c + d*x)**2*a*b**9 + 2*cos(c + d*x)*sin(c + d*x)**2*a**9*b 
 - 16*cos(c + d*x)*sin(c + d*x)**2*a**8*b**2*d*x + 6*cos(c + d*x)*sin(c + 
d*x)**2*a**7*b**3 + 48*cos(c + d*x)*sin(c + d*x)**2*a**6*b**4*d*x + 6*cos( 
c + d*x)*sin(c + d*x)**2*a**5*b**5 + 2*cos(c + d*x)*sin(c + d*x)**2*a**3*b 
**7 - 8*cos(c + d*x)*a**9*b - 24*cos(c + d*x)*a**7*b**3 - 24*cos(c + d*x)* 
a**5*b**5 - 8*cos(c + d*x)*a**3*b**7 - 24*log(tan((c + d*x)/2)**2 + 1)*sin 
(c + d*x)**3*a**8*b**2 + 32*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3*a 
**6*b**4 - 8*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**3*a**4*b**6 + 24*l 
og(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**8*b**2 - 8*log(tan((c + d*x)/2 
)**2 + 1)*sin(c + d*x)*a**6*b**4 - 80*log(tan((c + d*x)/2)**2*a - 2*tan((c 
 + d*x)/2)*b - a)*sin(c + d*x)**3*a**6*b**4 + 8*log(tan((c + d*x)/2)**2...