\(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [475]

Optimal result

Integrand size = 21, antiderivative size = 150 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {2 b \log (\sin (c+d x))}{a^3 d}+\frac {2 b^3 \left (2 a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b \left (a^2+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))} \] Output:

-(a^2-b^2)*x/(a^2+b^2)^2-2*b*ln(sin(d*x+c))/a^3/d+2*b^3*(2*a^2+b^2)*ln(a*c 
os(d*x+c)+b*sin(d*x+c))/a^3/(a^2+b^2)^2/d-b*(a^2+2*b^2)/a^2/(a^2+b^2)/d/(a 
+b*tan(d*x+c))-cot(d*x+c)/a/d/(a+b*tan(d*x+c))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {\cot (c+d x)}{a^2}-\frac {b^4}{a^3 \left (a^2+b^2\right ) (b+a \cot (c+d x))}+\frac {i \log (i-\cot (c+d x))}{2 (a-i b)^2}-\frac {i \log (i+\cot (c+d x))}{2 (a+i b)^2}-\frac {2 b^3 \left (2 a^2+b^2\right ) \log (b+a \cot (c+d x))}{a^3 \left (a^2+b^2\right )^2}}{d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4052, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 4013

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

Input:

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

Output:

-(Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x]))) - (((a^2*(a^2 - b^2)*x)/(a^2 + 
b^2) + (2*b*(a^2 + b^2)*Log[-Sin[c + d*x]])/(a*d) - (2*b^3*(2*a^2 + b^2)*L 
og[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a*(a^2 + b^2)) + 
(b*(a^2 + 2*b^2))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/a
 

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 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 = 1.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{a^{2} \tan \left (d x +c \right )}-\frac {2 b \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}-\frac {b^{3}}{\left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b^{3} \left (2 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}}{d}\)	\(140\)
default	\(\frac {\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {1}{a^{2} \tan \left (d x +c \right )}-\frac {2 b \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}-\frac {b^{3}}{\left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b^{3} \left (2 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}}{d}\)	\(140\)
parallelrisch	\(\frac {4 \left (a^{2}+\frac {b^{2}}{2}\right ) b^{3} \left (a +b \tan \left (d x +c \right )\right ) \ln \left (a +b \tan \left (d x +c \right )\right )+b \,a^{4} \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )-2 b \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\tan \left (d x +c \right )\right )-b \left (x \,a^{5} d -b^{2} a^{3} x d -a^{4} b -3 a^{2} b^{3}-2 b^{5}\right ) \tan \left (d x +c \right )-\left (\left (a^{2}+b^{2}\right )^{2} \cot \left (d x +c \right )+a^{2} d x \left (a -b \right ) \left (a +b \right )\right ) a^{2}}{\left (a +b \tan \left (d x +c \right )\right ) \left (a^{2}+b^{2}\right )^{2} a^{3} d}\)	\(199\)
norman	\(\frac {\frac {\left (-b^{2} a^{2}-2 b^{4}\right ) \tan \left (d x +c \right )}{a^{2} b d \left (a^{2}+b^{2}\right )}-\frac {1}{a d}-\frac {b \left (a^{2}-b^{2}\right ) x \tan \left (d x +c \right )^{2}}{a^{4}+2 b^{2} a^{2}+b^{4}}-\frac {\left (a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )}{a^{4}+2 b^{2} a^{2}+b^{4}}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {2 b \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}+\frac {2 b^{3} \left (2 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{3} d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}\)	\(243\)
risch	\(\frac {x}{2 i a b -a^{2}+b^{2}}+\frac {4 i b x}{a^{3}}+\frac {4 i b c}{a^{3} d}-\frac {8 i b^{3} x}{a \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {8 i b^{3} c}{a d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {4 i b^{5} x}{a^{3} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {4 i b^{5} c}{a^{3} d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {2 i \left (a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+a^{4}+2 b^{2} a^{2}+2 b^{4}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (i b +a \right ) \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) a^{2} d}-\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}+\frac {2 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{3} d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}\)	\(434\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (150) = 300\).

Time = 0.11 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.15 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} - {\left (a^{2} b^{4} - {\left (a^{5} b - a^{3} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left ({\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left ({\left (2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (a^{5} b + 2 \, a^{3} b^{3} + 2 \, a b^{5} + {\left (a^{6} - a^{4} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{{\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \tan \left (d x + c\right )} \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.88 (sec) , antiderivative size = 4083, normalized size of antiderivative = 27.22 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise(((-x - cot(c + d*x)/d)/a**2, Eq(b, 0)), ((x + 1/(d*tan(c + d*x)) 
 - 1/(3*d*tan(c + d*x)**3))/b**2, Eq(a, 0)), (-9*d*x*tan(c + d*x)**3/(4*a* 
*2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) 
 - 18*I*d*x*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + 
 d*x)**2 - 4*a**2*d*tan(c + d*x)) + 9*d*x*tan(c + d*x)/(4*a**2*d*tan(c + d 
*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 4*I*log(tan 
(c + d*x)**2 + 1)*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*t 
an(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 8*log(tan(c + d*x)**2 + 1)*tan(c 
 + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2 
*d*tan(c + d*x)) + 4*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*a**2*d*tan 
(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 8*I*l 
og(tan(c + d*x))*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*ta 
n(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 16*log(tan(c + d*x))*tan(c + d*x) 
**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan( 
c + d*x)) - 8*I*log(tan(c + d*x))*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 
 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 9*tan(c + d*x)**2/( 
4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d 
*x)) - 14*I*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d* 
x)**2 - 4*a**2*d*tan(c + d*x)) + 4/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d* 
tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)), Eq(b, -I*a)), (-9*d*x*tan(c +...
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.33 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}} - \frac {a^{3} + a b^{2} + {\left (a^{2} b + 2 \, b^{3}\right )} \tan \left (d x + c\right )}{{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{5} + a^{3} b^{2}\right )} \tan \left (d x + c\right )} - \frac {2 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{d} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.49 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} - \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} + \frac {2 \, {\left (2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{7} b d + 2 \, a^{5} b^{3} d + a^{3} b^{5} d} - \frac {2 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3} d} - \frac {a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 3 \, a^{2} b^{3} + 2 \, b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x + c\right ) + a\right )} a^{2} d \tan \left (d x + c\right )} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.40 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.22 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {\frac {1}{a}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^2\,b+2\,b^3\right )}{a^2\,\left (a^2+b^2\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {2\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {2\,b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (2\,a^2+b^2\right )}{a^3\,d\,{\left (a^2+b^2\right )}^2}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \] Input:

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

Output:

(log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b*2i - a^2 + b^2)) + log(tan(c + d*x) 
- 1i)/(2*d*(2*a*b - a^2*1i + b^2*1i)) - (1/a + (tan(c + d*x)*(a^2*b + 2*b^ 
3))/(a^2*(a^2 + b^2)))/(d*(a*tan(c + d*x) + b*tan(c + d*x)^2)) - (2*b*log( 
tan(c + d*x)))/(a^3*d) + (2*b^3*log(a + b*tan(c + d*x))*(2*a^2 + b^2))/(a^ 
3*d*(a^2 + b^2)^2)
 

Reduce [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 726, normalized size of antiderivative = 4.84 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**5*b**2 + 8*co 
s(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d 
*x)*a**3*b**4 + 4*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x) 
/2)*b - a)*sin(c + d*x)*a*b**6 - 4*cos(c + d*x)*log(tan((c + d*x)/2))*sin( 
c + d*x)*a**5*b**2 - 8*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)*a** 
3*b**4 - 4*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)*a*b**6 - cos(c 
+ d*x)*sin(c + d*x)*a**7 - 2*cos(c + d*x)*sin(c + d*x)*a**6*b*d*x - 4*cos( 
c + d*x)*sin(c + d*x)*a**5*b**2 + 2*cos(c + d*x)*sin(c + d*x)*a**4*b**3*d* 
x - 7*cos(c + d*x)*sin(c + d*x)*a**3*b**4 - 4*cos(c + d*x)*sin(c + d*x)*a* 
b**6 + 4*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**4*b**3 + 8*log(ta 
n((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**2*a**2*b**5 
+ 4*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**2* 
b**7 - 4*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**4*b**3 - 8*log(tan((c + 
d*x)/2))*sin(c + d*x)**2*a**2*b**5 - 4*log(tan((c + d*x)/2))*sin(c + d*x)* 
*2*b**7 + sin(c + d*x)**2*a**6*b - 2*sin(c + d*x)**2*a**5*b**2*d*x + 2*sin 
(c + d*x)**2*a**4*b**3 + 2*sin(c + d*x)**2*a**3*b**4*d*x + sin(c + d*x)**2 
*a**2*b**5 - 2*a**6*b - 4*a**4*b**3 - 2*a**2*b**5)/(2*sin(c + d*x)*a**3*b* 
d*(cos(c + d*x)*a**5 + 2*cos(c + d*x)*a**3*b**2 + cos(c + d*x)*a*b**4 + si 
n(c + d*x)*a**4*b + 2*sin(c + d*x)*a**2*b**3 + sin(c + d*x)*b**5))