\(\int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [476]

Optimal result

Integrand size = 21, antiderivative size = 189 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac {b^4 \left (5 a^2+3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^2 d}+\frac {b^2 \left (2 a^2+3 b^2\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))} \] Output:

2*a*b*x/(a^2+b^2)^2-(a^2-3*b^2)*ln(sin(d*x+c))/a^4/d-b^4*(5*a^2+3*b^2)*ln( 
a*cos(d*x+c)+b*sin(d*x+c))/a^4/(a^2+b^2)^2/d+b^2*(2*a^2+3*b^2)/a^3/(a^2+b^ 
2)/d/(a+b*tan(d*x+c))+3/2*b*cot(d*x+c)/a^2/d/(a+b*tan(d*x+c))-1/2*cot(d*x+ 
c)^2/a/d/(a+b*tan(d*x+c))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.77 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {-\frac {4 b \cot (c+d x)}{a^3}+\frac {\cot ^2(c+d x)}{a^2}+\frac {2 b^5}{a^4 \left (a^2+b^2\right ) (b+a \cot (c+d x))}-\frac {\log (i-\cot (c+d x))}{(a-i b)^2}-\frac {\log (i+\cot (c+d x))}{(a+i b)^2}+\frac {2 b^4 \left (5 a^2+3 b^2\right ) \log (b+a \cot (c+d x))}{a^4 \left (a^2+b^2\right )^2}}{2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.19, 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, 4133, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4133

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {2 \left (\frac {-\frac {\left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}+\frac {b^4 \left (5 a^2+3 b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {2 a^4 b x}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b^2 \left (2 a^2+3 b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{a}-\frac {3 b \cot (c+d x)}{a d (a+b \tan (c+d x))}}{2 a}-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}\)

\(\Big \downarrow \) 3956

\(\displaystyle -\frac {\frac {2 \left (\frac {\frac {b^4 \left (5 a^2+3 b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \log (-\sin (c+d x))}{a d}-\frac {2 a^4 b x}{a^2+b^2}}{a \left (a^2+b^2\right )}-\frac {b^2 \left (2 a^2+3 b^2\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{a}-\frac {3 b \cot (c+d x)}{a d (a+b \tan (c+d x))}}{2 a}-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}\)

\(\Big \downarrow \) 4013

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

Input:

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

Output:

-1/2*Cot[c + d*x]^2/(a*d*(a + b*Tan[c + d*x])) - ((-3*b*Cot[c + d*x])/(a*d 
*(a + b*Tan[c + d*x])) + (2*(((-2*a^4*b*x)/(a^2 + b^2) + ((a^2 - 3*b^2)*(a 
^2 + b^2)*Log[-Sin[c + d*x]])/(a*d) + (b^4*(5*a^2 + 3*b^2)*Log[a*Cos[c + d 
*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a*(a^2 + b^2)) - (b^2*(2*a^2 + 
3*b^2))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))))/a)/(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 4133

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(A*b^2 + a^2*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*Sim 
p[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n 
 + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*( 
m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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 = 1.45 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {-\frac {1}{2 a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (-a^{2}+3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {2 b}{a^{3} \tan \left (d x +c \right )}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {b^{4}}{a^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{4} \left (5 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(166\)
default	\(\frac {-\frac {1}{2 a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (-a^{2}+3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {2 b}{a^{3} \tan \left (d x +c \right )}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {b^{4}}{a^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{4} \left (5 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(166\)
parallelrisch	\(\frac {-10 b^{4} \left (a^{2}+\frac {3 b^{2}}{5}\right ) \left (a +b \tan \left (d x +c \right )\right ) \ln \left (a +b \tan \left (d x +c \right )\right )+a^{4} \left (a -b \right ) \left (a +b \right ) \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )-2 \left (a^{2}-3 b^{2}\right ) \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\tan \left (d x +c \right )\right )+\left (4 b^{2} a^{5} x d -4 a^{4} b^{3}-10 b^{5} a^{2}-6 b^{7}\right ) \tan \left (d x +c \right )-\left (a \left (a^{2}+b^{2}\right )^{2} \cot \left (d x +c \right )^{2}-3 b \left (a^{2}+b^{2}\right )^{2} \cot \left (d x +c \right )-4 b \,a^{4} x d \right ) a^{2}}{2 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{2}+b^{2}\right )^{2} a^{4} d}\)	\(224\)
norman	\(\frac {\frac {\left (-2 b^{2} a^{2}-3 b^{4}\right ) b \tan \left (d x +c \right )^{3}}{d \,a^{4} \left (a^{2}+b^{2}\right )}-\frac {1}{2 a d}+\frac {3 b \tan \left (d x +c \right )}{2 a^{2} d}+\frac {2 b \,a^{2} x \tan \left (d x +c \right )^{2}}{a^{4}+2 b^{2} a^{2}+b^{4}}+\frac {2 b^{2} a x \tan \left (d x +c \right )^{3}}{a^{4}+2 b^{2} a^{2}+b^{4}}}{\tan \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{2}-3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {b^{4} \left (5 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) a^{4} d}\)	\(266\)
risch	\(\frac {i x}{2 i a b -a^{2}+b^{2}}+\frac {2 i x}{a^{2}}+\frac {2 i c}{a^{2} d}-\frac {6 i b^{2} x}{a^{4}}-\frac {6 i b^{2} c}{a^{4} d}+\frac {10 i b^{4} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) a^{2}}+\frac {10 i b^{4} c}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) a^{2} d}+\frac {6 i b^{6} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) a^{4}}+\frac {6 i b^{6} c}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) a^{4} d}+\frac {2 i \left (2 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-4 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+4 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{2} b^{3}+3 b^{5}+2 a^{4} b \right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (-i a +b \right ) \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right ) a^{3} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}-\frac {5 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) a^{2} d}-\frac {3 b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) a^{4} d}\)	\(585\)

Input:

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

Output:

1/d*(-1/2/a^2/tan(d*x+c)^2+(-a^2+3*b^2)/a^4*ln(tan(d*x+c))+2/a^3*b/tan(d*x 
+c)+1/(a^2+b^2)^2*(1/2*(a^2-b^2)*ln(1+tan(d*x+c)^2)+2*a*b*arctan(tan(d*x+c 
)))+b^4/a^3/(a^2+b^2)/(a+b*tan(d*x+c))-b^4*(5*a^2+3*b^2)/a^4/(a^2+b^2)^2*l 
n(a+b*tan(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (185) = 370\).

Time = 0.13 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.04 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4} - {\left (4 \, a^{5} b^{2} d x - a^{6} b - 2 \, a^{4} b^{3} - 3 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{3} - {\left (4 \, a^{6} b d x - a^{7} + 2 \, a^{5} b^{2} + 7 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left ({\left (a^{6} b - a^{4} b^{3} - 5 \, a^{2} b^{5} - 3 \, b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left ({\left (5 \, a^{2} b^{5} + 3 \, b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (5 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \tan \left (d x + c\right )^{3} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \tan \left (d x + c\right )^{2}\right )}} \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.65 (sec) , antiderivative size = 5263, normalized size of antiderivative = 27.85 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise(((log(tan(c + d*x)**2 + 1)/(2*d) - log(tan(c + d*x))/d - 1/(2*d* 
tan(c + d*x)**2))/a**2, Eq(b, 0)), ((-log(tan(c + d*x)**2 + 1)/(2*d) + log 
(tan(c + d*x))/d + 1/(2*d*tan(c + d*x)**2) - 1/(4*d*tan(c + d*x)**4))/b**2 
, Eq(a, 0)), (-15*I*d*x*tan(c + d*x)**4/(4*a**2*d*tan(c + d*x)**4 + 8*I*a* 
*2*d*tan(c + d*x)**3 - 4*a**2*d*tan(c + d*x)**2) + 30*d*x*tan(c + d*x)**3/ 
(4*a**2*d*tan(c + d*x)**4 + 8*I*a**2*d*tan(c + d*x)**3 - 4*a**2*d*tan(c + 
d*x)**2) + 15*I*d*x*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**4 + 8*I*a**2*d 
*tan(c + d*x)**3 - 4*a**2*d*tan(c + d*x)**2) + 8*log(tan(c + d*x)**2 + 1)* 
tan(c + d*x)**4/(4*a**2*d*tan(c + d*x)**4 + 8*I*a**2*d*tan(c + d*x)**3 - 4 
*a**2*d*tan(c + d*x)**2) + 16*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/( 
4*a**2*d*tan(c + d*x)**4 + 8*I*a**2*d*tan(c + d*x)**3 - 4*a**2*d*tan(c + d 
*x)**2) - 8*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x 
)**4 + 8*I*a**2*d*tan(c + d*x)**3 - 4*a**2*d*tan(c + d*x)**2) - 16*log(tan 
(c + d*x))*tan(c + d*x)**4/(4*a**2*d*tan(c + d*x)**4 + 8*I*a**2*d*tan(c + 
d*x)**3 - 4*a**2*d*tan(c + d*x)**2) - 32*I*log(tan(c + d*x))*tan(c + d*x)* 
*3/(4*a**2*d*tan(c + d*x)**4 + 8*I*a**2*d*tan(c + d*x)**3 - 4*a**2*d*tan(c 
 + d*x)**2) + 16*log(tan(c + d*x))*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)* 
*4 + 8*I*a**2*d*tan(c + d*x)**3 - 4*a**2*d*tan(c + d*x)**2) - 15*I*tan(c + 
 d*x)**3/(4*a**2*d*tan(c + d*x)**4 + 8*I*a**2*d*tan(c + d*x)**3 - 4*a**2*d 
*tan(c + d*x)**2) + 22*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**4 + 8*I*...
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.27 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (5 \, a^{2} b^{4} + 3 \, b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {a^{4} + a^{2} b^{2} - 2 \, {\left (2 \, a^{2} b^{2} + 3 \, b^{4}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )}{{\left (a^{5} b + a^{3} b^{3}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{6} + a^{4} b^{2}\right )} \tan \left (d x + c\right )^{2}} - \frac {2 \, {\left (a^{2} - 3 \, b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.44 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {2 \, {\left (d x + c\right )} a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac {{\left (5 \, a^{2} b^{5} + 3 \, b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b d + 2 \, a^{6} b^{3} d + a^{4} b^{5} d} - \frac {{\left (a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4} d} - \frac {a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4} - 2 \, {\left (2 \, a^{5} b^{2} + 5 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x + c\right ) + a\right )} a^{4} d \tan \left (d x + c\right )^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {3\,b\,\mathrm {tan}\left (c+d\,x\right )}{2\,a^2}-\frac {1}{2\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2\,b^2+3\,b^4\right )}{a^3\,\left (a^2+b^2\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^3+a\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-3\,b^2\right )}{a^4\,d}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (5\,a^2\,b^4+3\,b^6\right )}{d\,\left (a^8+2\,a^6\,b^2+a^4\,b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\cot \left (d x +c \right )^{3}}{\left (a +\tan \left (d x +c \right ) b \right )^{2}}d x \] Input:

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

Output:

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