\(\int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [484]

Optimal result
Mathematica [C] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-2)]
Maxima [A] (verification not implemented)
Giac [A] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 19, antiderivative size = 168 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {\log (\sin (c+d x))}{a^3 d}-\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b^2}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.96 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.24 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {-\frac {a (a-i b) \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {2 \left (a^2+b^2\right ) \log (\tan (c+d x))}{a^2}-\frac {a (a+i b) \log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {b^2}{(a+b \tan (c+d x))^2}+\frac {4 a b^2}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 b^2}{a^2+a b \tan (c+d x)}}{2 a \left (a^2+b^2\right ) d} \] Input:

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

Output:

(-((a*(a - I*b)*Log[I - Tan[c + d*x]])/(a + I*b)^2) + (2*(a^2 + b^2)*Log[T 
an[c + d*x]])/a^2 - (a*(a + I*b)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b 
^2*(6*a^4 + 3*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)^2) 
+ b^2/(a + b*Tan[c + d*x])^2 + (4*a*b^2)/((a^2 + b^2)*(a + b*Tan[c + d*x]) 
) + (2*b^2)/(a^2 + a*b*Tan[c + d*x]))/(2*a*(a^2 + b^2)*d)
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {3042, 4052, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 4013

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

Input:

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

Output:

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

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 = 1.55 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.08

method result size
derivativedivides \(\frac {\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b^{2}}{2 a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (6 a^{4}+3 b^{2} a^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 a^{2} b +b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) \(182\)
default \(\frac {\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b^{2}}{2 a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (6 a^{4}+3 b^{2} a^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 a^{2} b +b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) \(182\)
parallelrisch \(\frac {-12 b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (a^{4}+\frac {1}{2} b^{2} a^{2}+\frac {1}{6} b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )-a^{4} \left (a^{2}-3 b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\sec \left (d x +c \right )^{2}\right )+2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\tan \left (d x +c \right )\right )-6 b \left (b^{2} \left (x \,a^{5} d -\frac {1}{3} b^{2} a^{3} x d +\frac {7}{6} a^{4} b +\frac {5}{3} a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \tan \left (d x +c \right )^{2}+2 b \left (x \,a^{5} d -\frac {1}{3} b^{2} a^{3} x d +\frac {2}{3} a^{4} b +a^{2} b^{3}+\frac {1}{3} b^{5}\right ) a \tan \left (d x +c \right )+a^{5} d x \left (a^{2}-\frac {b^{2}}{3}\right )\right )}{2 \left (a^{2}+b^{2}\right )^{3} a^{3} d \left (a +b \tan \left (d x +c \right )\right )^{2}}\) \(253\)
norman \(\frac {-\frac {b^{3} \left (3 a^{2}-b^{2}\right ) x \tan \left (d x +c \right )^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 b \left (-2 b^{2} a^{2}-b^{4}\right ) \tan \left (d x +c \right )}{d \,a^{2} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}+\frac {b^{2} \left (-7 b^{2} a^{2}-3 b^{4}\right ) \tan \left (d x +c \right )^{2}}{2 d \,a^{3} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {2 \left (3 a^{2}-b^{2}\right ) a \,b^{2} x \tan \left (d x +c \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{2} \left (6 a^{4}+3 b^{2} a^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}\) \(383\)
risch \(-\frac {i x}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {12 i a \,b^{2} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {12 i a \,b^{2} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {6 i b^{4} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a}+\frac {6 i b^{4} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a d}+\frac {2 i b^{6} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3}}+\frac {2 i b^{6} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}-\frac {2 i x}{a^{3}}-\frac {2 i c}{a^{3} d}-\frac {2 i b^{3} \left (-3 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+4 i a^{2} b +i b^{3}+4 a^{3}+a \,b^{2}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} a^{2} \left (i b +a \right )^{2} d \left (-i b +a \right )^{3}}-\frac {6 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a d}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\) \(621\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (166) = 332\).

Time = 0.14 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.94 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {9 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - 2 \, {\left (3 \, a^{7} b - a^{5} b^{3}\right )} d x - {\left (7 \, a^{4} b^{4} + a^{2} b^{6} + 2 \, {\left (3 \, a^{5} b^{3} - a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (6 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + {\left (6 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (4 \, a^{5} b^{3} - 3 \, a^{3} b^{5} - a b^{7} + 2 \, {\left (3 \, a^{6} b^{2} - a^{4} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \] Input:

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

Output:

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

Sympy [F(-2)]

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

integrate(cot(d*x+c)/(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 = 0.11 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.73 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (6 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {7 \, a^{3} b^{2} + 3 \, a b^{4} + 2 \, {\left (3 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )} - \frac {2 \, \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.62 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {{\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} - \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} - \frac {{\left (6 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b d + 3 \, a^{7} b^{3} d + 3 \, a^{5} b^{5} d + a^{3} b^{7} d} + \frac {\log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3} d} + \frac {7 \, a^{6} b^{2} + 10 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + 2 \, {\left (3 \, a^{5} b^{3} + 4 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{3} d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.52 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {\frac {7\,a^2\,b^2+3\,b^4}{2\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^2\,b^3+b^5\right )}{a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (6\,a^4+3\,a^2\,b^2+b^4\right )}{a^3\,d\,{\left (a^2+b^2\right )}^3}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )} \] Input:

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

Output:

(log(tan(c + d*x) - 1i)*1i)/(2*d*(a*b^2*3i + 3*a^2*b - a^3*1i - b^3)) + lo 
g(tan(c + d*x) + 1i)/(2*d*(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)) + log(tan(c 
 + d*x))/(a^3*d) + ((3*b^4 + 7*a^2*b^2)/(2*a*(a^4 + b^4 + 2*a^2*b^2)) + (t 
an(c + d*x)*(b^5 + 3*a^2*b^3))/(a^2*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2 + b^ 
2*tan(c + d*x)^2 + 2*a*b*tan(c + d*x))) - (b^2*log(a + b*tan(c + d*x))*(6* 
a^4 + b^4 + 3*a^2*b^2))/(a^3*d*(a^2 + b^2)^3)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 1290, normalized size of antiderivative = 7.68 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**7*b + 12*c 
os(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**5*b**3 - 24*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**5*b**3 - 12*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**3*b**5 - 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**7 + 4*cos(c + d*x)*log(tan( 
(c + d*x)/2))*sin(c + d*x)*a**7*b + 12*cos(c + d*x)*log(tan((c + d*x)/2))* 
sin(c + d*x)*a**5*b**3 + 12*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x 
)*a**3*b**5 + 4*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)*a*b**7 - 1 
2*cos(c + d*x)*sin(c + d*x)*a**6*b**2*d*x + 4*cos(c + d*x)*sin(c + d*x)*a* 
*4*b**4*d*x + 2*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**8 - 8*log( 
tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**6*b**2 + 6*log(tan((c + d*x)/2 
)**2 + 1)*sin(c + d*x)**2*a**4*b**4 - 2*log(tan((c + d*x)/2)**2 + 1)*a**8 
+ 6*log(tan((c + d*x)/2)**2 + 1)*a**6*b**2 + 12*log(tan((c + d*x)/2)**2*a 
- 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**2*a**6*b**2 - 6*log(tan((c + d*x 
)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**2*a**4*b**4 - 4*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**6 
- 2*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**2* 
b**8 - 12*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**6*b**2 
- 6*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**4*b**4 - 2...