\(\int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [493]

Optimal result

Integrand size = 19, antiderivative size = 226 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\log (\sin (c+d x))}{a^4 d}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^4 d}+\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \] Output:

-4*a*b*(a^2-b^2)*x/(a^2+b^2)^4+ln(sin(d*x+c))/a^4/d-b^2*(10*a^6+5*a^4*b^2+ 
4*a^2*b^4+b^6)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^4/(a^2+b^2)^4/d+1/3*b^2/a/( 
a^2+b^2)/d/(a+b*tan(d*x+c))^3+1/2*b^2*(3*a^2+b^2)/a^2/(a^2+b^2)^2/d/(a+b*t 
an(d*x+c))^2+b^2*(6*a^4+3*a^2*b^2+b^4)/a^3/(a^2+b^2)^3/d/(a+b*tan(d*x+c))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.40 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.08 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \left (-a^4 (a-i b)^4 \log (i-\tan (c+d x))+2 \left (a^2+b^2\right )^4 \log (\tan (c+d x))-a^4 (a+i b)^4 \log (i+\tan (c+d x))-2 b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a+b \tan (c+d x))\right )}{a^2 \left (a^2+b^2\right )^2}+\frac {2 a b^2 \left (a^2+b^2\right )}{(a+b \tan (c+d x))^3}+\frac {3 \left (3 a^2 b^2+b^4\right )}{(a+b \tan (c+d x))^2}+\frac {6 \left (6 a^4 b^2+3 a^2 b^4+b^6\right )}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{6 a^2 \left (a^2+b^2\right )^2 d} \] Input:

Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x])^4,x]
 

Output:

((3*(-(a^4*(a - I*b)^4*Log[I - Tan[c + d*x]]) + 2*(a^2 + b^2)^4*Log[Tan[c 
+ d*x]] - a^4*(a + I*b)^4*Log[I + Tan[c + d*x]] - 2*b^2*(10*a^6 + 5*a^4*b^ 
2 + 4*a^2*b^4 + b^6)*Log[a + b*Tan[c + d*x]]))/(a^2*(a^2 + b^2)^2) + (2*a* 
b^2*(a^2 + b^2))/(a + b*Tan[c + d*x])^3 + (3*(3*a^2*b^2 + b^4))/(a + b*Tan 
[c + d*x])^2 + (6*(6*a^4*b^2 + 3*a^2*b^4 + b^6))/(a*(a^2 + b^2)*(a + b*Tan 
[c + d*x])))/(6*a^2*(a^2 + b^2)^2*d)
 

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.24, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {3042, 4052, 27, 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.

\(\displaystyle \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (c+d x) (a+b \tan (c+d x))^4}dx\)

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4134

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2+b^2\right )^3 \int \cot (c+d x)dx}{a}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {4 a^4 b x \left (a^2-b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2+b^2\right )^3 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {4 a^4 b x \left (a^2-b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {-\frac {\left (a^2+b^2\right )^3 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {4 a^4 b x \left (a^2-b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {\frac {\frac {-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2+b^2\right )^3 \log (-\sin (c+d x))}{a d}-\frac {4 a^4 b x \left (a^2-b^2\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a \left (a^2+b^2\right )}+\frac {b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4013

\(\displaystyle \frac {b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {\frac {b^2 \left (3 a^2+b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {\left (a^2+b^2\right )^3 \log (-\sin (c+d x))}{a d}-\frac {4 a^4 b x \left (a^2-b^2\right )}{a^2+b^2}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}\)

Input:

Int[Cot[c + d*x]/(a + b*Tan[c + d*x])^4,x]
 

Output:

b^2/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + ((b^2*(3*a^2 + b^2))/(2*a 
*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((-4*a^4*b*(a^2 - b^2)*x)/(a^2 + 
 b^2) + ((a^2 + b^2)^3*Log[-Sin[c + d*x]])/(a*d) - (b^2*(10*a^6 + 5*a^4*b^ 
2 + 4*a^2*b^4 + b^6)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)* 
d))/(a*(a^2 + b^2)) + (b^2*(6*a^4 + 3*a^2*b^2 + b^4))/(a*(a^2 + b^2)*d*(a 
+ b*Tan[c + d*x])))/(a*(a^2 + b^2)))/(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 = 2.24 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {b^{2}}{3 a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{2 a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (6 a^{4}+3 b^{2} a^{2}+b^{4}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (10 a^{6}+5 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{4}}+\frac {\frac {\left (-a^{4}+6 b^{2} a^{2}-b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(246\)
default	\(\frac {\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {b^{2}}{3 a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{2 a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (6 a^{4}+3 b^{2} a^{2}+b^{4}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (10 a^{6}+5 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{4}}+\frac {\frac {\left (-a^{4}+6 b^{2} a^{2}-b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(246\)
parallelrisch	\(\frac {-20 b^{2} \left (a^{6}+\frac {1}{2} a^{4} b^{2}+\frac {2}{5} a^{2} b^{4}+\frac {1}{10} b^{6}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (a +b \tan \left (d x +c \right )\right )-a^{4} \left (a^{2}+2 a b -b^{2}\right ) \left (a^{2}-2 a b -b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\sec \left (d x +c \right )^{2}\right )+2 \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\tan \left (d x +c \right )\right )-8 b \left (b^{3} \left (-x \,a^{5} b^{2} d +\frac {47}{24} a^{6} b +\frac {27}{8} a^{4} b^{3}+\frac {15}{8} b^{5} a^{2}+a^{7} d x +\frac {11}{24} b^{7}\right ) \tan \left (d x +c \right )^{3}+3 b^{2} a \left (a^{7} d x -x \,a^{5} b^{2} d +\frac {35}{24} a^{6} b +\frac {21}{8} a^{4} b^{3}+\frac {37}{24} b^{5} a^{2}+\frac {3}{8} b^{7}\right ) \tan \left (d x +c \right )^{2}+3 b \,a^{2} \left (a^{7} d x -x \,a^{5} b^{2} d +\frac {5}{6} a^{6} b +\frac {19}{12} a^{4} b^{3}+b^{5} a^{2}+\frac {1}{4} b^{7}\right ) \tan \left (d x +c \right )+a^{8} d x \left (a -b \right ) \left (a +b \right )\right )}{2 \left (a^{2}+b^{2}\right )^{4} a^{4} d \left (a +b \tan \left (d x +c \right )\right )^{3}}\)	\(353\)
norman	\(\frac {\frac {b \left (-10 a^{4} b^{2}-9 a^{2} b^{4}-3 b^{6}\right ) \tan \left (d x +c \right )}{d \,a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {4 \left (a^{2}-b^{2}\right ) a^{4} b x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \tan \left (d x +c \right )^{2}}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )^{3}}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {b^{2} \left (-35 a^{4} b^{2}-28 a^{2} b^{4}-9 b^{6}\right ) \tan \left (d x +c \right )^{2}}{2 d \,a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{3} \left (-47 a^{4} b^{2}-34 a^{2} b^{4}-11 b^{6}\right ) \tan \left (d x +c \right )^{3}}{6 d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}-\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {b^{2} \left (10 a^{6}+5 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(589\)
risch	\(\frac {20 i a^{2} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {2 i x}{a^{4}}+\frac {8 i b^{6} x}{a^{2} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {20 i a^{2} b^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {10 i b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {2 i b^{8} x}{a^{4} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 b^{2} a^{2}-b^{4}}+\frac {8 i b^{6} c}{a^{2} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {10 i b^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {2 i b^{8} c}{a^{4} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {2 i c}{a^{4} d}+\frac {2 i b^{3} \left (-3 i a \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+9 i a \,b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-30 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a \,b^{5}+30 i a^{3} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-15 i a^{5} b \,{\mathrm e}^{2 i \left (d x +c \right )}-60 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-84 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-30 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+45 i a^{5} b \,{\mathrm e}^{4 i \left (d x +c \right )}-60 i a^{5} b -22 i a^{3} b^{3}-30 a^{6}+19 a^{4} b^{2}+8 a^{2} b^{4}+3 b^{6}\right )}{3 \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 )^{3} a^{3} \left (-i a +b \right )^{3} d \left (i a +b \right )^{4}}-\frac {10 a^{2} b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {5 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {4 b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{2} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {b^{8} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{4} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}\)	\(1035\)

Input:

int(cot(d*x+c)/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/a^4*ln(tan(d*x+c))+1/3*b^2/a/(a^2+b^2)/(a+b*tan(d*x+c))^3+1/2*b^2*( 
3*a^2+b^2)/a^2/(a^2+b^2)^2/(a+b*tan(d*x+c))^2+b^2*(6*a^4+3*a^2*b^2+b^4)/a^ 
3/(a^2+b^2)^3/(a+b*tan(d*x+c))-b^2*(10*a^6+5*a^4*b^2+4*a^2*b^4+b^6)/a^4/(a 
^2+b^2)^4*ln(a+b*tan(d*x+c))+1/(a^2+b^2)^4*(1/2*(-a^4+6*a^2*b^2-b^4)*ln(1+ 
tan(d*x+c)^2)+(-4*a^3*b+4*a*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. 793 vs. \(2 (222) = 444\).

Time = 0.19 (sec) , antiderivative size = 793, normalized size of antiderivative = 3.51 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)/(a+b*tan(d*x+c))^4,x, algorithm="fricas")
 

Output:

1/6*(75*a^7*b^4 + 42*a^5*b^6 + 11*a^3*b^8 - (47*a^6*b^5 + 6*a^4*b^7 + 3*a^ 
2*b^9 + 24*(a^7*b^4 - a^5*b^6)*d*x)*tan(d*x + c)^3 - 24*(a^10*b - a^8*b^3) 
*d*x - 3*(35*a^7*b^4 - 12*a^5*b^6 - 5*a^3*b^8 - 2*a*b^10 + 24*(a^8*b^3 - a 
^6*b^5)*d*x)*tan(d*x + c)^2 + 3*(a^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4*a^5*b^6 
+ a^3*b^8 + (a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*tan(d*x + 
 c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*tan(d*x + 
 c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6*a^6*b^5 + 4*a^4*b^7 + a^2*b^9)*tan(d*x + 
 c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - 3*(10*a^9*b^2 + 5*a^7*b^4 
+ 4*a^5*b^6 + a^3*b^8 + (10*a^6*b^5 + 5*a^4*b^7 + 4*a^2*b^9 + b^11)*tan(d* 
x + c)^3 + 3*(10*a^7*b^4 + 5*a^5*b^6 + 4*a^3*b^8 + a*b^10)*tan(d*x + c)^2 
+ 3*(10*a^8*b^3 + 5*a^6*b^5 + 4*a^4*b^7 + a^2*b^9)*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)) - 3*(20*a 
^8*b^3 - 37*a^6*b^5 - 18*a^4*b^7 - 5*a^2*b^9 + 24*(a^9*b^2 - a^7*b^4)*d*x) 
*tan(d*x + c))/((a^12*b^3 + 4*a^10*b^5 + 6*a^8*b^7 + 4*a^6*b^9 + a^4*b^11) 
*d*tan(d*x + c)^3 + 3*(a^13*b^2 + 4*a^11*b^4 + 6*a^9*b^6 + 4*a^7*b^8 + a^5 
*b^10)*d*tan(d*x + c)^2 + 3*(a^14*b + 4*a^12*b^3 + 6*a^10*b^5 + 4*a^8*b^7 
+ a^6*b^9)*d*tan(d*x + c) + (a^15 + 4*a^13*b^2 + 6*a^11*b^4 + 4*a^9*b^6 + 
a^7*b^8)*d)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \] Input:

integrate(cot(d*x+c)/(a+b*tan(d*x+c))**4,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.96 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, a^{6} b^{2} + 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {47 \, a^{6} b^{2} + 34 \, a^{4} b^{4} + 11 \, a^{2} b^{6} + 6 \, {\left (6 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (27 \, a^{5} b^{3} + 16 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \tan \left (d x + c\right )}{a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6} + {\left (a^{9} b^{3} + 3 \, a^{7} b^{5} + 3 \, a^{5} b^{7} + a^{3} b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} \tan \left (d x + c\right )} - \frac {6 \, \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \] Input:

integrate(cot(d*x+c)/(a+b*tan(d*x+c))^4,x, algorithm="maxima")
 

Output:

-1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^ 
6 + b^8) + 6*(10*a^6*b^2 + 5*a^4*b^4 + 4*a^2*b^6 + b^8)*log(b*tan(d*x + c) 
 + a)/(a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4*b^8) + 3*(a^4 - 6*a 
^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2 
*b^6 + b^8) - (47*a^6*b^2 + 34*a^4*b^4 + 11*a^2*b^6 + 6*(6*a^4*b^4 + 3*a^2 
*b^6 + b^8)*tan(d*x + c)^2 + 3*(27*a^5*b^3 + 16*a^3*b^5 + 5*a*b^7)*tan(d*x 
 + c))/(a^12 + 3*a^10*b^2 + 3*a^8*b^4 + a^6*b^6 + (a^9*b^3 + 3*a^7*b^5 + 3 
*a^5*b^7 + a^3*b^9)*tan(d*x + c)^3 + 3*(a^10*b^2 + 3*a^8*b^4 + 3*a^6*b^6 + 
 a^4*b^8)*tan(d*x + c)^2 + 3*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*ta 
n(d*x + c)) - 6*log(tan(d*x + c))/a^4)/d
 

Giac [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.65 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {4 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d} - \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d\right )}} - \frac {{\left (10 \, a^{6} b^{3} + 5 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{12} b d + 4 \, a^{10} b^{3} d + 6 \, a^{8} b^{5} d + 4 \, a^{6} b^{7} d + a^{4} b^{9} d} + \frac {\log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4} d} + \frac {47 \, a^{9} b^{2} + 81 \, a^{7} b^{4} + 45 \, a^{5} b^{6} + 11 \, a^{3} b^{8} + 6 \, {\left (6 \, a^{7} b^{4} + 9 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (27 \, a^{8} b^{3} + 43 \, a^{6} b^{5} + 21 \, a^{4} b^{7} + 5 \, a^{2} b^{9}\right )} \tan \left (d x + c\right )}{6 \, {\left (a^{2} + b^{2}\right )}^{4} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} a^{4} d} \] Input:

integrate(cot(d*x+c)/(a+b*tan(d*x+c))^4,x, algorithm="giac")
 

Output:

-4*(a^3*b - a*b^3)*(d*x + c)/(a^8*d + 4*a^6*b^2*d + 6*a^4*b^4*d + 4*a^2*b^ 
6*d + b^8*d) - 1/2*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1)/(a^8*d 
+ 4*a^6*b^2*d + 6*a^4*b^4*d + 4*a^2*b^6*d + b^8*d) - (10*a^6*b^3 + 5*a^4*b 
^5 + 4*a^2*b^7 + b^9)*log(abs(b*tan(d*x + c) + a))/(a^12*b*d + 4*a^10*b^3* 
d + 6*a^8*b^5*d + 4*a^6*b^7*d + a^4*b^9*d) + log(abs(tan(d*x + c)))/(a^4*d 
) + 1/6*(47*a^9*b^2 + 81*a^7*b^4 + 45*a^5*b^6 + 11*a^3*b^8 + 6*(6*a^7*b^4 
+ 9*a^5*b^6 + 4*a^3*b^8 + a*b^10)*tan(d*x + c)^2 + 3*(27*a^8*b^3 + 43*a^6* 
b^5 + 21*a^4*b^7 + 5*a^2*b^9)*tan(d*x + c))/((a^2 + b^2)^4*(b*tan(d*x + c) 
 + a)^3*a^4*d)
 

Mupad [B] (verification not implemented)

Time = 1.78 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.70 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {47\,a^4\,b^2+34\,a^2\,b^4+11\,b^6}{6\,a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (27\,a^4\,b^3+16\,a^2\,b^5+5\,b^7\right )}{2\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (6\,a^4\,b^4+3\,a^2\,b^6+b^8\right )}{a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^4\,d}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (10\,a^6+5\,a^4\,b^2+4\,a^2\,b^4+b^6\right )}{a^4\,d\,{\left (a^2+b^2\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \] Input:

int(cot(c + d*x)/(a + b*tan(c + d*x))^4,x)
 

Output:

((11*b^6 + 34*a^2*b^4 + 47*a^4*b^2)/(6*a*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^ 
2)) + (tan(c + d*x)*(5*b^7 + 16*a^2*b^5 + 27*a^4*b^3))/(2*a^2*(a^6 + b^6 + 
 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^2*(b^8 + 3*a^2*b^6 + 6*a^4*b^4))/ 
(a^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3 + b^3*tan(c + d*x)^3 + 
3*a*b^2*tan(c + d*x)^2 + 3*a^2*b*tan(c + d*x))) - (log(tan(c + d*x) + 1i)* 
1i)/(2*d*(4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i)) - log(tan(c + 
 d*x) - 1i)/(2*d*(a^3*b*4i - a*b^3*4i + a^4 + b^4 - 6*a^2*b^2)) + log(tan( 
c + d*x))/(a^4*d) - (b^2*log(a + b*tan(c + d*x))*(10*a^6 + b^6 + 4*a^2*b^4 
 + 5*a^4*b^2))/(a^4*d*(a^2 + b^2)^4)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 2606, normalized size of antiderivative = 11.53 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:

int(cot(d*x+c)/(a+b*tan(d*x+c))^4,x)
 

Output:

( - 6*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**11 + 54 
*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**9*b**2 - 114 
*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**7*b**4 + 18* 
cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**5*b**6 + 6*co 
s(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a**11 - 36*cos(c + d*x)*log(tan((c 
 + d*x)/2)**2 + 1)*a**9*b**2 + 6*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1) 
*a**7*b**4 - 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**9*b**2 + 150*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**7*b**4 + 66*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**6 + 66*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**8 + 18*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**10 + 60*cos(c + d*x)* 
log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**9*b**2 + 30*cos(c 
 + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**7*b**4 + 
24*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**5 
*b**6 + 6*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - 
a)*a**3*b**8 + 6*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**11 
+ 6*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**9*b**2 - 36*cos( 
c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**7*b**4 - 84*cos(c + d...