\(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) [288]

Optimal result

Integrand size = 31, antiderivative size = 287 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\left (a^2+b^2\right )^3}-\frac {(3 A b-a B) \log (\sin (c+d x))}{a^4 d}+\frac {b^2 \left (10 a^4 A b+9 a^2 A b^3+3 A b^5-6 a^5 B-3 a^3 b^2 B-a b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2 A+3 A b^2-a b B\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4 A+6 a^2 A b^2+3 A b^4-3 a^3 b B-a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:

-(A*a^3-3*A*a*b^2+3*B*a^2*b-B*b^3)*x/(a^2+b^2)^3-(3*A*b-B*a)*ln(sin(d*x+c) 
)/a^4/d+b^2*(10*A*a^4*b+9*A*a^2*b^3+3*A*b^5-6*B*a^5-3*B*a^3*b^2-B*a*b^4)*l 
n(a*cos(d*x+c)+b*sin(d*x+c))/a^4/(a^2+b^2)^3/d-1/2*b*(2*A*a^2+3*A*b^2-B*a* 
b)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-A*cot(d*x+c)/a/d/(a+b*tan(d*x+c))^2- 
b*(A*a^4+6*A*a^2*b^2+3*A*b^4-3*B*a^3*b-B*a*b^3)/a^3/(a^2+b^2)^2/d/(a+b*tan 
(d*x+c))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.28 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {A \cot (c+d x)}{a^3 d}+\frac {(A+i B) \log (i-\tan (c+d x))}{2 (i a-b)^3 d}-\frac {(3 A b-a B) \log (\tan (c+d x))}{a^4 d}-\frac {(i A+B) \log (i+\tan (c+d x))}{2 (a-i b)^3 d}+\frac {b^2 \left (10 a^4 A b+9 a^2 A b^3+3 A b^5-6 a^5 B-3 a^3 b^2 B-a b^4 B\right ) \log (a+b \tan (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b^2 (A b-a B)}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b^2 \left (4 a^2 A b+2 A b^3-3 a^3 B-a b^2 B\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Input:

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

Output:

-((A*Cot[c + d*x])/(a^3*d)) + ((A + I*B)*Log[I - Tan[c + d*x]])/(2*(I*a - 
b)^3*d) - ((3*A*b - a*B)*Log[Tan[c + d*x]])/(a^4*d) - ((I*A + B)*Log[I + T 
an[c + d*x]])/(2*(a - I*b)^3*d) + (b^2*(10*a^4*A*b + 9*a^2*A*b^3 + 3*A*b^5 
 - 6*a^5*B - 3*a^3*b^2*B - a*b^4*B)*Log[a + b*Tan[c + d*x]])/(a^4*(a^2 + b 
^2)^3*d) - (b^2*(A*b - a*B))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) 
- (b^2*(4*a^2*A*b + 2*A*b^3 - 3*a^3*B - a*b^2*B))/(a^3*(a^2 + b^2)^2*d*(a 
+ b*Tan[c + d*x]))
 

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4092, 3042, 4132, 27, 3042, 4132, 3042, 4134, 3042, 25, 3956, 4013}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4092

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4134

\(\displaystyle -\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^2 (3 A b-a B) \int \cot (c+d x)dx}{a}-\frac {b^2 \left (-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^3 x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4 A-3 a^3 b B+6 a^2 A b^2-a b^3 B+3 A b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^2 A-a b B+3 A b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^2 (3 A b-a B) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b^2 \left (-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^3 x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4 A-3 a^3 b B+6 a^2 A b^2-a b^3 B+3 A b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^2 A-a b B+3 A b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {-\frac {\left (a^2+b^2\right )^2 (3 A b-a B) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b^2 \left (-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {a^3 x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4 A-3 a^3 b B+6 a^2 A b^2-a b^3 B+3 A b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^2 A-a b B+3 A b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 3956

\(\displaystyle -\frac {\frac {\frac {-\frac {b^2 \left (-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {\left (a^2+b^2\right )^2 (3 A b-a B) \log (-\sin (c+d x))}{a d}+\frac {a^3 x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (a^4 A-3 a^3 b B+6 a^2 A b^2-a b^3 B+3 A b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b \left (2 a^2 A-a b B+3 A b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 4013

\(\displaystyle -\frac {\frac {b \left (2 a^2 A-a b B+3 A b^2\right )}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {b \left (a^4 A-3 a^3 b B+6 a^2 A b^2-a b^3 B+3 A b^4\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {\left (a^2+b^2\right )^2 (3 A b-a B) \log (-\sin (c+d x))}{a d}+\frac {a^3 x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{a^2+b^2}-\frac {b^2 \left (-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}\)

Input:

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

Output:

-((A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^2)) - ((b*(2*a^2*A + 3*A*b^2 
- a*b*B))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((a^3*(a^3*A - 3*a 
*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2) + ((a^2 + b^2)^2*(3*A*b - a*B)* 
Log[-Sin[c + d*x]])/(a*d) - (b^2*(10*a^4*A*b + 9*a^2*A*b^3 + 3*A*b^5 - 6*a 
^5*B - 3*a^3*b^2*B - a*b^4*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^ 
2 + b^2)*d))/(a*(a^2 + b^2)) + (b*(a^4*A + 6*a^2*A*b^2 + 3*A*b^4 - 3*a^3*b 
*B - a*b^3*B))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(a*(a^2 + b^2)))/a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4013

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* 
(x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si 
n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && 
 NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
 

rule 4092

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[b*(A*b - a*B)*(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[b* 
B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2 
)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n 
+ 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && 
 NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(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 = 1.03 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {A}{a^{3} \tan \left (d x +c \right )}+\frac {\left (-3 A b +B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}-\frac {b^{2} \left (4 A \,a^{2} b +2 A \,b^{3}-3 B \,a^{3}-B a \,b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{2} \left (10 A \,a^{4} b +9 A \,a^{2} b^{3}+3 A \,b^{5}-6 B \,a^{5}-3 B \,a^{3} b^{2}-B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{4}}-\frac {\left (A b -B a \right ) b^{2}}{2 \left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(289\)
default	\(\frac {\frac {\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {A}{a^{3} \tan \left (d x +c \right )}+\frac {\left (-3 A b +B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}-\frac {b^{2} \left (4 A \,a^{2} b +2 A \,b^{3}-3 B \,a^{3}-B a \,b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{2} \left (10 A \,a^{4} b +9 A \,a^{2} b^{3}+3 A \,b^{5}-6 B \,a^{5}-3 B \,a^{3} b^{2}-B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{4}}-\frac {\left (A b -B a \right ) b^{2}}{2 \left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(289\)
parallelrisch	\(\frac {20 \left (A \,a^{4} b +\frac {9}{10} A \,a^{2} b^{3}+\frac {3}{10} A \,b^{5}-\frac {3}{5} B \,a^{5}-\frac {3}{10} B \,a^{3} b^{2}-\frac {1}{10} B a \,b^{4}\right ) b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (a +b \tan \left (d x +c \right )\right )+3 a^{4} \left (A \,a^{2} b -\frac {1}{3} A \,b^{3}-\frac {1}{3} B \,a^{3}+B a \,b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\sec \left (d x +c \right )^{2}\right )-6 \left (A b -\frac {B a}{3}\right ) \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\tan \left (d x +c \right )\right )-2 b^{2} \left (-\frac {9 A \,b^{7}}{2}+\frac {3 B a \,b^{6}}{2}-13 A \,a^{2} b^{5}+5 B \,a^{3} b^{4}-\frac {21 a^{4} \left (\frac {2 B d x}{21}+A \right ) b^{3}}{2}-3 a^{5} \left (A d x -\frac {7 B}{6}\right ) b^{2}-2 a^{6} \left (-\frac {3 B d x}{2}+A \right ) b +A \,a^{7} d x \right ) \tan \left (d x +c \right )^{2}-4 a b \left (-3 A \,b^{7}+B a \,b^{6}-\frac {17 A \,a^{2} b^{5}}{2}+3 B \,a^{3} b^{4}-7 a^{4} \left (\frac {B d x}{7}+A \right ) b^{3}-3 a^{5} \left (A d x -\frac {2 B}{3}\right ) b^{2}-\frac {3 a^{6} \left (-2 B d x +A \right ) b}{2}+A \,a^{7} d x \right ) \tan \left (d x +c \right )-2 a^{3} \left (A \left (a^{2}+b^{2}\right )^{3} \cot \left (d x +c \right )+a^{3} d x \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right )\right )}{2 \left (a^{2}+b^{2}\right )^{3} a^{4} d \left (a +b \tan \left (d x +c \right )\right )^{2}}\)	\(435\)
norman	\(\frac {\frac {b \left (3 A \,a^{4} b +11 A \,a^{2} b^{3}+6 A \,b^{5}-4 B \,a^{3} b^{2}-2 B a \,b^{4}\right ) \tan \left (d x +c \right )^{2}}{d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {A}{a d}-\frac {b^{2} \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) x \tan \left (d x +c \right )^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (4 A \,a^{4} b +17 A \,a^{2} b^{3}+9 A \,b^{5}-7 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \tan \left (d x +c \right )^{3}}{2 d \,a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 b \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) a x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (10 A \,a^{4} b +9 A \,a^{2} b^{3}+3 A \,b^{5}-6 B \,a^{5}-3 B \,a^{3} b^{2}-B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) a^{4} d}-\frac {\left (3 A b -B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}\)	\(558\)
risch	\(\text {Expression too large to display}\)	\(1550\)

Input:

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBO 
SE)
 

Output:

1/d*(1/(a^2+b^2)^3*(1/2*(3*A*a^2*b-A*b^3-B*a^3+3*B*a*b^2)*ln(1+tan(d*x+c)^ 
2)+(-A*a^3+3*A*a*b^2-3*B*a^2*b+B*b^3)*arctan(tan(d*x+c)))-1/a^3*A/tan(d*x+ 
c)+(-3*A*b+B*a)/a^4*ln(tan(d*x+c))-b^2*(4*A*a^2*b+2*A*b^3-3*B*a^3-B*a*b^2) 
/(a^2+b^2)^2/a^3/(a+b*tan(d*x+c))+b^2*(10*A*a^4*b+9*A*a^2*b^3+3*A*b^5-6*B* 
a^5-3*B*a^3*b^2-B*a*b^4)/(a^2+b^2)^3/a^4*ln(a+b*tan(d*x+c))-1/2*(A*b-B*a)* 
b^2/(a^2+b^2)/a^2/(a+b*tan(d*x+c))^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 917 vs. \(2 (283) = 566\).

Time = 0.24 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.20 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-2)]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.58 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (6 \, B a^{5} b^{2} - 10 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 9 \, A a^{2} b^{5} + B a b^{6} - 3 \, A b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}} + \frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, A a^{6} + 4 \, A a^{4} b^{2} + 2 \, A a^{2} b^{4} + 2 \, {\left (A a^{4} b^{2} - 3 \, B a^{3} b^{3} + 6 \, A a^{2} b^{4} - B a b^{5} + 3 \, A b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, A a^{5} b - 7 \, B a^{4} b^{2} + 17 \, A a^{3} b^{3} - 3 \, B a^{2} b^{4} + 9 \, A a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \tan \left (d x + c\right )} - \frac {2 \, {\left (B a - 3 \, A b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.62 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} - \frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} - \frac {{\left (6 \, B a^{5} b^{3} - 10 \, A a^{4} b^{4} + 3 \, B a^{3} b^{5} - 9 \, A a^{2} b^{6} + B a b^{7} - 3 \, A b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b d + 3 \, a^{8} b^{3} d + 3 \, a^{6} b^{5} d + a^{4} b^{7} d} + \frac {{\left (B a - 3 \, A b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4} d} - \frac {2 \, A a^{9} + 6 \, A a^{7} b^{2} + 6 \, A a^{5} b^{4} + 2 \, A a^{3} b^{6} + 2 \, {\left (A a^{7} b^{2} - 3 \, B a^{6} b^{3} + 7 \, A a^{5} b^{4} - 4 \, B a^{4} b^{5} + 9 \, A a^{3} b^{6} - B a^{2} b^{7} + 3 \, A a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, A a^{8} b - 7 \, B a^{7} b^{2} + 21 \, A a^{6} b^{3} - 10 \, B a^{5} b^{4} + 26 \, A a^{4} b^{5} - 3 \, B a^{3} b^{6} + 9 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{4} d \tan \left (d x + c\right )} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.69 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.32 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-6\,B\,a^5+10\,A\,a^4\,b-3\,B\,a^3\,b^2+9\,A\,a^2\,b^3-B\,a\,b^4+3\,A\,b^5\right )}{a^4\,d\,{\left (a^2+b^2\right )}^3}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,A\,b-B\,a\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {A}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (A\,a^4\,b^2-3\,B\,a^3\,b^3+6\,A\,a^2\,b^4-B\,a\,b^5+3\,A\,b^6\right )}{a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,A\,a^4\,b-7\,B\,a^3\,b^2+17\,A\,a^2\,b^3-3\,B\,a\,b^4+9\,A\,b^5\right )}{2\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 3.01 (sec) , antiderivative size = 726, normalized size of antiderivative = 2.53 \[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(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**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))