\(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [544]

Optimal result

Integrand size = 23, antiderivative size = 192 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {b \left (a^2+3 b^2\right )}{a^2 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}} \] Output:

3*b*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d+I*arctanh((a+b*tan(d 
*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(3/2)/d-I*arctanh((a+b*tan(d*x+c))^(1/ 
2)/(a+I*b)^(1/2))/(a+I*b)^(3/2)/d-b*(a^2+3*b^2)/a^2/(a^2+b^2)/d/(a+b*tan(d 
*x+c))^(1/2)-cot(d*x+c)/a/d/(a+b*tan(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {i a^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2}}+\frac {i a^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2}}+\frac {b \left (a^2+3 b^2\right )}{\left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {a \cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}}{a^2 d} \] Input:

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

Output:

-(((-3*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] - (I*a^2*ArcTa 
nh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(a - I*b)^(3/2) + (I*a^2*ArcTa 
nh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(a + I*b)^(3/2) + (b*(a^2 + 3* 
b^2))/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) + (a*Cot[c + d*x])/Sqrt[a + b 
*Tan[c + d*x]])/(a^2*d))
 

Rubi [A] (warning: unable to verify)

Time = 1.63 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.16, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2+3 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {6 \left (a^2+b^2\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{d}+2 \left (\frac {a^2 (a+i b) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^2 (a-i b) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}-\frac {\frac {2 b \left (a^2+3 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {6 b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+2 \left (\frac {a^2 (a+i b) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {a^2 (a-i b) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}\)

Input:

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

Output:

-(Cot[c + d*x]/(a*d*Sqrt[a + b*Tan[c + d*x]])) - ((2*((a^2*(a + I*b)*ArcTa 
n[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) + (a^2*(a - I*b)*ArcTan[T 
an[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)) - (6*b*(a^2 + b^2)*ArcTanh[ 
Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d))/(a*(a^2 + b^2)) + (2*b*(a^ 
2 + 3*b^2))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]))/(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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

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

rule 4020

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

rule 4022

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2   Int[(a + b*Tan[e + f*x])^m*( 
1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2   Int[(a + b*Tan[e + f*x])^m 
*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c 
 - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]
 

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 4117

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/f   Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; 
FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
 

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 4136

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) 
+ (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2)   Int[(c + d*Tan[e + f*x])^ 
n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ 
(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)   Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ 
e + f*x]^2)/(a + b*Tan[e + f*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] & 
&  !GtQ[n, 0] &&  !LeQ[n, -1]
 

Maple [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2113 vs. \(2 (158) = 316\).

Time = 0.21 (sec) , antiderivative size = 4245, normalized size of antiderivative = 22.11 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[3,17,7]%%%}+%%%{8,[3,15,7]%%%}+%%%{28,[3,13,7]%%%}+ 
%%%{56,[3
 

Mupad [B] (verification not implemented)

Time = 1.85 (sec) , antiderivative size = 7971, normalized size of antiderivative = 41.52 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

log(72*a^14*b^25*d^4 - ((a + b*tan(c + d*x))^(1/2)*(144*a^14*b^26*d^5 + 86 
4*a^16*b^24*d^5 + 2048*a^18*b^22*d^5 + 2240*a^20*b^20*d^5 + 672*a^22*b^18* 
d^5 - 896*a^24*b^16*d^5 - 896*a^26*b^14*d^5 - 192*a^28*b^12*d^5 + 80*a^30* 
b^10*d^5 + 32*a^32*b^8*d^5) + (-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^ 
4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2 
)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(576*a^15 
*b^27*d^6 - ((a + b*tan(c + d*x))^(1/2)*(576*a^15*b^28*d^7 + 5184*a^17*b^2 
6*d^7 + 21568*a^19*b^24*d^7 + 53888*a^21*b^22*d^7 + 87808*a^23*b^20*d^7 + 
94976*a^25*b^18*d^7 + 66304*a^27*b^16*d^7 + 27008*a^29*b^14*d^7 + 4288*a^3 
1*b^12*d^7 - 832*a^33*b^10*d^7 - 320*a^35*b^8*d^7) - (-(((8*a^3*d^2 - 24*a 
*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) 
+ a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2 
*d^4)))^(1/2)*((a + b*tan(c + d*x))^(1/2)*(-(((8*a^3*d^2 - 24*a*b^2*d^2)^2 
/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 
 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/ 
2)*(512*a^18*b^28*d^9 + 5376*a^20*b^26*d^9 + 25344*a^22*b^24*d^9 + 70656*a 
^24*b^22*d^9 + 129024*a^26*b^20*d^9 + 161280*a^28*b^18*d^9 + 139776*a^30*b 
^16*d^9 + 82944*a^32*b^14*d^9 + 32256*a^34*b^12*d^9 + 7424*a^36*b^10*d^9 + 
 768*a^38*b^8*d^9) + 768*a^16*b^29*d^8 + 7680*a^18*b^27*d^8 + 34304*a^20*b 
^25*d^8 + 90112*a^22*b^23*d^8 + 154112*a^24*b^21*d^8 + 179200*a^26*b^19...
 

Reduce [F]

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

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

Output:

int((sqrt(tan(c + d*x)*b + a)*cot(c + d*x)**2)/(tan(c + d*x)**2*b**2 + 2*t 
an(c + d*x)*a*b + a**2),x)