3.6.52 \(\int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) [552]

3.6.52.1 Optimal result

Integrand size = 21, antiderivative size = 195 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}+\frac {2 b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 b^2 \left (3 a^2+b^2\right )}{a^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]

output

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

3.6.52.2 Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.22 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {2 \left (\frac {-\frac {3 \left (a^2+b^2\right )^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {3 a^2 (a+i b)^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{2 \sqrt {a-i b}}+\frac {3 a^2 (a-i b)^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{2 \sqrt {a+i b}}}{a \left (a^2+b^2\right )}+\frac {b^2}{(a+b \tan (c+d x))^{3/2}}+\frac {3 b^2 \left (3 a^2+b^2\right )}{a \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}\right )}{3 a \left (a^2+b^2\right ) d} \]

input

Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x])^(5/2),x]
 

output

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

3.6.52.3 Rubi [A] (warning: unable to verify)

Time = 1.70 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.26, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4136, 25, 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.

\(\displaystyle \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (c+d x) (a+b \tan (c+d x))^{5/2}}dx\)

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

input

Int[Cot[c + d*x]/(a + b*Tan[c + d*x])^(5/2),x]
 

output

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

3.6.52.3.1 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]
 

3.6.52.4 Maple [F(-1)]

Timed out.

hanged

input

int(cot(d*x+c)/(a+b*tan(d*x+c))^(5/2),x)
 

output

int(cot(d*x+c)/(a+b*tan(d*x+c))^(5/2),x)
 

3.6.52.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3430 vs. \(2 (163) = 326\).

Time = 0.49 (sec) , antiderivative size = 6876, normalized size of antiderivative = 35.26 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

input

integrate(cot(d*x+c)/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")
 

output

Too large to include
 

3.6.52.6 Sympy [F]

\[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]

input

integrate(cot(d*x+c)/(a+b*tan(d*x+c))**(5/2),x)
 

output

Integral(cot(c + d*x)/(a + b*tan(c + d*x))**(5/2), x)
 

3.6.52.7 Maxima [F]

\[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(cot(d*x+c)/(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")
 

output

integrate(cot(d*x + c)/(b*tan(d*x + c) + a)^(5/2), x)
 

3.6.52.8 Giac [F(-1)]

Timed out. \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]

input

integrate(cot(d*x+c)/(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")
 

output

Timed out
 

3.6.52.9 Mupad [B] (verification not implemented)

Time = 6.14 (sec) , antiderivative size = 13727, normalized size of antiderivative = 70.39 \[ \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

input

int(cot(c + d*x)/(a + b*tan(c + d*x))^(5/2),x)
 

output

(log((((a + b*tan(c + d*x))^(1/2)*(10304*a^20*b^34*d^5 - 512*a^16*b^38*d^5 
 - 544*a^18*b^36*d^5 - 64*a^14*b^40*d^5 + 66976*a^22*b^32*d^5 + 221312*a^2 
4*b^30*d^5 + 480480*a^26*b^28*d^5 + 741312*a^28*b^26*d^5 + 837408*a^30*b^2 
4*d^5 + 695552*a^32*b^22*d^5 + 416416*a^34*b^20*d^5 + 168896*a^36*b^18*d^5 
 + 37856*a^38*b^16*d^5 - 896*a^40*b^14*d^5 - 3424*a^42*b^12*d^5 - 960*a^44 
*b^10*d^5 - 96*a^46*b^8*d^5) - ((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6 
*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 1 
0*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10* 
a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(384*a^15*b^42*d^6 - ((((((20*a^2*b^8* 
d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) 
 + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^ 
8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(512*a^16* 
b^46*d^8 - ((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d 
^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10 
*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8* 
b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^18*b^46*d^9 + 9984*a^20* 
b^44*d^9 + 92160*a^22*b^42*d^9 + 535296*a^24*b^40*d^9 + 2193408*a^26*b^38* 
d^9 + 6736896*a^28*b^36*d^9 + 16084992*a^30*b^34*d^9 + 30551040*a^32*b^32* 
d^9 + 46844928*a^34*b^30*d^9 + 58499584*a^36*b^28*d^9 + 59744256*a^38*b^26 
*d^9 + 49900032*a^40*b^24*d^9 + 33945600*a^42*b^22*d^9 + 18643968*a^44*...