\(\int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [867]

Optimal result

Integrand size = 25, antiderivative size = 281 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {i \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a-b)^{3/2} d}-\frac {i \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a+b)^{3/2} d}+\frac {2 b^2 \left (5 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {8 b \sqrt {\cot (c+d x)}}{3 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d \sqrt {a+b \tan (c+d x)}} \] Output:

-I*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c 
)^(1/2)*tan(d*x+c)^(1/2)/(I*a-b)^(3/2)/d-I*arctanh((I*a+b)^(1/2)*tan(d*x+c 
)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/(I*a+b)^ 
(3/2)/d+2/3*b^2*(5*a^2+8*b^2)/a^3/(a^2+b^2)/d/cot(d*x+c)^(1/2)/(a+b*tan(d* 
x+c))^(1/2)+8/3*b*cot(d*x+c)^(1/2)/a^2/d/(a+b*tan(d*x+c))^(1/2)-2/3*cot(d* 
x+c)^(3/2)/a/d/(a+b*tan(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 3.90 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\sqrt {\cot (c+d x)} \left (-\frac {3 (-1)^{3/4} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{(-a+i b)^{3/2}}+\frac {3 (-1)^{3/4} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{(a+i b)^{3/2}}+\frac {-2 a^2 \left (a^2+b^2\right ) \cot (c+d x)+2 b \left (4 a \left (a^2+b^2\right )+b \left (5 a^2+8 b^2\right ) \tan (c+d x)\right )}{a^3 \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}\right )}{3 d} \] Input:

Integrate[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(3/2),x]
 

Output:

(Sqrt[Cot[c + d*x]]*((-3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt 
[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(-a + I*b)^( 
3/2) + (3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/ 
Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(a + I*b)^(3/2) + (-2*a^2*(a 
^2 + b^2)*Cot[c + d*x] + 2*b*(4*a*(a^2 + b^2) + b*(5*a^2 + 8*b^2)*Tan[c + 
d*x]))/(a^3*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])))/(3*d)
 

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.07, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 4729, 3042, 4052, 27, 3042, 4132, 27, 3042, 4133, 27, 3042, 4099, 3042, 4098, 104, 216, 219}

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.

\(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cot (c+d x)^{5/2}}{(a+b \tan (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 4729

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {1}{\tan (c+d x)^{5/2} (a+b \tan (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {-\frac {2 \int -\frac {3 a^2-8 b^2-8 b^2 \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}dx}{a}-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}}{3 a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\int \frac {3 a^2-8 b^2-8 b^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}dx}{a}-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}}{3 a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\int \frac {3 a^2-8 b^2-8 b^2 \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}dx}{a}-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}}{3 a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}\right )\)

\(\Big \downarrow \) 4133

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {2 \int \frac {3 \left (a^4-a^3 b \tan (c+d x)\right )}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a}-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}}{3 a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {3 \int \frac {a^4-a^3 b \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a}-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}}{3 a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {3 \int \frac {a^4-a^3 b \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a}-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}}{3 a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}\right )\)

\(\Big \downarrow \) 4099

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {1}{2} a^3 (a-i b) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^3 (a+i b) \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{a}}{3 a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {1}{2} a^3 (a-i b) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^3 (a+i b) \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{a}}{3 a}\right )\)

\(\Big \downarrow \) 4098

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {a^3 (a+i b) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}+\frac {a^3 (a-i b) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}\right )}{a \left (a^2+b^2\right )}}{a}}{3 a}\right )\)

\(\Big \downarrow \) 104

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {a^3 (a-i b) \int \frac {1}{\frac {(i a-b) \tan (c+d x)}{a+b \tan (c+d x)}+1}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}+\frac {a^3 (a+i b) \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}\right )}{a \left (a^2+b^2\right )}}{a}}{3 a}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {a^3 (a+i b) \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}+\frac {a^3 (a-i b) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}\right )}{a \left (a^2+b^2\right )}}{a}}{3 a}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {8 b}{a d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {a^3 (a-i b) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {a^3 (a+i b) \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}}\right )}{a \left (a^2+b^2\right )}}{a}}{3 a}\right )\)

Input:

Int[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(3/2),x]
 

Output:

Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-2/(3*a*d*Tan[c + d*x]^(3/2)*Sqrt[a 
 + b*Tan[c + d*x]]) - ((-8*b)/(a*d*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d 
*x]]) + ((3*((a^3*(a - I*b)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt 
[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + (a^3*(a + I*b)*ArcTanh[(Sqrt[I* 
a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)))/ 
(a*(a^2 + b^2)) - (2*b^2*(5*a^2 + 8*b^2)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2 
)*d*Sqrt[a + b*Tan[c + d*x]]))/a)/(3*a))
 

Defintions of rubi rules used

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

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

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 4098

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[A^2/f   Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* 
x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && 
 NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
 

rule 4099

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[(A + I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T 
an[e + f*x]), x], x] + Simp[(A - I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d 
*Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A 
, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 
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 4133

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*b^2 + a^2*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*Sim 
p[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n 
 + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*( 
m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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 4729

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a 
+ b*x])^m*(c*Tan[a + b*x])^m   Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ[u, 
x]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2460\) vs. \(2(233)=466\).

Time = 4.91 (sec) , antiderivative size = 2461, normalized size of antiderivative = 8.76

method	result	size
default	\(\text {Expression too large to display}\)	\(2461\)

Input:

int(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/12/d*2^(1/2)/a^3/(a^2+b^2)^(3/2)/(-b+(a^2+b^2)^(1/2))^(1/2)*(a+b*tan(d* 
x+c))^(1/2)*cot(d*x+c)^(5/2)/(a*cos(d*x+c)+b*sin(d*x+c))*(4*sin(d*x+c)*2^( 
1/2)*a^4*(-b+(a^2+b^2)^(1/2))^(1/2)*(a^2+b^2)^(1/2)-32*2^(1/2)*b^4*(-b+(a^ 
2+b^2)^(1/2))^(1/2)*(a^2+b^2)^(1/2)*sin(d*x+c)*tan(d*x+c)^2-16*2^(1/2)*a^3 
*(-b+(a^2+b^2)^(1/2))^(1/2)*(a^2+b^2)^(1/2)*b*sin(d*x+c)*tan(d*x+c)-2^(1/2 
)*(-24*sin(d*x+c)+20*tan(d*x+c)*sec(d*x+c))*b^2*a^2*(a^2+b^2)^(1/2)*(-b+(a 
^2+b^2)^(1/2))^(1/2)-16*2^(1/2)*b^3*(-b+(a^2+b^2)^(1/2))^(1/2)*(a^2+b^2)^( 
1/2)*a*sin(d*x+c)*tan(d*x+c)+(-3*cos(d*x+c)-3)*((a*cos(d*x+c)+b*sin(d*x+c) 
)*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^ 
2)^(1/2))^(1/2)*(a^2+b^2)^(1/2)*a^2*b*ln(-(a*cot(d*x+c)*cos(d*x+c)+2*2^(1/ 
2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2 
+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*a*cot(d*x+c)+2*(a^2+b^2)^(1/2)*cos(d*x+c)+ 
2*cos(d*x+c)*b-sin(d*x+c)*a+a*csc(d*x+c)-2*(a^2+b^2)^(1/2)-2*b)/(-1+cos(d* 
x+c)))*tan(d*x+c)^2+(-3*cos(d*x+c)-3)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x 
+c)/(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2)) 
^(1/2)*a^4*ln(-(a*cot(d*x+c)*cos(d*x+c)+2*2^(1/2)*((a*cos(d*x+c)+b*sin(d*x 
+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+ 
c)-2*a*cot(d*x+c)+2*(a^2+b^2)^(1/2)*cos(d*x+c)+2*cos(d*x+c)*b-sin(d*x+c)*a 
+a*csc(d*x+c)-2*(a^2+b^2)^(1/2)-2*b)/(-1+cos(d*x+c)))*tan(d*x+c)^2+(3*cos( 
d*x+c)+3)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7848 vs. \(2 (227) = 454\).

Time = 1.25 (sec) , antiderivative size = 7848, normalized size of antiderivative = 27.93 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**(5/2)/(a+b*tan(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate(cot(d*x + c)^(5/2)/(b*tan(d*x + c) + a)^(3/2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

int(cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(3/2),x)
 

Output:

int(cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\cot \left (d x +c \right )^{\frac {5}{2}}}{\left (a +\tan \left (d x +c \right ) b \right )^{\frac {3}{2}}}d x \] Input:

int(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x)
 

Output:

int(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x)