3.1.23 \(\int \frac {(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx\) [23]

3.1.23.1 Optimal result

Integrand size = 25, antiderivative size = 111 \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx=\frac {e^{5/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d}-\frac {e^{5/2} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{a d} \]

output

e^(5/2)*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a/d-1/2*e^(5/2)*arctan(1/2*(e 
^(1/2)-cot(d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d*x+c))^(1/2))/a/d*2^(1/2)-2*e^2 
*(e*cot(d*x+c))^(1/2)/a/d
 

3.1.23.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(296\) vs. \(2(111)=222\).

Time = 0.92 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.67 \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx=\frac {e \left (8 e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-4 \left (-e^2\right )^{3/4} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )-2 \sqrt {2} e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+2 \sqrt {2} e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+4 \left (-e^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )-16 e \sqrt {e \cot (c+d x)}-\sqrt {2} e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )+\sqrt {2} e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )\right )}{8 a d} \]

input

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

output

(e*(8*e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]] - 4*(-e^2)^(3/4)*ArcTan 
[Sqrt[e*Cot[c + d*x]]/(-e^2)^(1/4)] - 2*Sqrt[2]*e^(3/2)*ArcTan[1 - (Sqrt[2 
]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] + 2*Sqrt[2]*e^(3/2)*ArcTan[1 + (Sqrt[2]*S 
qrt[e*Cot[c + d*x]])/Sqrt[e]] + 4*(-e^2)^(3/4)*ArcTanh[Sqrt[e*Cot[c + d*x] 
]/(-e^2)^(1/4)] - 16*e*Sqrt[e*Cot[c + d*x]] - Sqrt[2]*e^(3/2)*Log[Sqrt[e] 
+ Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]] + Sqrt[2]*e^(3/2)*L 
og[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]]))/(8*a*d 
)
 

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

Time = 0.90 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4049, 27, 3042, 4136, 3042, 4015, 218, 4117, 27, 73, 216}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{a-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 4049

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4015

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

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\frac {1}{2} a e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {e^{5/2} \arctan \left (\frac {a^2 e^3-a^2 e^3 \cot (c+d x)}{\sqrt {2} a^2 e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} d}}{a}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {e^{5/2} \arctan \left (\frac {a^2 e^3-a^2 e^3 \cot (c+d x)}{\sqrt {2} a^2 e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} d}+\frac {e^{5/2} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}}{a}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{a d}\)

input

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

output

-(((e^(5/2)*ArcTan[Cot[c + d*x]/Sqrt[e]])/d + (e^(5/2)*ArcTan[(a^2*e^3 - a 
^2*e^3*Cot[c + d*x])/(Sqrt[2]*a^2*e^(5/2)*Sqrt[e*Cot[c + d*x]])])/(Sqrt[2] 
*d))/a) - (2*e^2*Sqrt[e*Cot[c + d*x]])/(a*d)
 

3.1.23.3.1 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 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 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 218

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

rule 3042

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

rule 4015

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[-2*(d^2/f)   Subst[Int[1/(2*c*d + b*x^2), x], x, (c 
- d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && 
 EqQ[c^2 - d^2, 0]
 

rule 4049

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 - 2)*((c 
+ d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) 
 Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*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] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !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 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]
 

3.1.23.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(311\) vs. \(2(92)=184\).

Time = 0.10 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.81

method	result	size
derivativedivides	\(-\frac {2 e^{2} \left (\sqrt {e \cot \left (d x +c \right )}-\frac {e \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{2}-\frac {\sqrt {e}\, \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2}\right )}{d a}\)	\(312\)
default	\(-\frac {2 e^{2} \left (\sqrt {e \cot \left (d x +c \right )}-\frac {e \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{2}-\frac {\sqrt {e}\, \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2}\right )}{d a}\)	\(312\)

input

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

output

-2/d/a*e^2*((e*cot(d*x+c))^(1/2)-1/2*e*(1/8/e*(e^2)^(1/4)*2^(1/2)*(ln((e*c 
ot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x 
+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2 
)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot 
(d*x+c))^(1/2)+1))+1/8/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*( 
e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot( 
d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d* 
x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-1/2 
*e^(1/2)*arctan((e*cot(d*x+c))^(1/2)/e^(1/2)))
 

3.1.23.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.60 \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx=\left [\frac {\sqrt {2} \sqrt {-e} e^{2} \log \left ({\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) - \sqrt {2}\right )} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 2 \, \sqrt {-e} e^{2} \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) + 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) - 8 \, e^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{4 \, a d}, -\frac {\sqrt {2} e^{\frac {5}{2}} \arctan \left (-\frac {{\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) - 2 \, e^{\frac {5}{2}} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right ) + 4 \, e^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \, a d}\right ] \]

input

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

output

[1/4*(sqrt(2)*sqrt(-e)*e^2*log((sqrt(2)*cos(2*d*x + 2*c) + sqrt(2)*sin(2*d 
*x + 2*c) - sqrt(2))*sqrt(-e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2* 
c)) - 2*e*sin(2*d*x + 2*c) + e) + 2*sqrt(-e)*e^2*log((e*cos(2*d*x + 2*c) - 
 e*sin(2*d*x + 2*c) + 2*sqrt(-e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 
 2*c))*sin(2*d*x + 2*c) + e)/(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)) - 
8*e^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(a*d), -1/2*(sqrt(2 
)*e^(5/2)*arctan(-1/2*(sqrt(2)*cos(2*d*x + 2*c) - sqrt(2)*sin(2*d*x + 2*c) 
 + sqrt(2))*sqrt(e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/(e*cos 
(2*d*x + 2*c) + e)) - 2*e^(5/2)*arctan(sqrt((e*cos(2*d*x + 2*c) + e)/sin(2 
*d*x + 2*c))/sqrt(e)) + 4*e^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2* 
c)))/(a*d)]
 

3.1.23.6 Sympy [F]

\[ \int \frac {(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx=\frac {\int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}{\cot {\left (c + d x \right )} + 1}\, dx}{a} \]

input

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

output

Integral((e*cot(c + d*x))**(5/2)/(cot(c + d*x) + 1), x)/a
 

3.1.23.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.1.23.8 Giac [F]

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

input

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

output

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

3.1.23.9 Mupad [B] (verification not implemented)

Time = 12.93 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.11 \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+a \cot (c+d x)} \, dx=\frac {e^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{a\,d}-\frac {2\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{a\,d}+\frac {\sqrt {2}\,e^{5/2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{4\,a\,d} \]

input

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

output

(e^(5/2)*atan((e*cot(c + d*x))^(1/2)/e^(1/2)))/(a*d) - (2*e^2*(e*cot(c + d 
*x))^(1/2))/(a*d) + (2^(1/2)*e^(5/2)*(2*atan((2^(1/2)*(e*cot(c + d*x))^(1/ 
2))/(2*e^(1/2))) + 2*atan((2^(1/2)*(e*cot(c + d*x))^(1/2))/(2*e^(1/2)) + ( 
2^(1/2)*(e*cot(c + d*x))^(3/2))/(2*e^(3/2)))))/(4*a*d)