3.1.36 \(\int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx\) [36]

3.1.36.1 Optimal result

Integrand size = 25, antiderivative size = 164 \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx=\frac {5 e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac {e \sqrt {e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )} \]

output

5/8*e^(3/2)*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a^3/d+1/4*e^(3/2)*arctan( 
1/2*(e^(1/2)-cot(d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d*x+c))^(1/2))/a^3/d*2^(1/ 
2)-1/4*e*(e*cot(d*x+c))^(1/2)/a/d/(a+a*cot(d*x+c))^2+1/8*e*(e*cot(d*x+c))^ 
(1/2)/d/(a^3+a^3*cot(d*x+c))
 

3.1.36.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 4.13 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.23 \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx=-\frac {e \left (70 \sqrt {e} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+\frac {20 e \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )}{\sqrt [4]{-e^2}}-10 \sqrt {2} \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+10 \sqrt {2} \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-\frac {20 e \text {arctanh}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )}{\sqrt [4]{-e^2}}-80 \sqrt {e \cot (c+d x)}-\frac {10 \sqrt {e \cot (c+d x)} (3+5 \cot (c+d x))}{(1+\cot (c+d x))^2}+\frac {16 (e \cot (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},-\cot (c+d x)\right )}{e^2}-5 \sqrt {2} \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )+5 \sqrt {2} \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )\right )}{80 a^3 d} \]

input

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

output

-1/80*(e*(70*Sqrt[e]*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]] + (20*e*ArcTan[S 
qrt[e*Cot[c + d*x]]/(-e^2)^(1/4)])/(-e^2)^(1/4) - 10*Sqrt[2]*Sqrt[e]*ArcTa 
n[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] + 10*Sqrt[2]*Sqrt[e]*ArcTan[ 
1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]] - (20*e*ArcTanh[Sqrt[e*Cot[c + 
 d*x]]/(-e^2)^(1/4)])/(-e^2)^(1/4) - 80*Sqrt[e*Cot[c + d*x]] - (10*Sqrt[e* 
Cot[c + d*x]]*(3 + 5*Cot[c + d*x]))/(1 + Cot[c + d*x])^2 + (16*(e*Cot[c + 
d*x])^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -Cot[c + d*x]])/e^2 - 5*Sqrt[2] 
*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]] 
] + 5*Sqrt[2]*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e* 
Cot[c + d*x]]]))/(a^3*d)
 

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

Time = 1.33 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4050, 27, 3042, 4132, 25, 3042, 4136, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4050

\(\displaystyle -\frac {\int \frac {-3 a \cot ^2(c+d x) e^2+a e^2-4 a \cot (c+d x) e^2}{2 \sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)^2}dx}{4 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {-3 a \cot ^2(c+d x) e^2+a e^2-4 a \cot (c+d x) e^2}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)^2}dx}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle -\frac {-\frac {\int -\frac {a^3 e^3+a^3 \cot ^2(c+d x) e^3-8 a^3 \cot (c+d x) e^3}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int \frac {a^3 e^3+a^3 \cot ^2(c+d x) e^3-8 a^3 \cot (c+d x) e^3}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

\(\displaystyle -\frac {\frac {5 a^3 e^3 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx+\frac {\int -\frac {8 \left (e^3 a^4+e^3 \cot (c+d x) a^4\right )}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {5 a^3 e^3 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx-\frac {4 \int \frac {e^3 a^4+e^3 \cot (c+d x) a^4}{\sqrt {e \cot (c+d x)}}dx}{a^2}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 \int \frac {a^4 e^3-a^4 e^3 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 4015

\(\displaystyle -\frac {\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {8 a^6 e^6 \int \frac {1}{-2 e^6 a^8-\left (a^4 e^3-a^4 e^3 \cot (c+d x)\right )^2 \tan (c+d x)}d\frac {a^4 e^3-a^4 e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}}{d}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\frac {5 a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 \sqrt {2} a^2 e^{5/2} \arctan \left (\frac {a^4 e^3-a^4 e^3 \cot (c+d x)}{\sqrt {2} a^4 e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {\frac {\frac {5 a^3 e^3 \int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 \sqrt {2} a^2 e^{5/2} \arctan \left (\frac {a^4 e^3-a^4 e^3 \cot (c+d x)}{\sqrt {2} a^4 e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {5 a^2 e^3 \int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 \sqrt {2} a^2 e^{5/2} \arctan \left (\frac {a^4 e^3-a^4 e^3 \cot (c+d x)}{\sqrt {2} a^4 e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {-\frac {10 a^2 e^2 \int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{d}-\frac {4 \sqrt {2} a^2 e^{5/2} \arctan \left (\frac {a^4 e^3-a^4 e^3 \cot (c+d x)}{\sqrt {2} a^4 e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {\frac {10 a^2 e^{5/2} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}-\frac {4 \sqrt {2} a^2 e^{5/2} \arctan \left (\frac {a^4 e^3-a^4 e^3 \cot (c+d x)}{\sqrt {2} a^4 e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{2 a^3 e}-\frac {e \sqrt {e \cot (c+d x)}}{d (a \cot (c+d x)+a)}}{8 a^2}-\frac {e \sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2}\)

input

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

output

-1/4*(e*Sqrt[e*Cot[c + d*x]])/(a*d*(a + a*Cot[c + d*x])^2) - (((10*a^2*e^( 
5/2)*ArcTan[Cot[c + d*x]/Sqrt[e]])/d - (4*Sqrt[2]*a^2*e^(5/2)*ArcTan[(a^4* 
e^3 - a^4*e^3*Cot[c + d*x])/(Sqrt[2]*a^4*e^(5/2)*Sqrt[e*Cot[c + d*x]])])/d 
)/(2*a^3*e) - (e*Sqrt[e*Cot[c + d*x]])/(d*(a + a*Cot[c + d*x])))/(8*a^2)
 

3.1.36.3.1 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 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 4050

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

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]
 

3.1.36.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs. \(2(135)=270\).

Time = 0.04 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.13

method	result	size
derivativedivides	\(-\frac {2 e^{4} \left (\frac {\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{2}}-\frac {\frac {\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}-\frac {e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {5 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{2}}\right )}{d \,a^{3}}\)	\(349\)
default	\(-\frac {2 e^{4} \left (\frac {\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{2}}-\frac {\frac {\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}-\frac {e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {5 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{2}}\right )}{d \,a^{3}}\)	\(349\)

input

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

output

-2/d/a^3*e^4*(1/4/e^2*(1/8/e*(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) 
)+1/8/(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)))-1/4/e^2*((1/4*(e*cot 
(d*x+c))^(3/2)-1/4*e*(e*cot(d*x+c))^(1/2))/(e*cot(d*x+c)+e)^2+5/4/e^(1/2)* 
arctan((e*cot(d*x+c))^(1/2)/e^(1/2))))
 

3.1.36.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.25 \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx=\left [\frac {2 \, {\left (\sqrt {2} e \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2} e\right )} \sqrt {-e} \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 ) + 5 \, {\left (e \sin \left (2 \, d x + 2 \, c\right ) + e\right )} \sqrt {-e} \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 ) + {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e \sin \left (2 \, d x + 2 \, c\right ) - e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{16 \, {\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}, \frac {4 \, {\left (\sqrt {2} e \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2} e\right )} \sqrt {e} \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 ) + 10 \, {\left (e \sin \left (2 \, d x + 2 \, c\right ) + e\right )} \sqrt {e} \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 ) + {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e \sin \left (2 \, d x + 2 \, c\right ) - e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{16 \, {\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}\right ] \]

input

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

output

[1/16*(2*(sqrt(2)*e*sin(2*d*x + 2*c) + sqrt(2)*e)*sqrt(-e)*log(-(sqrt(2)*c 
os(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) + 5*(e*si 
n(2*d*x + 2*c) + e)*sqrt(-e)*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)) + (e*cos(2*d*x + 2*c) 
+ e*sin(2*d*x + 2*c) - e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) 
/(a^3*d*sin(2*d*x + 2*c) + a^3*d), 1/16*(4*(sqrt(2)*e*sin(2*d*x + 2*c) + s 
qrt(2)*e)*sqrt(e)*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)) + 10*(e*sin(2*d*x + 2*c) + e)*sqrt(e)*arctan(s 
qrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/sqrt(e)) + (e*cos(2*d*x + 2 
*c) + e*sin(2*d*x + 2*c) - e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2* 
c)))/(a^3*d*sin(2*d*x + 2*c) + a^3*d)]
 

3.1.36.6 Sympy [F]

\[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx=\frac {\int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\cot ^{3}{\left (c + d x \right )} + 3 \cot ^{2}{\left (c + d x \right )} + 3 \cot {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

input

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

output

Integral((e*cot(c + d*x))**(3/2)/(cot(c + d*x)**3 + 3*cot(c + d*x)**2 + 3* 
cot(c + d*x) + 1), x)/a**3
 

3.1.36.7 Maxima [F(-2)]

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

input

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

3.1.36.8 Giac [F]

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

input

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

output

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

3.1.36.9 Mupad [B] (verification not implemented)

Time = 13.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.09 \[ \int \frac {(e \cot (c+d x))^{3/2}}{(a+a \cot (c+d x))^3} \, dx=\frac {5\,e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{8\,a^3\,d}-\frac {\frac {e^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{8}-\frac {e^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{8}}{d\,a^3\,e^2\,{\mathrm {cot}\left (c+d\,x\right )}^2+2\,d\,a^3\,e^2\,\mathrm {cot}\left (c+d\,x\right )+d\,a^3\,e^2}-\frac {\sqrt {2}\,e^{3/2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{8\,a^3\,d} \]

input

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

output

(5*e^(3/2)*atan((e*cot(c + d*x))^(1/2)/e^(1/2)))/(8*a^3*d) - ((e^3*(e*cot( 
c + d*x))^(1/2))/8 - (e^2*(e*cot(c + d*x))^(3/2))/8)/(a^3*d*e^2 + a^3*d*e^ 
2*cot(c + d*x)^2 + 2*a^3*d*e^2*cot(c + d*x)) - (2^(1/2)*e^(3/2)*(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)))))/ 
(8*a^3*d)