3.3.31 \(\int \frac {\cot ^{\frac {5}{2}}(a+b \log (c x^n))}{x} \, dx\) [231]

3.3.31.1 Optimal result

Integrand size = 19, antiderivative size = 201 \[ \int \frac {\cot ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {2 \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \]

output

-2/3*cot(a+b*ln(c*x^n))^(3/2)/b/n+1/2*arctan(-1+2^(1/2)*cot(a+b*ln(c*x^n)) 
^(1/2))/b/n*2^(1/2)+1/2*arctan(1+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/b/n*2^( 
1/2)+1/4*ln(1+cot(a+b*ln(c*x^n))-2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/b/n*2^( 
1/2)-1/4*ln(1+cot(a+b*ln(c*x^n))+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/b/n*2^( 
1/2)
 

3.3.31.2 Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.60 \[ \int \frac {\cot ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {-3 \arctan \left (\sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{-\cot \left (a+b \log \left (c x^n\right )\right )}+3 \text {arctanh}\left (\sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{-\cot \left (a+b \log \left (c x^n\right )\right )}+2 \cot ^{\frac {7}{4}}\left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt [4]{\cot \left (a+b \log \left (c x^n\right )\right )}} \]

input

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

output

-1/3*(-3*ArcTan[(-Cot[a + b*Log[c*x^n]]^2)^(1/4)]*(-Cot[a + b*Log[c*x^n]]) 
^(1/4) + 3*ArcTanh[(-Cot[a + b*Log[c*x^n]]^2)^(1/4)]*(-Cot[a + b*Log[c*x^n 
]])^(1/4) + 2*Cot[a + b*Log[c*x^n]]^(7/4))/(b*n*Cot[a + b*Log[c*x^n]]^(1/4 
))
 

3.3.31.3 Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {3039, 3042, 3954, 3042, 3957, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {\cot ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \cot ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3954

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2}}\right )\right )}{b}-\frac {2 \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b}}{n}\)

input

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

output

((-2*Cot[a + b*Log[c*x^n]]^(3/2))/(3*b) + (2*((-(ArcTan[1 - Sqrt[2]*Sqrt[C 
ot[a + b*Log[c*x^n]]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c* 
x^n]]]]/Sqrt[2])/2 + (Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a 
+ b*Log[c*x^n]]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] 
 + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]))/2))/b)/n
 

3.3.31.3.1 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 217

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

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3039

Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst 
[[3]]   Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /;  !FalseQ[lst]] /; 
NonsumQ[u]
 

rule 3042

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

rule 3954

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d 
*x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2   Int[(b*Tan[c + d*x])^(n - 2), x] 
, x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

3.3.31.4 Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\frac {-\frac {2 {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\)	\(139\)
default	\(\frac {-\frac {2 {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\)	\(139\)

input

int(cot(a+b*ln(c*x^n))^(5/2)/x,x,method=_RETURNVERBOSE)
 

output

1/n/b*(-2/3*cot(a+b*ln(c*x^n))^(3/2)+1/4*2^(1/2)*(ln((1+cot(a+b*ln(c*x^n)) 
-2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))/(1+cot(a+b*ln(c*x^n))+2^(1/2)*cot(a+b*l 
n(c*x^n))^(1/2)))+2*arctan(1+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))+2*arctan(-1 
+2^(1/2)*cot(a+b*ln(c*x^n))^(1/2))))
 

3.3.31.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 696, normalized size of antiderivative = 3.46 \[ \int \frac {\cot ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

input

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

output

-1/6*(3*b*n*(-1/(b^4*n^4))^(1/4)*log((b*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*l 
og(x) + 2*b*log(c) + 2*a) + b*n*(-1/(b^4*n^4))^(1/4) + sqrt((cos(2*b*n*log 
(x) + 2*b*log(c) + 2*a) + 1)/sin(2*b*n*log(x) + 2*b*log(c) + 2*a))*sin(2*b 
*n*log(x) + 2*b*log(c) + 2*a))/(cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)) 
*sin(2*b*n*log(x) + 2*b*log(c) + 2*a) - 3*b*n*(-1/(b^4*n^4))^(1/4)*log(-(b 
*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n*(-1/(b^ 
4*n^4))^(1/4) - sqrt((cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)/sin(2*b*n* 
log(x) + 2*b*log(c) + 2*a))*sin(2*b*n*log(x) + 2*b*log(c) + 2*a))/(cos(2*b 
*n*log(x) + 2*b*log(c) + 2*a) + 1))*sin(2*b*n*log(x) + 2*b*log(c) + 2*a) + 
 3*I*b*n*(-1/(b^4*n^4))^(1/4)*log((I*b*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*lo 
g(x) + 2*b*log(c) + 2*a) + I*b*n*(-1/(b^4*n^4))^(1/4) + sqrt((cos(2*b*n*lo 
g(x) + 2*b*log(c) + 2*a) + 1)/sin(2*b*n*log(x) + 2*b*log(c) + 2*a))*sin(2* 
b*n*log(x) + 2*b*log(c) + 2*a))/(cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1) 
)*sin(2*b*n*log(x) + 2*b*log(c) + 2*a) - 3*I*b*n*(-1/(b^4*n^4))^(1/4)*log( 
(-I*b*n*(-1/(b^4*n^4))^(1/4)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) - I*b*n* 
(-1/(b^4*n^4))^(1/4) + sqrt((cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)/sin 
(2*b*n*log(x) + 2*b*log(c) + 2*a))*sin(2*b*n*log(x) + 2*b*log(c) + 2*a))/( 
cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1))*sin(2*b*n*log(x) + 2*b*log(c) + 
 2*a) + 4*sqrt((cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)/sin(2*b*n*log(x) 
 + 2*b*log(c) + 2*a))*(cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1))/(b*n*...
 

3.3.31.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

input

integrate(cot(a+b*ln(c*x**n))**(5/2)/x,x)
 

output

Timed out
 

3.3.31.7 Maxima [F]

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

input

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

output

integrate(cot(b*log(c*x^n) + a)^(5/2)/x, x)
 

3.3.31.8 Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.3.31.9 Mupad [B] (verification not implemented)

Time = 28.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.39 \[ \int \frac {\cot ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2\,{\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}{3\,b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n} \]

input

int(cot(a + b*log(c*x^n))^(5/2)/x,x)
 

output

((-1)^(1/4)*atan((-1)^(1/4)*cot(a + b*log(c*x^n))^(1/2)))/(b*n) - (2*cot(a 
 + b*log(c*x^n))^(3/2))/(3*b*n) - ((-1)^(1/4)*atanh((-1)^(1/4)*cot(a + b*l 
og(c*x^n))^(1/2)))/(b*n)