Integrand size = 19, antiderivative size = 199 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {2}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \]
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)+2/b/n/cot(a+b*ln(c*x^n))^(1/2 )
Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2+\arctan \left (\sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}-\text {arctanh}\left (\sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}} \]
(2 + ArcTan[(-Cot[a + b*Log[c*x^n]]^2)^(1/4)]*(-Cot[a + b*Log[c*x^n]]^2)^( 1/4) - ArcTanh[(-Cot[a + b*Log[c*x^n]]^2)^(1/4)]*(-Cot[a + b*Log[c*x^n]]^2 )^(1/4))/(b*n*Sqrt[Cot[a + b*Log[c*x^n]]])
Time = 0.44 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {3039, 3042, 3955, 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.
\(\displaystyle \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \frac {1}{\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\left (-\tan \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )\right )^{3/2}}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3955 |
\(\displaystyle \frac {\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}-\int \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}-\int \sqrt {-\tan \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\frac {\int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{\cot ^2\left (a+b \log \left (c x^n\right )\right )+1}d\cot \left (a+b \log \left (c x^n\right )\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {2 \int \frac {\cot \left (a+b \log \left (c x^n\right )\right )}{\cot ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \int \frac {\cot \left (a+b \log \left (c x^n\right )\right )+1}{\cot ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2} \int \frac {1-\cot \left (a+b \log \left (c x^n\right )\right )}{\cot ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \int \frac {1}{\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )-\frac {1}{2} \int \frac {1-\cot \left (a+b \log \left (c x^n\right )\right )}{\cot ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\cot \left (a+b \log \left (c x^n\right )\right )-1}d\left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot \left (a+b \log \left (c x^n\right )\right )-1}d\left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\cot \left (a+b \log \left (c x^n\right )\right )}{\cot ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\cot \left (a+b \log \left (c x^n\right )\right )}{\cot ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}{\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2}}\right )\right )}{b}+\frac {2}{b \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
(2/(b*Sqrt[Cot[a + b*Log[c*x^n]]]) + (2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[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*L og[c*x^n]]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Co t[a + b*Log[c*x^n]]]/(2*Sqrt[2]))/2))/b)/n
3.3.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]]))
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]
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]
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]
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]
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]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
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]
Time = 0.98 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {\frac {2}{\sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\) | \(139\) |
default | \(\frac {\frac {2}{\sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\) | \(139\) |
1/n/b*(2/cot(a+b*ln(c*x^n))^(1/2)+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*ln(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))))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 756, normalized size of antiderivative = 3.80 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \]
-1/2*((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))*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)) - (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))*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)) + (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) )*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)) + (-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))*lo g((-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)/s in(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)) - 4*sqrt((cos(2*b*n*log(x)...
\[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \cot ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
\[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \cot \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
Time = 28.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x \cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2}{b\,n\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n} \]