Integrand size = 19, antiderivative size = 68 \[ \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right ),2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
-2/3*(sin(1/2*a+1/4*Pi+1/2*b*ln(c*x^n))^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*ln (c*x^n))*EllipticF(cos(1/2*a+1/4*Pi+1/2*b*ln(c*x^n)),2^(1/2))/b/n-2/3*cos( a+b*ln(c*x^n))/b/n/sin(a+b*ln(c*x^n))^(3/2)
Time = 0.17 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \left (\operatorname {EllipticF}\left (\frac {1}{4} \left (2 a-\pi +2 b \log \left (c x^n\right )\right ),2\right )-\frac {\cos \left (a+b \log \left (c x^n\right )\right )}{\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\right )}{3 b n} \]
(2*(EllipticF[(2*a - Pi + 2*b*Log[c*x^n])/4, 2] - Cos[a + b*Log[c*x^n]]/Si n[a + b*Log[c*x^n]]^(3/2)))/(3*b*n)
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3039, 3042, 3116, 3042, 3120}
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.
\(\displaystyle \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\sin \left (a+b \log \left (c x^n\right )\right )^{5/2}}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{3 b}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\) |
((2*EllipticF[(a - Pi/2 + b*Log[c*x^n])/2, 2])/(3*b) - (2*Cos[a + b*Log[c* x^n]])/(3*b*Sin[a + b*Log[c*x^n]]^(3/2)))/n
3.1.68.3.1 Defintions of rubi rules used
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_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Time = 0.88 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.93
method | result | size |
derivativedivides | \(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )-2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3 n {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\) | \(131\) |
default | \(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )-2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3 n {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\) | \(131\) |
1/3/n/sin(a+b*ln(c*x^n))^(3/2)*((sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*l n(c*x^n))+2)^(1/2)*(-sin(a+b*ln(c*x^n)))^(1/2)*EllipticF((sin(a+b*ln(c*x^n ))+1)^(1/2),1/2*2^(1/2))*sin(a+b*ln(c*x^n))-2*cos(a+b*ln(c*x^n))^2)/cos(a+ b*ln(c*x^n))/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {{\left (\sqrt {2} \sqrt {-i} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - \sqrt {2} \sqrt {-i}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + {\left (\sqrt {2} \sqrt {i} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - \sqrt {2} \sqrt {i}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + 2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sqrt {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}{3 \, {\left (b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )}} \]
1/3*((sqrt(2)*sqrt(-I)*cos(b*n*log(x) + b*log(c) + a)^2 - sqrt(2)*sqrt(-I) )*weierstrassPInverse(4, 0, cos(b*n*log(x) + b*log(c) + a) + I*sin(b*n*log (x) + b*log(c) + a)) + (sqrt(2)*sqrt(I)*cos(b*n*log(x) + b*log(c) + a)^2 - sqrt(2)*sqrt(I))*weierstrassPInverse(4, 0, cos(b*n*log(x) + b*log(c) + a) - I*sin(b*n*log(x) + b*log(c) + a)) + 2*cos(b*n*log(x) + b*log(c) + a)*sq rt(sin(b*n*log(x) + b*log(c) + a)))/(b*n*cos(b*n*log(x) + b*log(c) + a)^2 - b*n)
Timed out. \[ \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
Time = 26.77 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left ({\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {3}{2};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{b\,n\,{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \]