Integrand size = 19, antiderivative size = 94 \[ \int \frac {\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{b n} \]
-2*cos(a+b*ln(c*x^n))*csc(a+b*ln(c*x^n))^(1/2)/b/n+2*(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))*EllipticE(cos(1/2 *a+1/4*Pi+1/2*b*ln(c*x^n)),2^(1/2))*csc(a+b*ln(c*x^n))^(1/2)*sin(a+b*ln(c* x^n))^(1/2)/b/n
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77 \[ \int \frac {\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (\cos \left (a+b \log \left (c x^n\right )\right )-E\left (\left .\frac {1}{4} \left (-2 a+\pi -2 b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
(-2*Sqrt[Csc[a + b*Log[c*x^n]]]*(Cos[a + b*Log[c*x^n]] - EllipticE[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]]))/(b*n)
Time = 0.36 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3039, 3042, 4255, 3042, 4258, 3042, 3119}
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.
| \(\displaystyle \int \frac {\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (a+b \log \left (c x^n\right )\right )^{3/2}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {-\int \frac {1}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\int \frac {1}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{b}}{n}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {-\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{b}}{n}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b}}{n}\) |
((-2*Cos[a + b*Log[c*x^n]]*Sqrt[Csc[a + b*Log[c*x^n]]])/b - (2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticE[(a - Pi/2 + b*Log[c*x^n])/2, 2]*Sqrt[Sin[a + b* Log[c*x^n]]])/b)/n
3.4.11.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Time = 1.32 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.02
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(190\) |
| default | \(\frac {2 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(190\) |
1/n*(2*(sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*ln(c*x^n))+2)^(1/2)*(-sin( a+b*ln(c*x^n)))^(1/2)*EllipticE((sin(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))- (sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*ln(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))-2*cos(a +b*ln(c*x^n))^2)/cos(a+b*ln(c*x^n))/sin(a+b*ln(c*x^n))^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18 \[ \int \frac {\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + \frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sqrt {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}}{b n} \]
-(sqrt(2*I)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*n*log(x) + b*log(c) + a) + I*sin(b*n*log(x) + b*log(c) + a))) + sqrt(-2*I)*weierst rassZeta(4, 0, 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)/sqrt (sin(b*n*log(x) + b*log(c) + a)))/(b*n)
\[ \int \frac {\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\csc ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
\[ \int \frac {\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x} \,d x } \]
Timed out. \[ \int \frac {\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}}{x} \,d x \]