Integrand size = 19, antiderivative size = 111 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right )}{5 b n \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]
2/5*cosh(a+b*ln(c*x^n))/b/n/csch(a+b*ln(c*x^n))^(3/2)-6/5*I*(sin(1/2*I*a+1 /4*Pi+1/2*I*b*ln(c*x^n))^2)^(1/2)/sin(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n))*El lipticE(cos(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n)),2^(1/2))/b/n/csch(a+b*ln(c*x ^n))^(1/2)/(I*sinh(a+b*ln(c*x^n)))^(1/2)
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \left (\cosh \left (a+b \log \left (c x^n\right )\right )-3 \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) E\left (\left .\frac {1}{4} \left (-2 i a+\pi -2 i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
(2*(Cosh[a + b*Log[c*x^n]] - 3*Csch[a + b*Log[c*x^n]]^2*EllipticE[((-2*I)* a + Pi - (2*I)*b*Log[c*x^n])/4, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]]))/(5*b*n *Csch[a + b*Log[c*x^n]]^(3/2))
Time = 0.41 (sec) , antiderivative size = 109, 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, 4256, 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.
\(\displaystyle \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \frac {1}{\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\left (i \csc \left (i a+i b \log \left (c x^n\right )\right )\right )^{5/2}}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3}{5} \int \frac {1}{\sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3}{5} \int \frac {1}{\sqrt {i \csc \left (i a+i b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3 \int \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{5 \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3 \int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{5 \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
((2*Cosh[a + b*Log[c*x^n]])/(5*b*Csch[a + b*Log[c*x^n]]^(3/2)) + (((6*I)/5 )*EllipticE[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2])/(b*Sqrt[Csch[a + b*Log[c* x^n]]]*Sqrt[I*Sinh[a + b*Log[c*x^n]]]))/n
3.2.75.3.1 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[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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 = 0.93 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.05
method | result | size |
derivativedivides | \(\frac {-\frac {6 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{5}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(227\) |
default | \(\frac {-\frac {6 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{5}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(227\) |
1/n*(-6/5*(1-I*sinh(a+b*ln(c*x^n)))^(1/2)*2^(1/2)*(1+I*sinh(a+b*ln(c*x^n)) )^(1/2)*(I*sinh(a+b*ln(c*x^n)))^(1/2)*EllipticE((1-I*sinh(a+b*ln(c*x^n)))^ (1/2),1/2*2^(1/2))+3/5*(1-I*sinh(a+b*ln(c*x^n)))^(1/2)*2^(1/2)*(1+I*sinh(a +b*ln(c*x^n)))^(1/2)*(I*sinh(a+b*ln(c*x^n)))^(1/2)*EllipticF((1-I*sinh(a+b *ln(c*x^n)))^(1/2),1/2*2^(1/2))+2/5*cosh(a+b*ln(c*x^n))^4-2/5*cosh(a+b*ln( c*x^n))^2)/cosh(a+b*ln(c*x^n))/sinh(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 = 602, normalized size of antiderivative = 5.42 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \]
1/20*(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^6 + 6*cosh(b*n*log(x) + b*l og(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^5 + sinh(b*n*log(x) + b*log(c) + a)^6 + (15*cosh(b*n*log(x) + b*log(c) + a)^2 + 11)*sinh(b*n*log(x) + b*l og(c) + a)^4 + 11*cosh(b*n*log(x) + b*log(c) + a)^4 + 4*(5*cosh(b*n*log(x) + b*log(c) + a)^3 + 11*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + (15*cosh(b*n*log(x) + b*log(c) + a)^4 + 66*cosh(b*n*log (x) + b*log(c) + a)^2 - 13)*sinh(b*n*log(x) + b*log(c) + a)^2 - 13*cosh(b* n*log(x) + b*log(c) + a)^2 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^5 + 22*c osh(b*n*log(x) + b*log(c) + a)^3 - 13*cosh(b*n*log(x) + b*log(c) + a))*sin h(b*n*log(x) + b*log(c) + a) + 1)*sqrt((cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a))/(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*co sh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*l og(x) + b*log(c) + a)^2 - 1)) + 24*(sqrt(2)*cosh(b*n*log(x) + b*log(c) + a )^3 + 3*sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)^2*sinh(b*n*log(x) + b*log( c) + a) + 3*sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*lo g(c) + a)^2 + sqrt(2)*sinh(b*n*log(x) + b*log(c) + a)^3)*weierstrassZeta(4 , 0, weierstrassPInverse(4, 0, cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n* log(x) + b*log(c) + a))))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*c osh(b*n*log(x) + b*log(c) + a)^2*sinh(b*n*log(x) + b*log(c) + a) + 3*b*n*c osh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n*...
\[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \operatorname {csch}^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
\[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x\,{\left (\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \]