\(\int \frac {1}{x \text {csch}^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [175]

Optimal result

Integrand size = 19, antiderivative size = 111 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right )}{5 b n \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

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)/si
n(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n))*EllipticE(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)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3854, 3856, 2719} \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{5 b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \]

[In]

Int[1/(x*Csch[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(2*Cosh[a + b*Log[c*x^n]])/(5*b*n*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*n*Sqrt[Csch[a + b*Log[c*x^n]]]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])

Rule 2719

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]

Rule 3854

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
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \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 {3 \text {Subst}\left (\int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{5 n} \\ & = \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 {3 \text {Subst}\left (\int \sqrt {i \sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 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 )}} \\ & = \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 )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \left (\cosh \left (a+b \log \left (c x^n\right )\right )-3 \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) E\left (\left .\frac {1}{4} \left (-2 i a+\pi -2 i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Integrate[1/(x*Csch[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(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]*S
qrt[I*Sinh[a + b*Log[c*x^n]]]))/(5*b*n*Csch[a + b*Log[c*x^n]]^(3/2))

Maple [A] (verified)

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	\(\frac {-\frac {6 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{5}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(227\)

[In]

int(1/x/csch(a+b*ln(c*x^n))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

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 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \]

[In]

integrate(1/x/csch(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

1/20*(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^6 + 6*cosh(b*n*log(x) + b*log(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
*log(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*l
og(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*cosh(b*n*log(x) + b*log(c) + a)^3 - 13*cosh(b*n*log(x) + b*log(c) + a
))*sinh(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*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + si
nh(b*n*log(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*log(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*cosh(b*n*log(x) + b*log(c) + a)^2*sinh(b*n*log(x) + b*log(c) + a) + 3*b*n*cosh(b*n*log(x) + b*log(c
) + a)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n*sinh(b*n*log(x) + b*log(c) + a)^3)

Sympy [F]

\[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \operatorname {csch}^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

[In]

integrate(1/x/csch(a+b*ln(c*x**n))**(5/2),x)

[Out]

Integral(1/(x*csch(a + b*log(c*x**n))**(5/2)), x)

Maxima [F]

\[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x/csch(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x*csch(b*log(c*x^n) + a)^(5/2)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x/csch(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x\,{\left (\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \]

[In]

int(1/(x*(1/sinh(a + b*log(c*x^n)))^(5/2)),x)

[Out]

int(1/(x*(1/sinh(a + b*log(c*x^n)))^(5/2)), x)