\(\int \frac {1}{x \sqrt {\sinh (a+b \log (c x^n))}} \, dx\) [282]

Optimal result

Integrand size = 19, antiderivative size = 72 \[ \int \frac {1}{x \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right ),2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

2*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))*EllipticF(cos(1/2*I*
a+1/4*Pi+1/2*I*b*ln(c*x^n)),2^(1/2))*(I*sinh(a+b*ln(c*x^n)))^(1/2)/b/n/sinh(a+b*ln(c*x^n))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2721, 2720} \[ \int \frac {1}{x \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \]

[In]

Int[1/(x*Sqrt[Sinh[a + b*Log[c*x^n]]]),x]

[Out]

((-2*I)*EllipticF[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(b*n*Sqrt[Sinh[a + b*Log
[c*x^n]]])

Rule 2720

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]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\frac {1}{4} \left (-2 i a+\pi -2 i b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]

[In]

Integrate[1/(x*Sqrt[Sinh[a + b*Log[c*x^n]]]),x]

[Out]

(-2*EllipticF[((-2*I)*a + Pi - (2*I)*b*Log[c*x^n])/4, 2]*Sqrt[Sinh[a + b*Log[c*x^n]]])/(b*n*Sqrt[I*Sinh[a + b*
Log[c*x^n]]])

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.67

method	result	size
derivativedivides	\(\frac {i \sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(120\)
default	\(\frac {i \sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(120\)

[In]

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

[Out]

I/n*(-I*(sinh(a+b*ln(c*x^n))+I))^(1/2)*2^(1/2)*(-I*(-sinh(a+b*ln(c*x^n))+I))^(1/2)*(I*sinh(a+b*ln(c*x^n)))^(1/
2)*EllipticF((-I*(sinh(a+b*ln(c*x^n))+I))^(1/2),1/2*2^(1/2))/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 = 39, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \, \sqrt {2} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{b n} \]

[In]

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

[Out]

2*sqrt(2)*weierstrassPInverse(4, 0, cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a))/(b*n)

Sympy [F]

\[ \int \frac {1}{x \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{x \sqrt {\sinh {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]

[In]

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

[Out]

Integral(1/(x*sqrt(sinh(a + b*log(c*x**n)))), x)

Maxima [F]

\[ \int \frac {1}{x \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\sinh \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/(x*sqrt(sinh(b*log(c*x^n) + a))), x)

Giac [F]

\[ \int \frac {1}{x \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\sinh \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/(x*sqrt(sinh(b*log(c*x^n) + a))), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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