\(\int \frac {1}{x \sqrt {\cosh (a+b \log (c x^n))}} \, dx\) [255]

Optimal result

Integrand size = 19, antiderivative size = 28 \[ \int \frac {1}{x \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]

[Out]

-2*I*(cosh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/cosh(1/2*a+1/2*b*ln(c*x^n))*EllipticF(I*sinh(1/2*a+1/2*b*ln(c*x^n))
,2^(1/2))/b/n

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2720} \[ \int \frac {1}{x \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(64)=128\).

Time = 0.20 (sec) , antiderivative size = 183, normalized size of antiderivative = 6.54

method	result	size
derivativedivides	\(\frac {2 \sqrt {\left (-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, b}\)	\(183\)
default	\(\frac {2 \sqrt {\left (-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, b}\)	\(183\)

[In]

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

[Out]

2/n*((-1+2*cosh(1/2*a+1/2*b*ln(c*x^n))^2)*sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-sinh(1/2*a+1/2*b*ln(c*x^n))^2
)^(1/2)*(-2*cosh(1/2*a+1/2*b*ln(c*x^n))^2+1)^(1/2)/(2*sinh(1/2*a+1/2*b*ln(c*x^n))^4+sinh(1/2*a+1/2*b*ln(c*x^n)
)^2)^(1/2)*EllipticF(cosh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))/sinh(1/2*a+1/2*b*ln(c*x^n))/(-1+2*cosh(1/2*a+1/2*b*l
n(c*x^n))^2)^(1/2)/b

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{b n} \]

[In]

integrate(1/x/cosh(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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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