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

Optimal. Leaf size=28 \[ -\frac {2 i F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[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

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Rubi [A]  time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2641} \[ -\frac {2 i F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.00 \[ -\frac {2 i F\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[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)

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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \sqrt {\cosh \left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\cosh \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.30, size = 183, normalized size = 6.54 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-\left (\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+1}\, \EllipticF \left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {2 \left (\sinh ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sinh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\cosh \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x\,\sqrt {\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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