3.82 \(\int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx\)

Optimal. Leaf size=46 \[ -\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cosh (x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2663, 2661} \[ -\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Cosh[x]],x]

[Out]

((-2*I)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/Sqrt[a + b*Cosh[x]]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 46, normalized size = 1.00 \[ -\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Cosh[x]],x]

[Out]

((-2*I)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/Sqrt[a + b*Cosh[x]]

________________________________________________________________________________________

fricas [F]  time = 1.32, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \cosh \relax (x) + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(b*cosh(x) + a), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \cosh \relax (x) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*cosh(x) + a), x)

________________________________________________________________________________________

maple [B]  time = 0.36, size = 146, normalized size = 3.17 \[ \frac {2 \sqrt {\left (2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {\frac {2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )}{\sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 b \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((2*b*cosh(1/2*x)^2+a-b)*sinh(1/2*x)^2)^(1/2)/(-2*b/(a-b))^(1/2)*((2*b*cosh(1/2*x)^2+a-b)/(a-b))^(1/2)*(-sin
h(1/2*x)^2)^(1/2)/(2*b*sinh(1/2*x)^4+(a+b)*sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(
-2*(a-b)/b)^(1/2))/sinh(1/2*x)/(2*b*sinh(1/2*x)^2+a+b)^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \cosh \relax (x) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*cosh(x) + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a+b\,\mathrm {cosh}\relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x))^(1/2),x)

[Out]

int(1/(a + b*cosh(x))^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \cosh {\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*cosh(x)), x)

________________________________________________________________________________________