3.18 \(\int \sqrt {a \cosh (x)} \, dx\)

Optimal. Leaf size=27 \[ -\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{\sqrt {\cosh (x)}} \]

[Out]

-2*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))*(a*cosh(x))^(1/2)/cosh(x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2640, 2639} \[ -\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{\sqrt {\cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Cosh[x]],x]

[Out]

((-2*I)*Sqrt[a*Cosh[x]]*EllipticE[(I/2)*x, 2])/Sqrt[Cosh[x]]

Rule 2639

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

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{\sqrt {\cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Cosh[x]],x]

[Out]

((-2*I)*Sqrt[a*Cosh[x]]*EllipticE[(I/2)*x, 2])/Sqrt[Cosh[x]]

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x)}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cosh(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cosh(x)), x)

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maple [B]  time = 0.37, size = 118, normalized size = 4.37 \[ \frac {\sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, a \sqrt {2}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-2 \EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)*a*2^(1/2)*(-2*cosh(1/2*x)^2+1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*(Ellip
ticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))-2*EllipticE(2^(1/2)*cosh(1/2*x),1/2*2^(1/2)))/(a*(2*sinh(1/2*x)^4+sinh(1
/2*x)^2))^(1/2)/sinh(1/2*x)/(a*(2*cosh(1/2*x)^2-1))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cosh(x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {a\,\mathrm {cosh}\relax (x)} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*cosh(x))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*cosh(x)), x)

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