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

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Rubi [A]  time = 0.0150689, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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]

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]

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*}

Mathematica [A]  time = 0.0092948, size = 27, normalized size = 1. \[ -\frac{2 i E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{\sqrt{\cosh (x)}} \]

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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Maple [B]  time = 0.043, size = 118, normalized size = 4.4 \begin{align*}{a\sqrt{2}\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+1}\sqrt{- \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}} \left ({\it EllipticF} \left ( \cosh \left ({\frac{x}{2}} \right ) \sqrt{2},{\frac{\sqrt{2}}{2}} \right ) -2\,{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) \right ){\frac{1}{\sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}

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(cosh(1/2*x)*2^(1/2),1/2*2^(1/2))-2*EllipticE(cosh(1/2*x)*2^(1/2),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., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cosh \left (x\right )}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a \cosh \left (x\right )}, x\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cosh{\left (x \right )}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cosh \left (x\right )}\,{d x} \end{align*}

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)