\(\int \sqrt {a \cosh (x)} \, dx\) [18]

Optimal result

Integrand size = 8, antiderivative size = 27 \[ \int \sqrt {a \cosh (x)} \, dx=-\frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {\cosh (x)}} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2721, 2719} \[ \int \sqrt {a \cosh (x)} \, dx=-\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{\sqrt {\cosh (x)}} \]

[In]

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

[Out]

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

Rule 2719

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

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \cosh (x)} \, dx=-\frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {\cosh (x)}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs. \(2(38)=76\).

Time = 0.52 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.37

method	result	size
default	\(\frac {\sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) \sinh \left (\frac {x}{2}\right )^{2}}\, a \sqrt {2}\, \sqrt {-2 \cosh \left (\frac {x}{2}\right )^{2}+1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \left (\operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-2 \operatorname {EllipticE}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right )}}\)	\(118\)
risch	\(\sqrt {2}\, \sqrt {a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}}+\frac {\left (-\frac {4 \left (a \,{\mathrm e}^{2 x}+a \right )}{a \sqrt {{\mathrm e}^{x} \left (a \,{\mathrm e}^{2 x}+a \right )}}+\frac {2 i \sqrt {-i \left ({\mathrm e}^{x}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{x}-i\right )}\, \sqrt {i {\mathrm e}^{x}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{x}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{x}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {a \,{\mathrm e}^{3 x}+a \,{\mathrm e}^{x}}}\right ) \sqrt {2}\, \sqrt {a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}}\, \sqrt {a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{x}}}{2+2 \,{\mathrm e}^{2 x}}\)	\(171\)

[In]

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

[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)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \sqrt {a \cosh (x)} \, dx=-2 \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - 2 \, \sqrt {a \cosh \left (x\right )} \]

[In]

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

[Out]

-2*sqrt(2)*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) - 2*sqrt(a*cosh(x))

Sympy [F]

\[ \int \sqrt {a \cosh (x)} \, dx=\int \sqrt {a \cosh {\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(a*cosh(x)), x)

Maxima [F]

\[ \int \sqrt {a \cosh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(a*cosh(x)), x)

Giac [F]

\[ \int \sqrt {a \cosh (x)} \, dx=\int { \sqrt {a \cosh \left (x\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(a*cosh(x)), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a \cosh (x)} \, dx=\int \sqrt {a\,\mathrm {cosh}\left (x\right )} \,d x \]

[In]

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

[Out]

int((a*cosh(x))^(1/2), x)