\(\int \sqrt {a \cosh (x)+b \sinh (x)} \, dx\) [590]

Optimal result

Integrand size = 13, antiderivative size = 65 \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]

[Out]

-2*I*(cos(1/2*I*x-1/2*arctan(a,-I*b))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(a,-I*b))*EllipticE(sin(1/2*I*x-1/2*arcta
n(a,-I*b)),2^(1/2))*(a*cosh(x)+b*sinh(x))^(1/2)/((a*cosh(x)+b*sinh(x))/(a^2-b^2)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3157, 2719} \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=-\frac {2 i \sqrt {a \cosh (x)+b \sinh (x)} E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]

[In]

Int[Sqrt[a*Cosh[x] + b*Sinh[x]],x]

[Out]

((-2*I)*EllipticE[(I*x - ArcTan[a, (-I)*b])/2, 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/Sqrt[(a*Cosh[x] + b*Sinh[x])/Sq
rt[a^2 - b^2]]

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 3157

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[c + d*x] +
b*Sin[c + d*x])^n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n, Int[Cos[c + d*x - ArcTan[a, b]]^n, x]
, x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) &&  !(GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a \cosh (x)+b \sinh (x)} \int \sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )} \, dx}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \\ & = -\frac {2 i E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.51 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.17 \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=\frac {b \left (-a^2+b^2\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cosh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )\right ) \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )+\sqrt {-\sinh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )} \left (2 a^3 \sqrt {1-\frac {b^2}{a^2}} \cosh (x)-2 a \left (a^2-b^2\right ) \cosh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )+2 a^2 b \sqrt {1-\frac {b^2}{a^2}} \sinh (x)+a^2 b \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )-b^3 \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )\right )}{a b \sqrt {1-\frac {b^2}{a^2}} \sqrt {a \cosh (x)+b \sinh (x)} \sqrt {-\sinh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )}} \]

[In]

Integrate[Sqrt[a*Cosh[x] + b*Sinh[x]],x]

[Out]

(b*(-a^2 + b^2)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cosh[x + ArcTanh[b/a]]^2]*Sinh[x + ArcTanh[b/a]] + Sqrt
[-Sinh[x + ArcTanh[b/a]]^2]*(2*a^3*Sqrt[1 - b^2/a^2]*Cosh[x] - 2*a*(a^2 - b^2)*Cosh[x + ArcTanh[b/a]] + 2*a^2*
b*Sqrt[1 - b^2/a^2]*Sinh[x] + a^2*b*Sinh[x + ArcTanh[b/a]] - b^3*Sinh[x + ArcTanh[b/a]]))/(a*b*Sqrt[1 - b^2/a^
2]*Sqrt[a*Cosh[x] + b*Sinh[x]]*Sqrt[-Sinh[x + ArcTanh[b/a]]^2])

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.51

method	result	size
default	\(-\frac {\sqrt {a^{2}-b^{2}}\, \cosh \left (x \right )}{\sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\)	\(33\)
risch	\(\sqrt {2}\, \sqrt {\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) {\mathrm e}^{-x}}+\frac {\left (2 a -2 b \right ) \left (-\frac {2 \left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right )}{\left (a -b \right ) \sqrt {\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) {\mathrm e}^{x}}}+\frac {\left (-\frac {a +b}{a -b}+\frac {2 a +2 b}{a -b}\right ) \sqrt {-\left (a +b \right ) \left (a -b \right )}\, \sqrt {\frac {\left ({\mathrm e}^{x}+\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}}{a +b}\right ) \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}\, \sqrt {-\frac {2 \left ({\mathrm e}^{x}-\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}}{a +b}\right ) \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}\, \sqrt {-\frac {{\mathrm e}^{x} \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}\, \left (-\frac {2 \sqrt {-\left (a +b \right ) \left (a -b \right )}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}}{a +b}\right ) \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}, \frac {\sqrt {2}}{2}\right )}{a +b}+\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {\sqrt {-\left (a +b \right ) \left (a -b \right )}}{a +b}\right ) \left (a +b \right )}{\sqrt {-\left (a +b \right ) \left (a -b \right )}}}, \frac {\sqrt {2}}{2}\right )}{a +b}\right )}{\left (a +b \right ) \sqrt {a \,{\mathrm e}^{3 x}+{\mathrm e}^{3 x} b +a \,{\mathrm e}^{x}-{\mathrm e}^{x} b}}\right ) \sqrt {2}\, \sqrt {\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) {\mathrm e}^{-x}}\, \sqrt {\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) {\mathrm e}^{x}}}{2 a \,{\mathrm e}^{2 x}+2 b \,{\mathrm e}^{2 x}+2 a -2 b}\)	\(455\)

[In]

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

[Out]

-(a^2-b^2)^(1/2)/(-sinh(x)*(a^2-b^2)^(1/2))^(1/2)*cosh(x)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx=-2 \, \sqrt {2} \sqrt {a + b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - 2 \, \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(a*cosh(x) + b*sinh(x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(a*cosh(x) + b*sinh(x)), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(a*cosh(x) + b*sinh(x)), x)

Mupad [F(-1)]

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

[In]

int((a*cosh(x) + b*sinh(x))^(1/2),x)

[Out]

int((a*cosh(x) + b*sinh(x))^(1/2), x)