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

Optimal. Leaf size=65 \[ -\frac {2 i \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} F\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt {a \cosh (x)+b \sinh (x)}} \]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3078, 2641} \[ -\frac {2 i \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} F\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt {a \cosh (x)+b \sinh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cosh[x] + b*Sinh[x]],x]

[Out]

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

Rule 2641

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

Rule 3078

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

________________________________________________________________________________________

Mathematica [C]  time = 0.10, size = 81, normalized size = 1.25 \[ \frac {2 \sqrt {a \cosh (x)+b \sinh (x)} \sqrt {\cosh ^2\left (\tanh ^{-1}\left (\frac {a}{b}\right )+x\right )} \text {sech}\left (\tanh ^{-1}\left (\frac {a}{b}\right )+x\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\sinh ^2\left (x+\tanh ^{-1}\left (\frac {a}{b}\right )\right )\right )}{b \sqrt {1-\frac {a^2}{b^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cosh[x] + b*Sinh[x]],x]

[Out]

(2*Sqrt[Cosh[x + ArcTanh[a/b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, -Sinh[x + ArcTanh[a/b]]^2]*Sech[x + Arc
Tanh[a/b]]*Sqrt[a*Cosh[x] + b*Sinh[x]])/(Sqrt[1 - a^2/b^2]*b)

________________________________________________________________________________________

fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {a \cosh \relax (x) + b \sinh \relax (x)}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(a*cosh(x) + b*sinh(x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.41, size = 97, normalized size = 1.49 \[ \frac {\sqrt {-\sqrt {a^{2}-b^{2}}\, \left (\sinh ^{3}\relax (x )\right )}\, \arctan \left (\frac {\sqrt {\sinh \relax (x ) \sqrt {a^{2}-b^{2}}}\, \cosh \relax (x )}{\sqrt {-\sqrt {a^{2}-b^{2}}\, \left (\sinh ^{3}\relax (x )\right )}}\right )}{\sqrt {\sinh \relax (x ) \sqrt {a^{2}-b^{2}}}\, \sinh \relax (x ) \sqrt {-\sinh \relax (x ) \sqrt {a^{2}-b^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(a^2-b^2)^(1/2)*sinh(x)^3)^(1/2)/(sinh(x)*(a^2-b^2)^(1/2))^(1/2)*arctan((sinh(x)*(a^2-b^2)^(1/2))^(1/2)*cosh
(x)/(-(a^2-b^2)^(1/2)*sinh(x)^3)^(1/2))/sinh(x)/(-sinh(x)*(a^2-b^2)^(1/2))^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*cosh(x) + b*sinh(x)), x)

________________________________________________________________________________________