3.86 \(\int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx\)

Optimal. Leaf size=100 \[ \frac {2 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2752, 2663, 2661, 2655, 2653} \[ \frac {2 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/Sqrt[a + b*Cosh[x]],x]

[Out]

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

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.38, size = 73, normalized size = 0.73 \[ -\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} \left ((a+b) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-a F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )\right )}{b \sqrt {a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \relax (x)}{\sqrt {b \cosh \relax (x) + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.42, size = 181, normalized size = 1.81 \[ \frac {2 \left (\EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-2 \EllipticE \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )\right ) \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {\frac {2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b}{a -b}}\, \sqrt {\left (2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{\sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 b \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\relax (x)}{\sqrt {a+b\,\mathrm {cosh}\relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(a + b*cosh(x))^(1/2),x)

[Out]

int(cosh(x)/(a + b*cosh(x))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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