\(\int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx\) [41]

Optimal result

Integrand size = 14, antiderivative size = 53 \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}}+\frac {2 \sinh (x)}{\sqrt {a-a \cosh (x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2830, 2728, 212} \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\frac {2 \sinh (x)}{\sqrt {a-a \cosh (x)}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}} \]

[In]

Int[Cosh[x]/Sqrt[a - a*Cosh[x]],x]

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a - a*Cosh[x]])])/Sqrt[a]) + (2*Sinh[x])/Sqrt[a - a*Cosh[x]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

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

Rule 2830

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\frac {2 \left (2 \cosh \left (\frac {x}{2}\right )-\log \left (\cosh \left (\frac {x}{4}\right )\right )+\log \left (\sinh \left (\frac {x}{4}\right )\right )\right ) \sinh \left (\frac {x}{2}\right )}{\sqrt {a-a \cosh (x)}} \]

[In]

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

[Out]

(2*(2*Cosh[x/2] - Log[Cosh[x/4]] + Log[Sinh[x/4]])*Sinh[x/2])/Sqrt[a - a*Cosh[x]]

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {\sinh \left (\frac {x}{2}\right ) \left (4 \cosh \left (\frac {x}{2}\right )+\ln \left (\cosh \left (\frac {x}{2}\right )-1\right )-\ln \left (\cosh \left (\frac {x}{2}\right )+1\right )\right )}{\sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\)	\(40\)

[In]

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

[Out]

sinh(1/2*x)*(4*cosh(1/2*x)+ln(cosh(1/2*x)-1)-ln(cosh(1/2*x)+1))/(-2*sinh(1/2*x)^2*a)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (42) = 84\).

Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.74 \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\frac {\sqrt {2} a \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}} \sqrt {-\frac {1}{a}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - \cosh \left (x\right ) - \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - 2 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}}{a} \]

[In]

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

[Out]

(sqrt(2)*a*sqrt(-1/a)*log((2*sqrt(2)*sqrt(1/2)*sqrt(-a/(cosh(x) + sinh(x)))*sqrt(-1/a)*(cosh(x) + sinh(x)) - c
osh(x) - sinh(x) - 1)/(cosh(x) + sinh(x) - 1)) - 2*sqrt(1/2)*sqrt(-a/(cosh(x) + sinh(x)))*(cosh(x) + sinh(x) +
1))/a

Sympy [F]

\[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\int \frac {\cosh {\left (x \right )}}{\sqrt {- a \left (\cosh {\left (x \right )} - 1\right )}}\, dx \]

[In]

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

[Out]

Integral(cosh(x)/sqrt(-a*(cosh(x) - 1)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.40 \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=-\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a e^{x}}}{\sqrt {a}}\right )}{\sqrt {a} \mathrm {sgn}\left (-e^{x} + 1\right )} - \frac {\sqrt {2}}{\sqrt {-a e^{x}} \mathrm {sgn}\left (-e^{x} + 1\right )} + \frac {\sqrt {2} \sqrt {-a e^{x}}}{a \mathrm {sgn}\left (-e^{x} + 1\right )} \]

[In]

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

[Out]

-2*sqrt(2)*arctan(sqrt(-a*e^x)/sqrt(a))/(sqrt(a)*sgn(-e^x + 1)) - sqrt(2)/(sqrt(-a*e^x)*sgn(-e^x + 1)) + sqrt(
2)*sqrt(-a*e^x)/(a*sgn(-e^x + 1))

Mupad [F(-1)]

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

[In]

int(cosh(x)/(a - a*cosh(x))^(1/2),x)

[Out]

int(cosh(x)/(a - a*cosh(x))^(1/2), x)