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

Optimal. Leaf size=51 \[ \frac {2 \sinh (x)}{\sqrt {a \cosh (x)+a}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a \cosh (x)+a}}\right )}{\sqrt {a}} \]

[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)

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Rubi [A]  time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2751, 2649, 206} \[ \frac {2 \sinh (x)}{\sqrt {a \cosh (x)+a}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a \cosh (x)+a}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[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 206

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

Rule 2649

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 2751

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*Sin
[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*} \int \frac {\cosh (x)}{\sqrt {a+a \cosh (x)}} \, dx &=\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 \operatorname {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} \tan ^{-1}\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*}

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Mathematica [A]  time = 0.02, size = 34, normalized size = 0.67 \[ -\frac {2 \cosh \left (\frac {x}{2}\right ) \left (\tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )\right )-2 \sinh \left (\frac {x}{2}\right )\right )}{\sqrt {a (\cosh (x)+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cosh[x/2]*(ArcTan[Sinh[x/2]] - 2*Sinh[x/2]))/Sqrt[a*(1 + Cosh[x])]

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fricas [A]  time = 0.52, size = 62, normalized size = 1.22 \[ \frac {2 \, {\left (\sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \relax (x) + \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x) - 1\right )} - \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \relax (x) + \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{\sqrt {a}}\right )\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt(1/2)*sqrt(a/(cosh(x) + sinh(x)))*(cosh(x) + sinh(x) - 1) - sqrt(2)*sqrt(a)*arctan(sqrt(2)*sqrt(1/2)*sq
rt(a/(cosh(x) + sinh(x)))*(cosh(x) + sinh(x))/sqrt(a)))/a

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giac [C]  time = 0.17, size = 46, normalized size = 0.90 \[ -\sqrt {2} {\left (\frac {2 \, {\left (-i \, \arctan \left (-i\right ) + 1\right )}}{\sqrt {-a}} + \frac {2 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{\sqrt {a}} - \frac {e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {a}} + \frac {e^{\left (-\frac {1}{2} \, x\right )}}{\sqrt {a}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.27, size = 92, normalized size = 1.80 \[ \frac {\cosh \left (\frac {x}{2}\right ) \sqrt {a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (2 \sqrt {a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {-a}+\ln \left (\frac {2 \sqrt {a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) a \right ) \sqrt {2}}{a \sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.11, size = 114, normalized size = 2.24 \[ -\sqrt {2} {\left (\frac {\arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{\sqrt {a}} - \frac {e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {a} e^{x} + \sqrt {a}}\right )} + \frac {1}{3} \, \sqrt {2} {\left (\frac {3 \, \arctan \left (e^{\left (-\frac {1}{2} \, x\right )}\right )}{\sqrt {a}} - \frac {2 \, e^{\left (-\frac {1}{2} \, x\right )}}{\sqrt {a}} - \frac {e^{\left (-\frac {1}{2} \, x\right )}}{\sqrt {a} e^{\left (-x\right )} + \sqrt {a}}\right )} + \frac {3 \, \sqrt {2} \sqrt {a} e^{\left (\frac {3}{2} \, x\right )} - \sqrt {2} \sqrt {a} e^{\left (-\frac {1}{2} \, x\right )}}{3 \, {\left (a e^{x} + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*(arctan(e^(1/2*x))/sqrt(a) - e^(1/2*x)/(sqrt(a)*e^x + sqrt(a))) + 1/3*sqrt(2)*(3*arctan(e^(-1/2*x))/s
qrt(a) - 2*e^(-1/2*x)/sqrt(a) - e^(-1/2*x)/(sqrt(a)*e^(-x) + sqrt(a))) + 1/3*(3*sqrt(2)*sqrt(a)*e^(3/2*x) - sq
rt(2)*sqrt(a)*e^(-1/2*x))/(a*e^x + a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(a + a*cosh(x))^(1/2),x)

[Out]

int(cosh(x)/(a + a*cosh(x))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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