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]

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

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Rubi [A]  time = 0.0452953, antiderivative size = 51, normalized size of antiderivative = 1., 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 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)]

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 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])

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*}

Mathematica [A]  time = 0.0200889, 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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Maple [B]  time = 0.085, size = 92, normalized size = 1.8 \begin{align*}{\frac{\sqrt{2}}{a}\cosh \left ({\frac{x}{2}} \right ) \sqrt{ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}a} \left ( \ln \left ( 2\,{\frac{\sqrt{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}\sqrt{-a}-a}{\cosh \left ( x/2 \right ) }} \right ) a+2\,\sqrt{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}} \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{x}{2}} \right ) \right ) ^{2}}}}} \end{align*}

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)*(sinh(1/2*x)^2*a)^(1/2)*(ln(2/cosh(1/2*x)*((sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2)-a))*a+2*(sinh(1/2*x)
^2*a)^(1/2)*(-a)^(1/2))/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 = 1.8469, size = 154, normalized size = 3.02 \begin{align*} -\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 )}} \end{align*}

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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Fricas [A]  time = 1.78399, size = 228, normalized size = 4.47 \begin{align*} \frac{2 \,{\left (\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 )} - \sqrt{2} \sqrt{a} \arctan \left (\frac{\sqrt{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 )\right )}}{\sqrt{a}}\right )\right )}}{a} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (x \right )}}{\sqrt{a \left (\cosh{\left (x \right )} + 1\right )}}\, dx \end{align*}

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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Giac [C]  time = 1.2217, size = 76, normalized size = 1.49 \begin{align*} -\frac{1}{4} \, \sqrt{2}{\left (\frac{8 \, \arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{\sqrt{a}} - \frac{4 \, e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{a}} + \frac{4 \, e^{\left (-\frac{1}{2} \, x\right )}}{\sqrt{a}} - \frac{-8 i \, \sqrt{-a} \arctan \left (-i\right ) + 8 \, \sqrt{-a}}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*(8*arctan(e^(1/2*x))/sqrt(a) - 4*e^(1/2*x)/sqrt(a) + 4*e^(-1/2*x)/sqrt(a) - (-8*I*sqrt(-a)*arctan
(-I) + 8*sqrt(-a))/a)