3.595 \(\int \frac{\cosh (x)}{\sqrt{\cosh (2 x)}} \, dx\)

Optimal. Leaf size=15 \[ \frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]

[Out]

ArcSinh[Sqrt[2]*Sinh[x]]/Sqrt[2]

________________________________________________________________________________________

Rubi [A]  time = 0.0185004, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4356, 215} \[ \frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/Sqrt[Cosh[2*x]],x]

[Out]

ArcSinh[Sqrt[2]*Sinh[x]]/Sqrt[2]

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{\cosh (x)}{\sqrt{\cosh (2 x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+2 x^2}} \, dx,x,\sinh (x)\right )\\ &=\frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0107721, size = 15, normalized size = 1. \[ \frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/Sqrt[Cosh[2*x]],x]

[Out]

ArcSinh[Sqrt[2]*Sinh[x]]/Sqrt[2]

________________________________________________________________________________________

Maple [B]  time = 0.048, size = 63, normalized size = 4.2 \begin{align*}{\frac{\sqrt{2}}{4\,\sinh \left ( x \right ) }\sqrt{ \left ( 2\, \left ( \cosh \left ( x \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}\ln \left ( \sqrt{2} \left ( \sinh \left ( x \right ) \right ) ^{2}+\sqrt{2\, \left ( \sinh \left ( x \right ) \right ) ^{4}+ \left ( \sinh \left ( x \right ) \right ) ^{2}}+{\frac{\sqrt{2}}{4}} \right ){\frac{1}{\sqrt{2\, \left ( \cosh \left ( x \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (x\right )}{\sqrt{\cosh \left (2 \, x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(x)/sqrt(cosh(2*x)), x)

________________________________________________________________________________________

Fricas [B]  time = 2.30835, size = 1636, normalized size = 109.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*log(-(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + (28*cosh(x)^2 - 3)*sinh(x)^6 - 3*cosh(x)^6 + 2
*(28*cosh(x)^3 - 9*cosh(x))*sinh(x)^5 + 5*(14*cosh(x)^4 - 9*cosh(x)^2 + 1)*sinh(x)^4 + 5*cosh(x)^4 + 4*(14*cos
h(x)^5 - 15*cosh(x)^3 + 5*cosh(x))*sinh(x)^3 + (28*cosh(x)^6 - 45*cosh(x)^4 + 30*cosh(x)^2 - 4)*sinh(x)^2 + sq
rt(2)*(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 - 1)*sinh(x)^4 - 3*cosh(x)^4 + 4*(5*cosh(x
)^3 - 3*cosh(x))*sinh(x)^3 + (15*cosh(x)^4 - 18*cosh(x)^2 + 4)*sinh(x)^2 + 4*cosh(x)^2 + 2*(3*cosh(x)^5 - 6*co
sh(x)^3 + 4*cosh(x))*sinh(x) - 4)*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) -
4*cosh(x)^2 + 2*(4*cosh(x)^7 - 9*cosh(x)^5 + 10*cosh(x)^3 - 4*cosh(x))*sinh(x) + 4)/(cosh(x)^6 + 6*cosh(x)^5*s
inh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh
(x)^6)) + 1/8*sqrt(2)*log((cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 + 1)*sinh(x)^2 + sqrt(2)
*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) +
sinh(x)^2)) + cosh(x)^2 + 2*(2*cosh(x)^3 + cosh(x))*sinh(x) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (x \right )}}{\sqrt{\cosh{\left (2 x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)/sqrt(cosh(2*x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (x\right )}{\sqrt{\cosh \left (2 \, x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(x)/sqrt(cosh(2*x)), x)