3.261 \(\int \frac{1}{\sqrt{\cos (x)} \sqrt{1+\cos (x)}} \, dx\)

Optimal. Leaf size=16 \[ \sqrt{2} \sin ^{-1}\left (\frac{\sin (x)}{\cos (x)+1}\right ) \]

[Out]

Sqrt[2]*ArcSin[Sin[x]/(1 + Cos[x])]

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Rubi [A]  time = 0.0433364, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2781, 216} \[ \sqrt{2} \sin ^{-1}\left (\frac{\sin (x)}{\cos (x)+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[Cos[x]]*Sqrt[1 + Cos[x]]),x]

[Out]

Sqrt[2]*ArcSin[Sin[x]/(1 + Cos[x])]

Rule 2781

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

Rule 216

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

Rubi steps

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

Mathematica [A]  time = 0.0249956, size = 30, normalized size = 1.88 \[ \frac{2 \cos \left (\frac{x}{2}\right ) \tan ^{-1}\left (\frac{\sin \left (\frac{x}{2}\right )}{\sqrt{\cos (x)}}\right )}{\sqrt{\cos (x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[Cos[x]]*Sqrt[1 + Cos[x]]),x]

[Out]

(2*ArcTan[Sin[x/2]/Sqrt[Cos[x]]]*Cos[x/2])/Sqrt[1 + Cos[x]]

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Maple [B]  time = 0.096, size = 36, normalized size = 2.3 \begin{align*} -{\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}\sqrt{2+2\,\cos \left ( x \right ) }\arcsin \left ({\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ){\frac{1}{\sqrt{\cos \left ( x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^(1/2)/(cos(x)+1)^(1/2),x)

[Out]

-1/cos(x)^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*(2+2*cos(x))^(1/2)*arcsin((-1+cos(x))/sin(x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^(1/2)/(1+cos(x))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 1.93107, size = 116, normalized size = 7.25 \begin{align*} \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{\cos \left (x\right ) + 1} \sqrt{\cos \left (x\right )} \sin \left (x\right )}{2 \,{\left (\cos \left (x\right )^{2} + \cos \left (x\right )\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^(1/2)/(1+cos(x))^(1/2),x, algorithm="fricas")

[Out]

sqrt(2)*arctan(1/2*sqrt(2)*sqrt(cos(x) + 1)*sqrt(cos(x))*sin(x)/(cos(x)^2 + cos(x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)**(1/2)/(1+cos(x))**(1/2),x)

[Out]

Integral(1/(sqrt(cos(x) + 1)*sqrt(cos(x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cos \left (x\right ) + 1} \sqrt{\cos \left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(x)^(1/2)/(1+cos(x))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(cos(x) + 1)*sqrt(cos(x))), x)