3.3.61 \(\int \frac {1}{\sqrt {\cos (x)} \sqrt {1+\cos (x)}} \, dx\) [261]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {2860, 222} \begin {gather*} \sqrt {2} \text {ArcSin}\left (\frac {\sin (x)}{\cos (x)+1}\right ) \end {gather*}

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 222

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]

Rule 2860

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\cos (x)} \sqrt {1+\cos (x)}} \, dx &=-\left (\sqrt {2} \text {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 = 0.03, size = 30, normalized size = 1.88 \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sin \left (\frac {x}{2}\right )}{\sqrt {\cos (x)}}\right ) \cos \left (\frac {x}{2}\right )}{\sqrt {1+\cos (x)}} \end {gather*}

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]]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(14)=28\).
time = 0.15, size = 36, normalized size = 2.25

method	result	size
default	\(-\frac {\sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \sqrt {2 \cos \left (x \right )+2}\, \arcsin \left (\frac {-1+\cos \left (x \right )}{\sin \left (x \right )}\right )}{\sqrt {\cos \left (x \right )}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(x)^(1/2)/(cos(x)+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found %i

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
time = 0.43, size = 31, normalized size = 1.94 \begin {gather*} \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 {gather*}

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)))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\cos {\left (x \right )} + 1} \sqrt {\cos {\left (x \right )}}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{\sqrt {\cos \left (x\right )}\,\sqrt {\cos \left (x\right )+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________