3.1.67 \(\int \frac {\cos (x)}{\sqrt {1+\cos ^2(x)}} \, dx\) [67]

Optimal. Leaf size=9 \[ \text {ArcSin}\left (\frac {\sin (x)}{\sqrt {2}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3265, 222} \begin {gather*} \text {ArcSin}\left (\frac {\sin (x)}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]/Sqrt[1 + Cos[x]^2],x]

[Out]

ArcSin[Sin[x]/Sqrt[2]]

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 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} \text {ArcSin}\left (\frac {\sin (x)}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/Sqrt[1 + Cos[x]^2],x]

[Out]

ArcSin[Sin[x]/Sqrt[2]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(8)=16\).
time = 0.27, size = 33, normalized size = 3.67

method	result	size
default	\(-\frac {\sqrt {\left (1+\cos ^{2}\left (x \right )\right ) \left (\sin ^{2}\left (x \right )\right )}\, \arcsin \left (\cos ^{2}\left (x \right )\right )}{2 \sin \left (x \right ) \sqrt {1+\cos ^{2}\left (x \right )}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.47, size = 8, normalized size = 0.89 \begin {gather*} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(1/2*sqrt(2)*sin(x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (8) = 16\).
time = 0.38, size = 49, normalized size = 5.44 \begin {gather*} \frac {1}{2} \, \arctan \left (\frac {\sqrt {\cos \left (x\right )^{2} + 1} \cos \left (x\right )^{2} \sin \left (x\right ) - \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan((sqrt(cos(x)^2 + 1)*cos(x)^2*sin(x) - cos(x)*sin(x))/(cos(x)^4 + cos(x)^2 - 1)) + 1/2*arctan(sin(x)
/cos(x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (x \right )}}{\sqrt {\cos ^{2}{\left (x \right )} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)/sqrt(cos(x)**2 + 1), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (8) = 16\).
time = 0.43, size = 23, normalized size = 2.56 \begin {gather*} \frac {1}{2} \, \sqrt {-\sin \left (x\right )^{2} + 2} \sin \left (x\right ) + \arcsin \left (\frac {1}{2} \, \sqrt {2} \sin \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.11 \begin {gather*} \int \frac {\cos \left (x\right )}{\sqrt {{\cos \left (x\right )}^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(cos(x)^2 + 1)^(1/2),x)

[Out]

int(cos(x)/(cos(x)^2 + 1)^(1/2), x)

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