3.2.20 \(\int \frac {\cos (\sqrt {x})}{\sqrt {x}} \, dx\) [120]

Optimal. Leaf size=8 \[ 2 \sin \left (\sqrt {x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3461, 2717} \begin {gather*} 2 \sin \left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[Sqrt[x]]/Sqrt[x],x]

[Out]

2*Sin[Sqrt[x]]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=2 \text {Subst}\left (\int \cos (x) \, dx,x,\sqrt {x}\right )\\ &=2 \sin \left (\sqrt {x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} 2 \sin \left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[Sqrt[x]]/Sqrt[x],x]

[Out]

2*Sin[Sqrt[x]]

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Maple [A]
time = 0.03, size = 7, normalized size = 0.88

method result size
derivativedivides \(2 \sin \left (\sqrt {x}\right )\) \(7\)
default \(2 \sin \left (\sqrt {x}\right )\) \(7\)
meijerg \(2 \sin \left (\sqrt {x}\right )\) \(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.41, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \sin \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(sqrt(x))

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Fricas [A]
time = 0.60, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \sin \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(sqrt(x))

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Sympy [A]
time = 0.11, size = 7, normalized size = 0.88 \begin {gather*} 2 \sin {\left (\sqrt {x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(sqrt(x))

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Giac [A]
time = 0.45, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \sin \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(sqrt(x))

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Mupad [B]
time = 0.04, size = 6, normalized size = 0.75 \begin {gather*} 2\,\sin \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

not solved

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