3.1.86 \(\int \frac {\pi \sin (\pi \sqrt {x})}{\sqrt {x}} \, dx\) [86]

Optimal. Leaf size=10 \[ -2 \cos \left (\pi \sqrt {x}\right ) \]

[Out]

-2*cos(Pi*x^(1/2))

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 3460, 2718} \begin {gather*} -2 \cos \left (\pi \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Pi*Sin[Pi*Sqrt[x]])/Sqrt[x],x]

[Out]

-2*Cos[Pi*Sqrt[x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2718

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

Rule 3460

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[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} &=\pi \int \frac {\sin \left (\pi \sqrt {x}\right )}{\sqrt {x}} \, dx\\ &=(2 \pi ) \text {Subst}\left (\int \sin (\pi x) \, dx,x,\sqrt {x}\right )\\ &=-2 \cos \left (\pi \sqrt {x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} -2 \cos \left (\pi \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Pi*Sin[Pi*Sqrt[x]])/Sqrt[x],x]

[Out]

-2*Cos[Pi*Sqrt[x]]

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Maple [A]
time = 0.06, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(-2 \cos \left (\pi \sqrt {x}\right )\)	\(9\)
default	\(-2 \cos \left (\pi \sqrt {x}\right )\)	\(9\)
meijerg	\(2 \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (\pi \sqrt {x}\right )}{\sqrt {\pi }}\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Pi*sin(Pi*x^(1/2))/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*cos(Pi*x^(1/2))

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Maxima [A]
time = 0.33, size = 8, normalized size = 0.80 \begin {gather*} -2 \, \cos \left (\pi \sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(pi*sqrt(x))

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Fricas [A]
time = 0.60, size = 8, normalized size = 0.80 \begin {gather*} -2 \, \cos \left (\pi \sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(pi*sqrt(x))

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Sympy [A]
time = 0.12, size = 10, normalized size = 1.00 \begin {gather*} - 2 \cos {\left (\pi \sqrt {x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(pi*sin(pi*x**(1/2))/x**(1/2),x)

[Out]

-2*cos(pi*sqrt(x))

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Giac [A]
time = 0.53, size = 8, normalized size = 0.80 \begin {gather*} -2 \, \cos \left (\pi \sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(pi*sqrt(x))

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Mupad [B]
time = 0.17, size = 8, normalized size = 0.80 \begin {gather*} -2\,\cos \left (\Pi \,\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((Pi*sin(Pi*x^(1/2)))/x^(1/2),x)

[Out]

-2*cos(Pi*x^(1/2))

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Chatgpt [F] Failed to verify
time = 1.00, size = 8, normalized size = 0.80 \begin {gather*} -\cos \left (\pi \sqrt {x}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(Pi*sin(Pi*x^(1/2))/x^(1/2),x)

[Out]

-cos(Pi*x^(1/2))

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