3.2.25 \(\int \frac {\sin ^3(\sqrt {x})}{\sqrt {x}} \, dx\) [125]

Optimal. Leaf size=21 \[ -2 \cos \left (\sqrt {x}\right )+\frac {2}{3} \cos ^3\left (\sqrt {x}\right ) \]

[Out]

-2*cos(x^(1/2))+2/3*cos(x^(1/2))^3

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3460, 2713} \begin {gather*} \frac {2}{3} \cos ^3\left (\sqrt {x}\right )-2 \cos \left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[Sqrt[x]]^3/Sqrt[x],x]

[Out]

-2*Cos[Sqrt[x]] + (2*Cos[Sqrt[x]]^3)/3

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 {align*} \int \frac {\sin ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \text {Subst}\left (\int \sin ^3(x) \, dx,x,\sqrt {x}\right )\\ &=-\left (2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (\sqrt {x}\right )\right )\right )\\ &=-2 \cos \left (\sqrt {x}\right )+\frac {2}{3} \cos ^3\left (\sqrt {x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 1.10 \begin {gather*} -\frac {3}{2} \cos \left (\sqrt {x}\right )+\frac {1}{6} \cos \left (3 \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[Sqrt[x]]^3/Sqrt[x],x]

[Out]

(-3*Cos[Sqrt[x]])/2 + Cos[3*Sqrt[x]]/6

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Maple [A]
time = 0.03, size = 15, normalized size = 0.71

method	result	size
derivativedivides	\(-\frac {2 \left (2+\sin ^{2}\left (\sqrt {x}\right )\right ) \cos \left (\sqrt {x}\right )}{3}\)	\(15\)
default	\(-\frac {2 \left (2+\sin ^{2}\left (\sqrt {x}\right )\right ) \cos \left (\sqrt {x}\right )}{3}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x^(1/2))^3/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(2+sin(x^(1/2))^2)*cos(x^(1/2))

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Maxima [A]
time = 0.31, size = 15, normalized size = 0.71 \begin {gather*} \frac {2}{3} \, \cos \left (\sqrt {x}\right )^{3} - 2 \, \cos \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/2))^3/x^(1/2),x, algorithm="maxima")

[Out]

2/3*cos(sqrt(x))^3 - 2*cos(sqrt(x))

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Fricas [A]
time = 0.38, size = 15, normalized size = 0.71 \begin {gather*} \frac {2}{3} \, \cos \left (\sqrt {x}\right )^{3} - 2 \, \cos \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/2))^3/x^(1/2),x, algorithm="fricas")

[Out]

2/3*cos(sqrt(x))^3 - 2*cos(sqrt(x))

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Sympy [A]
time = 0.18, size = 29, normalized size = 1.38 \begin {gather*} - 2 \sin ^{2}{\left (\sqrt {x} \right )} \cos {\left (\sqrt {x} \right )} - \frac {4 \cos ^{3}{\left (\sqrt {x} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x**(1/2))**3/x**(1/2),x)

[Out]

-2*sin(sqrt(x))**2*cos(sqrt(x)) - 4*cos(sqrt(x))**3/3

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Giac [A]
time = 4.02, size = 15, normalized size = 0.71 \begin {gather*} \frac {2}{3} \, \cos \left (\sqrt {x}\right )^{3} - 2 \, \cos \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x^(1/2))^3/x^(1/2),x, algorithm="giac")

[Out]

2/3*cos(sqrt(x))^3 - 2*cos(sqrt(x))

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Mupad [B]
time = 4.74, size = 14, normalized size = 0.67 \begin {gather*} \frac {2\,\cos \left (\sqrt {x}\right )\,\left ({\cos \left (\sqrt {x}\right )}^2-3\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x^(1/2))^3/x^(1/2),x)

[Out]

(2*cos(x^(1/2))*(cos(x^(1/2))^2 - 3))/3

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