3.125 \(\int \frac{\sin ^3(\sqrt{x})}{\sqrt{x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0204478, antiderivative size = 21, normalized size of antiderivative = 1., 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 = {3379, 2633} \[ \frac{2}{3} \cos ^3\left (\sqrt{x}\right )-2 \cos \left (\sqrt{x}\right ) \]

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 3379

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]))

Rule 2633

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]

Rubi steps

\begin{align*} \int \frac{\sin ^3\left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \sin ^3(x) \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{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*}

Mathematica [A]  time = 0.0231795, size = 23, normalized size = 1.1 \[ \frac{1}{6} \cos \left (3 \sqrt{x}\right )-\frac{3 \cos \left (\sqrt{x}\right )}{2} \]

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 = 0.009, size = 15, normalized size = 0.7 \begin{align*} -{\frac{2}{3} \left ( 2+ \left ( \sin \left ( \sqrt{x} \right ) \right ) ^{2} \right ) \cos \left ( \sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.940897, size = 20, normalized size = 0.95 \begin{align*} \frac{2}{3} \, \cos \left (\sqrt{x}\right )^{3} - 2 \, \cos \left (\sqrt{x}\right ) \end{align*}

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 = 1.65846, size = 50, normalized size = 2.38 \begin{align*} \frac{2}{3} \, \cos \left (\sqrt{x}\right )^{3} - 2 \, \cos \left (\sqrt{x}\right ) \end{align*}

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 = 0.902545, size = 29, normalized size = 1.38 \begin{align*} - 2 \sin ^{2}{\left (\sqrt{x} \right )} \cos{\left (\sqrt{x} \right )} - \frac{4 \cos ^{3}{\left (\sqrt{x} \right )}}{3} \end{align*}

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 = 1.09923, size = 20, normalized size = 0.95 \begin{align*} \frac{2}{3} \, \cos \left (\sqrt{x}\right )^{3} - 2 \, \cos \left (\sqrt{x}\right ) \end{align*}

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))