\(\int \arcsin (\frac {\sqrt [3]{x}}{3}) \, dx\) [143]

Optimal result

Integrand size = 10, antiderivative size = 45 \[ \int \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \, dx=9 \sqrt {9-x^{2/3}}-\frac {1}{3} \left (9-x^{2/3}\right )^{3/2}+x \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \]

[Out]

9*(9-x^(2/3))^(1/2)-1/3*(9-x^(2/3))^(3/2)+x*arcsin(1/3*x^(1/3))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4924, 12, 272, 45} \[ \int \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \, dx=x \arcsin \left (\frac {\sqrt [3]{x}}{3}\right )-\frac {1}{3} \left (9-x^{2/3}\right )^{3/2}+9 \sqrt {9-x^{2/3}} \]

[In]

Int[ArcSin[x^(1/3)/3],x]

[Out]

9*Sqrt[9 - x^(2/3)] - (9 - x^(2/3))^(3/2)/3 + x*ArcSin[x^(1/3)/3]

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4924

Int[ArcSin[u_], x_Symbol] :> Simp[x*ArcSin[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/Sqrt[1 - u^2]), x], x] /;
InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = x \arcsin \left (\frac {\sqrt [3]{x}}{3}\right )-\int \frac {\sqrt [3]{x}}{3 \sqrt {9-x^{2/3}}} \, dx \\ & = x \arcsin \left (\frac {\sqrt [3]{x}}{3}\right )-\frac {1}{3} \int \frac {\sqrt [3]{x}}{\sqrt {9-x^{2/3}}} \, dx \\ & = x \arcsin \left (\frac {\sqrt [3]{x}}{3}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {x}{\sqrt {9-x}} \, dx,x,x^{2/3}\right ) \\ & = x \arcsin \left (\frac {\sqrt [3]{x}}{3}\right )-\frac {1}{2} \text {Subst}\left (\int \left (\frac {9}{\sqrt {9-x}}-\sqrt {9-x}\right ) \, dx,x,x^{2/3}\right ) \\ & = 9 \sqrt {9-x^{2/3}}-\frac {1}{3} \left (9-x^{2/3}\right )^{3/2}+x \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \, dx=-\frac {1}{3} \left (-18-x^{2/3}\right ) \sqrt {9-x^{2/3}}+x \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \]

[In]

Integrate[ArcSin[x^(1/3)/3],x]

[Out]

-1/3*((-18 - x^(2/3))*Sqrt[9 - x^(2/3)]) + x*ArcSin[x^(1/3)/3]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(x \arcsin \left (\frac {x^{\frac {1}{3}}}{3}\right )+x^{\frac {2}{3}} \sqrt {-\frac {x^{\frac {2}{3}}}{9}+1}+18 \sqrt {-\frac {x^{\frac {2}{3}}}{9}+1}\)	\(34\)
default	\(x \arcsin \left (\frac {x^{\frac {1}{3}}}{3}\right )+x^{\frac {2}{3}} \sqrt {-\frac {x^{\frac {2}{3}}}{9}+1}+18 \sqrt {-\frac {x^{\frac {2}{3}}}{9}+1}\)	\(34\)
parts	\(x \arcsin \left (\frac {x^{\frac {1}{3}}}{3}\right )+\frac {x^{\frac {2}{3}} \sqrt {9-x^{\frac {2}{3}}}}{3}+6 \sqrt {9-x^{\frac {2}{3}}}\)	\(35\)

[In]

int(arcsin(1/3*x^(1/3)),x,method=_RETURNVERBOSE)

[Out]

x*arcsin(1/3*x^(1/3))+x^(2/3)*(-1/9*x^(2/3)+1)^(1/2)+18*(-1/9*x^(2/3)+1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 4.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.56 \[ \int \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \, dx=x \arcsin \left (\frac {1}{3} \, x^{\frac {1}{3}}\right ) + \frac {1}{3} \, {\left (x^{\frac {2}{3}} + 18\right )} \sqrt {-x^{\frac {2}{3}} + 9} \]

[In]

integrate(arcsin(1/3*x^(1/3)),x, algorithm="fricas")

[Out]

x*arcsin(1/3*x^(1/3)) + 1/3*(x^(2/3) + 18)*sqrt(-x^(2/3) + 9)

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \, dx=\frac {x^{\frac {2}{3}} \sqrt {9 - x^{\frac {2}{3}}}}{3} + x \operatorname {asin}{\left (\frac {\sqrt [3]{x}}{3} \right )} + 6 \sqrt {9 - x^{\frac {2}{3}}} \]

[In]

integrate(asin(1/3*x**(1/3)),x)

[Out]

x**(2/3)*sqrt(9 - x**(2/3))/3 + x*asin(x**(1/3)/3) + 6*sqrt(9 - x**(2/3))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73 \[ \int \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \, dx=x \arcsin \left (\frac {1}{3} \, x^{\frac {1}{3}}\right ) + x^{\frac {2}{3}} \sqrt {-\frac {1}{9} \, x^{\frac {2}{3}} + 1} + 18 \, \sqrt {-\frac {1}{9} \, x^{\frac {2}{3}} + 1} \]

[In]

integrate(arcsin(1/3*x^(1/3)),x, algorithm="maxima")

[Out]

x*arcsin(1/3*x^(1/3)) + x^(2/3)*sqrt(-1/9*x^(2/3) + 1) + 18*sqrt(-1/9*x^(2/3) + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \, dx=x^{\frac {1}{3}} {\left (x^{\frac {2}{3}} - 9\right )} \arcsin \left (\frac {1}{3} \, x^{\frac {1}{3}}\right ) - 9 \, {\left (-\frac {1}{9} \, x^{\frac {2}{3}} + 1\right )}^{\frac {3}{2}} + 9 \, x^{\frac {1}{3}} \arcsin \left (\frac {1}{3} \, x^{\frac {1}{3}}\right ) + 27 \, \sqrt {-\frac {1}{9} \, x^{\frac {2}{3}} + 1} \]

[In]

integrate(arcsin(1/3*x^(1/3)),x, algorithm="giac")

[Out]

x^(1/3)*(x^(2/3) - 9)*arcsin(1/3*x^(1/3)) - 9*(-1/9*x^(2/3) + 1)^(3/2) + 9*x^(1/3)*arcsin(1/3*x^(1/3)) + 27*sq
rt(-1/9*x^(2/3) + 1)

Mupad [B] (verification not implemented)

Time = 14.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.56 \[ \int \arcsin \left (\frac {\sqrt [3]{x}}{3}\right ) \, dx=x\,\mathrm {asin}\left (\frac {x^{1/3}}{3}\right )+\frac {\sqrt {9-x^{2/3}}\,\left (x^{2/3}+18\right )}{3} \]

[In]

int(asin(x^(1/3)/3),x)

[Out]

x*asin(x^(1/3)/3) + ((9 - x^(2/3))^(1/2)*(x^(2/3) + 18))/3