3.3.91 \(\int x^2 \sqrt {\frac {1-x^3}{1+x^3}} \, dx\) [291]

Optimal. Leaf size=53 \[ \frac {1}{3} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )-\frac {2}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{1+x^3}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1981, 1979, 294, 210} \begin {gather*} \frac {1}{3} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right )-\frac {2}{3} \text {ArcTan}\left (\sqrt {\frac {1-x^3}{x^3+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[(1 - x^3)/(1 + x^3)],x]

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 1979

Int[(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Denominator[p]
}, Dist[q*e*((b*c - a*d)/n), Subst[Int[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n - 1)/(b*e - d*x^q)^(1/n + 1)),
x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n
]

Rule 1981

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

Rubi steps

\begin {align*} \int x^2 \sqrt {\frac {1-x^3}{1+x^3}} \, dx &=-\left (\frac {4}{3} \text {Subst}\left (\int \frac {x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\right )\\ &=\frac {1}{3} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ &=\frac {1}{3} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )-\frac {2}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 85, normalized size = 1.60 \begin {gather*} \frac {\sqrt {\frac {1-x^3}{1+x^3}} \left (\sqrt {1-x^3} \left (1+x^3\right )-2 \sqrt {1+x^3} \tan ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {1+x^3}}\right )\right )}{3 \sqrt {1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[(1 - x^3)/(1 + x^3)],x]

[Out]

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

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Maple [A]
time = 0.23, size = 68, normalized size = 1.28

method	result	size
risch	\(\frac {\left (x^{3}+1\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}}{3}-\frac {\arcsin \left (x^{3}\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}\, \sqrt {-\left (x^{3}-1\right ) \left (x^{3}+1\right )}}{3 \left (x^{3}-1\right )}\)	\(68\)
trager	\(\left (\frac {x^{3}}{3}+\frac {1}{3}\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}\, x^{3}+x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}\right )}{3}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*((-x^3+1)/(x^3+1))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*((-x^3+1)/(x^3+1))^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2*sqrt(-(x^3 - 1)/(x^3 + 1)), x)

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Fricas [A]
time = 0.35, size = 55, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, {\left (x^{3} + 1\right )} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}} - \frac {2}{3} \, \arctan \left (\frac {{\left (x^{3} + 1\right )} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}} - 1}{x^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*((-x^3+1)/(x^3+1))^(1/2),x, algorithm="fricas")

[Out]

1/3*(x^3 + 1)*sqrt(-(x^3 - 1)/(x^3 + 1)) - 2/3*arctan(((x^3 + 1)*sqrt(-(x^3 - 1)/(x^3 + 1)) - 1)/x^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {- \frac {\left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{3} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*((-x**3+1)/(x**3+1))**(1/2),x)

[Out]

Integral(x**2*sqrt(-(x - 1)*(x**2 + x + 1)/(x**3 + 1)), x)

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Giac [A]
time = 5.52, size = 22, normalized size = 0.42 \begin {gather*} \frac {1}{3} \, {\left (\sqrt {-x^{6} + 1} + \arcsin \left (x^{3}\right )\right )} \mathrm {sgn}\left (x^{3} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*((-x^3+1)/(x^3+1))^(1/2),x, algorithm="giac")

[Out]

1/3*(sqrt(-x^6 + 1) + arcsin(x^3))*sgn(x^3 + 1)

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Mupad [B]
time = 2.67, size = 56, normalized size = 1.06 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\sqrt {-\frac {x^3-1}{x^3+1}}\right )}{3}-\frac {2\,\sqrt {-\frac {x^3-1}{x^3+1}}}{\frac {3\,\left (x^3-1\right )}{x^3+1}-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-(x^3 - 1)/(x^3 + 1))^(1/2),x)

[Out]

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

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