3.15.90 \(\int x (1+x^6)^{2/3} \, dx\) [1490]

Optimal. Leaf size=104 \[ \frac {1}{6} x^2 \left (1+x^6\right )^{2/3}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1+x^6}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (-x^2+\sqrt [3]{1+x^6}\right )+\frac {1}{18} \log \left (x^4+x^2 \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]

[Out]

1/6*x^2*(x^6+1)^(2/3)+1/9*arctan(3^(1/2)*x^2/(x^2+2*(x^6+1)^(1/3)))*3^(1/2)-1/9*ln(-x^2+(x^6+1)^(1/3))+1/18*ln
(x^4+x^2*(x^6+1)^(1/3)+(x^6+1)^(2/3))

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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 0.66, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {281, 201, 245} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} \left (x^6+1\right )^{2/3} x^2-\frac {1}{6} \log \left (x^2-\sqrt [3]{x^6+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(1 + x^6)^(2/3),x]

[Out]

(x^2*(1 + x^6)^(2/3))/6 + ArcTan[(1 + (2*x^2)/(1 + x^6)^(1/3))/Sqrt[3]]/(3*Sqrt[3]) - Log[x^2 - (1 + x^6)^(1/3
)]/6

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int x \left (1+x^6\right )^{2/3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \left (1+x^3\right )^{2/3} \, dx,x,x^2\right )\\ &=\frac {1}{6} x^2 \left (1+x^6\right )^{2/3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{6} x^2 \left (1+x^6\right )^{2/3}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{1+x^6}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (x^2-\sqrt [3]{1+x^6}\right )\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 98, normalized size = 0.94 \begin {gather*} \frac {1}{18} \left (3 x^2 \left (1+x^6\right )^{2/3}+2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1+x^6}}\right )-2 \log \left (-x^2+\sqrt [3]{1+x^6}\right )+\log \left (x^4+x^2 \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(1 + x^6)^(2/3),x]

[Out]

(3*x^2*(1 + x^6)^(2/3) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))] - 2*Log[-x^2 + (1 + x^6)^(1
/3)] + Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)])/18

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 3.27, size = 17, normalized size = 0.16

method	result	size
meijerg	\(\frac {x^{2} \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{6}\right )}{2}\)	\(17\)
risch	\(\frac {x^{2} \left (x^{6}+1\right )^{\frac {2}{3}}}{6}+\frac {x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{6}\right )}{3}\)	\(30\)
trager	\(\frac {x^{2} \left (x^{6}+1\right )^{\frac {2}{3}}}{6}+\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{4}+4 x^{6}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{9}-\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{9}+\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{9}\)	\(324\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^6+1)^(2/3),x,method=_RETURNVERBOSE)

[Out]

1/2*x^2*hypergeom([-2/3,1/3],[4/3],-x^6)

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Maxima [A]
time = 0.48, size = 94, normalized size = 0.90 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) + \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{4} {\left (\frac {x^{6} + 1}{x^{6}} - 1\right )}} + \frac {1}{18} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{9} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^6+1)^(2/3),x, algorithm="maxima")

[Out]

-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3)/x^2 + 1)) + 1/6*(x^6 + 1)^(2/3)/(x^4*((x^6 + 1)/x^6 - 1)) +
1/18*log((x^6 + 1)^(1/3)/x^2 + (x^6 + 1)^(2/3)/x^4 + 1) - 1/9*log((x^6 + 1)^(1/3)/x^2 - 1)

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Fricas [A]
time = 0.35, size = 94, normalized size = 0.90 \begin {gather*} \frac {1}{6} \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x^{2} - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {1}{9} \, \log \left (-\frac {x^{2} - {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{18} \, \log \left (\frac {x^{4} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^6+1)^(2/3),x, algorithm="fricas")

[Out]

1/6*(x^6 + 1)^(2/3)*x^2 - 1/9*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 + 1)^(1/3))/x^2) - 1/9*log(-(x^
2 - (x^6 + 1)^(1/3))/x^2) + 1/18*log((x^4 + (x^6 + 1)^(1/3)*x^2 + (x^6 + 1)^(2/3))/x^4)

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Sympy [C] Result contains complex when optimal does not.
time = 0.51, size = 31, normalized size = 0.30 \begin {gather*} \frac {x^{2} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x**6+1)**(2/3),x)

[Out]

x**2*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), x**6*exp_polar(I*pi))/(6*gamma(4/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^6+1)^(2/3),x, algorithm="giac")

[Out]

integrate((x^6 + 1)^(2/3)*x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (x^6+1\right )}^{2/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^6 + 1)^(2/3),x)

[Out]

int(x*(x^6 + 1)^(2/3), x)

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