3.17.64 \(\int \frac {x^{13}}{\sqrt [3]{1+x^6}} \, dx\)

Optimal. Leaf size=112 \[ -\frac {1}{27} \log \left (\sqrt [3]{x^6+1}-x^2\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6+1}+x^2}\right )}{9 \sqrt {3}}+\frac {1}{36} \left (x^6+1\right )^{2/3} \left (3 x^8-4 x^2\right )+\frac {1}{54} \log \left (\left (x^6+1\right )^{2/3}+x^4+\sqrt [3]{x^6+1} x^2\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 85, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 321, 239} \begin {gather*} \frac {1}{12} \left (x^6+1\right )^{2/3} x^8-\frac {1}{9} \left (x^6+1\right )^{2/3} x^2-\frac {1}{18} \log \left (x^2-\sqrt [3]{x^6+1}\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^13/(1 + x^6)^(1/3),x]

[Out]

-1/9*(x^2*(1 + x^6)^(2/3)) + (x^8*(1 + x^6)^(2/3))/12 + ArcTan[(1 + (2*x^2)/(1 + x^6)^(1/3))/Sqrt[3]]/(9*Sqrt[
3]) - Log[x^2 - (1 + x^6)^(1/3)]/18

Rule 239

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 275

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]

Rule 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^{13}}{\sqrt [3]{1+x^6}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{12} x^8 \left (1+x^6\right )^{2/3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=-\frac {1}{9} x^2 \left (1+x^6\right )^{2/3}+\frac {1}{12} x^8 \left (1+x^6\right )^{2/3}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=-\frac {1}{9} x^2 \left (1+x^6\right )^{2/3}+\frac {1}{12} x^8 \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 )}{9 \sqrt {3}}-\frac {1}{18} \log \left (x^2-\sqrt [3]{1+x^6}\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 113, normalized size = 1.01 \begin {gather*} \frac {1}{108} \left (9 \left (x^6+1\right )^{2/3} x^8-12 \left (x^6+1\right )^{2/3} x^2-4 \log \left (1-\frac {x^2}{\sqrt [3]{x^6+1}}\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )+2 \log \left (\frac {x^4}{\left (x^6+1\right )^{2/3}}+\frac {x^2}{\sqrt [3]{x^6+1}}+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^13/(1 + x^6)^(1/3),x]

[Out]

(-12*x^2*(1 + x^6)^(2/3) + 9*x^8*(1 + x^6)^(2/3) + 4*Sqrt[3]*ArcTan[(1 + (2*x^2)/(1 + x^6)^(1/3))/Sqrt[3]] - 4
*Log[1 - x^2/(1 + x^6)^(1/3)] + 2*Log[1 + x^4/(1 + x^6)^(2/3) + x^2/(1 + x^6)^(1/3)])/108

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IntegrateAlgebraic [A]  time = 2.69, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{36} \left (1+x^6\right )^{2/3} \left (-4 x^2+3 x^8\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1+x^6}}\right )}{9 \sqrt {3}}-\frac {1}{27} \log \left (-x^2+\sqrt [3]{1+x^6}\right )+\frac {1}{54} \log \left (x^4+x^2 \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^13/(1 + x^6)^(1/3),x]

[Out]

((1 + x^6)^(2/3)*(-4*x^2 + 3*x^8))/36 + ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/(9*Sqrt[3]) - Log[-x^2
+ (1 + x^6)^(1/3)]/27 + Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/54

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fricas [A]  time = 0.73, size = 102, normalized size = 0.91 \begin {gather*} \frac {1}{36} \, {\left (3 \, x^{8} - 4 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} - \frac {1}{27} \, \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}{27} \, \log \left (-\frac {x^{2} - {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{54} \, \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^13/(x^6+1)^(1/3),x, algorithm="fricas")

[Out]

1/36*(3*x^8 - 4*x^2)*(x^6 + 1)^(2/3) - 1/27*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 + 1)^(1/3))/x^2)
- 1/27*log(-(x^2 - (x^6 + 1)^(1/3))/x^2) + 1/54*log((x^4 + (x^6 + 1)^(1/3)*x^2 + (x^6 + 1)^(2/3))/x^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{13}}{{\left (x^{6} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(x^6+1)^(1/3),x, algorithm="giac")

[Out]

integrate(x^13/(x^6 + 1)^(1/3), x)

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maple [C]  time = 3.85, size = 17, normalized size = 0.15

method	result	size
meijerg	\(\frac {x^{14} \hypergeom \left (\left [\frac {1}{3}, \frac {7}{3}\right ], \left [\frac {10}{3}\right ], -x^{6}\right )}{14}\)	\(17\)
risch	\(\frac {x^{2} \left (3 x^{6}-4\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{36}+\frac {x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{6}\right )}{9}\)	\(37\)
trager	\(\frac {x^{2} \left (3 x^{6}-4\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{36}-\frac {\ln \left (-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-2 x^{6}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}+1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{27}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{4}-2 x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right )}{27}\)	\(228\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(x^6+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/14*x^14*hypergeom([1/3,7/3],[10/3],-x^6)

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maxima [A]  time = 0.42, size = 122, normalized size = 1.09 \begin {gather*} -\frac {1}{27} \, \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 {\frac {7 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} - \frac {4 \, {\left (x^{6} + 1\right )}^{\frac {5}{3}}}{x^{10}}}{36 \, {\left (\frac {2 \, {\left (x^{6} + 1\right )}}{x^{6}} - \frac {{\left (x^{6} + 1\right )}^{2}}{x^{12}} - 1\right )}} + \frac {1}{54} \, \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}{27} \, \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^13/(x^6+1)^(1/3),x, algorithm="maxima")

[Out]

-1/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3)/x^2 + 1)) - 1/36*(7*(x^6 + 1)^(2/3)/x^4 - 4*(x^6 + 1)^(5/3
)/x^10)/(2*(x^6 + 1)/x^6 - (x^6 + 1)^2/x^12 - 1) + 1/54*log((x^6 + 1)^(1/3)/x^2 + (x^6 + 1)^(2/3)/x^4 + 1) - 1
/27*log((x^6 + 1)^(1/3)/x^2 - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{13}}{{\left (x^6+1\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(x^6 + 1)^(1/3),x)

[Out]

int(x^13/(x^6 + 1)^(1/3), x)

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sympy [C]  time = 1.31, size = 29, normalized size = 0.26 \begin {gather*} \frac {x^{14} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13/(x**6+1)**(1/3),x)

[Out]

x**14*gamma(7/3)*hyper((1/3, 7/3), (10/3,), x**6*exp_polar(I*pi))/(6*gamma(10/3))

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