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

Optimal. Leaf size=97 \[ \frac {1}{27} \log \left (\sqrt [3]{x^6+1}-1\right )-\frac {1}{54} \log \left (\left (x^6+1\right )^{2/3}+\sqrt [3]{x^6+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^6+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^6+1\right )^{2/3} \left (4 x^6-3\right )}{36 x^{12}} \]

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Rubi [A]  time = 0.05, antiderivative size = 86, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 51, 55, 618, 204, 31} \begin {gather*} \frac {\left (x^6+1\right )^{2/3}}{9 x^6}+\frac {1}{18} \log \left (1-\sqrt [3]{x^6+1}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^6+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\left (x^6+1\right )^{2/3}}{12 x^{12}}-\frac {\log (x)}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^13*(1 + x^6)^(1/3)),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 204

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

Rule 266

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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt [3]{1+x}} \, dx,x,x^6\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{12 x^{12}}-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{1+x}} \, dx,x,x^6\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (1+x^6\right )^{2/3}}{9 x^6}+\frac {1}{27} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^6\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (1+x^6\right )^{2/3}}{9 x^6}-\frac {\log (x)}{9}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^6}\right )+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^6}\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (1+x^6\right )^{2/3}}{9 x^6}-\frac {\log (x)}{9}+\frac {1}{18} \log \left (1-\sqrt [3]{1+x^6}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^6}\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (1+x^6\right )^{2/3}}{9 x^6}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\log (x)}{9}+\frac {1}{18} \log \left (1-\sqrt [3]{1+x^6}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 26, normalized size = 0.27 \begin {gather*} -\frac {1}{4} \left (x^6+1\right )^{2/3} \, _2F_1\left (\frac {2}{3},3;\frac {5}{3};x^6+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^13*(1 + x^6)^(1/3)),x]

[Out]

-1/4*((1 + x^6)^(2/3)*Hypergeometric2F1[2/3, 3, 5/3, 1 + x^6])

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IntegrateAlgebraic [A]  time = 0.10, size = 97, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^6\right )^{2/3} \left (-3+4 x^6\right )}{36 x^{12}}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{27} \log \left (-1+\sqrt [3]{1+x^6}\right )-\frac {1}{54} \log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x^6)^(2/3)*(-3 + 4*x^6))/(36*x^12) + ArcTan[1/Sqrt[3] + (2*(1 + x^6)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) + Log[-
1 + (1 + x^6)^(1/3)]/27 - Log[1 + (1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/54

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fricas [A]  time = 0.98, size = 86, normalized size = 0.89 \begin {gather*} \frac {4 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 2 \, x^{12} \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + 4 \, x^{12} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) + 3 \, {\left (4 \, x^{6} - 3\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{108 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*(4*sqrt(3)*x^12*arctan(2/3*sqrt(3)*(x^6 + 1)^(1/3) + 1/3*sqrt(3)) - 2*x^12*log((x^6 + 1)^(2/3) + (x^6 +
1)^(1/3) + 1) + 4*x^12*log((x^6 + 1)^(1/3) - 1) + 3*(4*x^6 - 3)*(x^6 + 1)^(2/3))/x^12

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giac [A]  time = 0.18, size = 78, normalized size = 0.80 \begin {gather*} \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {4 \, {\left (x^{6} + 1\right )}^{\frac {5}{3}} - 7 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{36 \, x^{12}} - \frac {1}{54} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{27} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 6.66, size = 86, normalized size = 0.89

method	result	size
risch	\(\frac {4 x^{12}+x^{6}-3}{36 x^{12} \left (x^{6}+1\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{54 \pi }\)	\(86\)
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {28 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {10}{3}\right ], \left [2, 4\right ], -x^{6}\right )}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {4 \left (\frac {9}{4}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )\right ) \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{12}}+\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi }\)	\(90\)
trager	\(\frac {\left (x^{6}+1\right )^{\frac {2}{3}} \left (4 x^{6}-3\right )}{36 x^{12}}-\frac {\ln \left (-\frac {5670704 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}+3479919 x^{6}[phone] \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-5670704 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}+6057568 \left (x^{6}+1\right )^{\frac {2}{3}}+9021462 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+6057568 \left (x^{6}+1\right )^{\frac {1}{3}}+4639892}{x^{6}}\right )}{27}-\frac {2 \ln \left (-\frac {5670704 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}+3479919 x^{6}[phone] \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-5670704 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}+6057568 \left (x^{6}+1\right )^{\frac {2}{3}}+9021462 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+6057568 \left (x^{6}+1\right )^{\frac {1}{3}}+4639892}{x^{6}}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{27}+\frac {2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {1415351072 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}-1046043282 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}-369307790 x^{6}-3200009934 \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-1415351072 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-3200009934 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}-569431707 \left (x^{6}+1\right )^{\frac {2}{3}}-3907685470 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-569431707 \left (x^{6}+1\right )^{\frac {1}{3}}-923269475}{x^{6}}\right )}{27}\)	\(448\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(4*x^12+x^6-3)/x^12/(x^6+1)^(1/3)+1/54/Pi*3^(1/2)*GAMMA(2/3)*(-2/9*Pi*3^(1/2)/GAMMA(2/3)*x^6*hypergeom([1
,1,4/3],[2,2],-x^6)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+6*ln(x))*Pi*3^(1/2)/GAMMA(2/3))

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maxima [A]  time = 0.42, size = 93, normalized size = 0.96 \begin {gather*} \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {4 \, {\left (x^{6} + 1\right )}^{\frac {5}{3}} - 7 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{36 \, {\left (2 \, x^{6} - {\left (x^{6} + 1\right )}^{2} + 1\right )}} - \frac {1}{54} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{27} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/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) + 1)) - 1/36*(4*(x^6 + 1)^(5/3) - 7*(x^6 + 1)^(2/3))/(2*x^6
- (x^6 + 1)^2 + 1) - 1/54*log((x^6 + 1)^(2/3) + (x^6 + 1)^(1/3) + 1) + 1/27*log((x^6 + 1)^(1/3) - 1)

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mupad [B]  time = 0.99, size = 118, normalized size = 1.22 \begin {gather*} \frac {\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{81}-\frac {1}{81}\right )}{27}+\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{81}-9\,{\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )-\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{81}-9\,{\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\frac {\frac {7\,{\left (x^6+1\right )}^{2/3}}{36}-\frac {{\left (x^6+1\right )}^{5/3}}{9}}{2\,x^6-{\left (x^6+1\right )}^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^13*(x^6 + 1)^(1/3)),x)

[Out]

log((x^6 + 1)^(1/3)/81 - 1/81)/27 + log((x^6 + 1)^(1/3)/81 - 9*((3^(1/2)*1i)/54 - 1/54)^2)*((3^(1/2)*1i)/54 -
1/54) - log((x^6 + 1)^(1/3)/81 - 9*((3^(1/2)*1i)/54 + 1/54)^2)*((3^(1/2)*1i)/54 + 1/54) + ((7*(x^6 + 1)^(2/3))
/36 - (x^6 + 1)^(5/3)/9)/(2*x^6 - (x^6 + 1)^2 + 1)

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sympy [C]  time = 1.49, size = 31, normalized size = 0.32 \begin {gather*} - \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 x^{14} \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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