3.13.46 \(\int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx\)

Optimal. Leaf size=90 \[ -\frac {\left (x^6+1\right )^{2/3}}{6 x^6}-\frac {1}{18} \log \left (\sqrt [3]{x^6+1}-1\right )+\frac {1}{36} \log \left (\left (x^6+1\right )^{2/3}+\sqrt [3]{x^6+1}+1\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^6+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{6 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 70, normalized size of antiderivative = 0.78, number of steps used = 6, 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}}{6 x^6}-\frac {1}{12} \log \left (1-\sqrt [3]{x^6+1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^6+1}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {\log (x)}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(1 + x^6)^(1/3)),x]

[Out]

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

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^7 \sqrt [3]{1+x^6}} \, dx &=\frac {1}{6} \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}}{6 x^6}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^6\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{6 x^6}+\frac {\log (x)}{6}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^6}\right )-\frac {1}{12} \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}}{6 x^6}+\frac {\log (x)}{6}-\frac {1}{12} \log \left (1-\sqrt [3]{1+x^6}\right )+\frac {1}{6} \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}}{6 x^6}-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {\log (x)}{6}-\frac {1}{12} \log \left (1-\sqrt [3]{1+x^6}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 26, normalized size = 0.29 \begin {gather*} \frac {1}{4} \left (x^6+1\right )^{2/3} \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};x^6+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(1 + x^6)^(1/3)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.09, size = 90, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^6\right )^{2/3}}{6 x^6}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {1}{18} \log \left (-1+\sqrt [3]{1+x^6}\right )+\frac {1}{36} \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^7*(1 + x^6)^(1/3)),x]

[Out]

-1/6*(1 + x^6)^(2/3)/x^6 - ArcTan[1/Sqrt[3] + (2*(1 + x^6)^(1/3))/Sqrt[3]]/(6*Sqrt[3]) - Log[-1 + (1 + x^6)^(1
/3)]/18 + Log[1 + (1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/36

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fricas [A]  time = 0.45, size = 79, normalized size = 0.88 \begin {gather*} -\frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) + 6 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{36 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+1)^(1/3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.31, size = 66, normalized size = 0.73 \begin {gather*} -\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} + \frac {1}{36} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{18} \, \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^7/(x^6+1)^(1/3),x, algorithm="giac")

[Out]

-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3) + 1)) - 1/6*(x^6 + 1)^(2/3)/x^6 + 1/36*log((x^6 + 1)^(2/3)
+ (x^6 + 1)^(1/3) + 1) - 1/18*log((x^6 + 1)^(1/3) - 1)

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maple [C]  time = 7.64, size = 76, normalized size = 0.84

method	result	size
risch	\(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{6 x^{6}}-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{36 \pi }\)	\(76\)
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {4 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {7}{3}\right ], \left [2, 3\right ], -x^{6}\right )}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (2-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )\right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi }\)	\(77\)
trager	\(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{6 x^{6}}-\frac {\ln \left (\frac {738615580 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{6}[phone] x^{6}-3200009934 \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-738615580 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}[phone] \left (x^{6}+1\right )^{\frac {2}{3}}[phone] \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-1600004967 \left (x^{6}+1\right )^{\frac {1}{3}}[phone]}{x^{6}}\right )}{18}+\frac {\RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \ln \left (\frac {676735492 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{6}-645795448 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{6}-553961685 x^{6}[phone] \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-676735492 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}-1138863414 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}-1030573260 \left (x^{6}+1\right )^{\frac {2}{3}}-1722778774 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )[phone] \left (x^{6}+1\right )^{\frac {1}{3}}-738615580}{x^{6}}\right )}{9}\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^6+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-1/6*(x^6+1)^(2/3)/x^6-1/36/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 = 66, normalized size = 0.73 \begin {gather*} -\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} + \frac {1}{36} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{18} \, \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^7/(x^6+1)^(1/3),x, algorithm="maxima")

[Out]

-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3) + 1)) - 1/6*(x^6 + 1)^(2/3)/x^6 + 1/36*log((x^6 + 1)^(2/3)
+ (x^6 + 1)^(1/3) + 1) - 1/18*log((x^6 + 1)^(1/3) - 1)

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mupad [B]  time = 0.94, size = 92, normalized size = 1.02 \begin {gather*} -\frac {\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{36}-\frac {1}{36}\right )}{18}-\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{36}-9\,{\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2\right )\,\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )+\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{36}-9\,{\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2\right )\,\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )-\frac {{\left (x^6+1\right )}^{2/3}}{6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(x^6 + 1)^(1/3)),x)

[Out]

log((x^6 + 1)^(1/3)/36 - 9*((3^(1/2)*1i)/36 + 1/36)^2)*((3^(1/2)*1i)/36 + 1/36) - log((x^6 + 1)^(1/3)/36 - 9*(
(3^(1/2)*1i)/36 - 1/36)^2)*((3^(1/2)*1i)/36 - 1/36) - log((x^6 + 1)^(1/3)/36 - 1/36)/18 - (x^6 + 1)^(2/3)/(6*x
^6)

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sympy [C]  time = 1.01, size = 31, normalized size = 0.34 \begin {gather*} - \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 x^{8} \Gamma \left (\frac {7}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**6+1)**(1/3),x)

[Out]

-gamma(4/3)*hyper((1/3, 4/3), (7/3,), exp_polar(I*pi)/x**6)/(6*x**8*gamma(7/3))

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