3.15.61 \(\int \frac {(-1+x^3)^{2/3} (1+x^3)}{x^3} \, dx\)

Optimal. Leaf size=103 \[ -\frac {1}{9} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{3 \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (2 x^3-3\right )}{6 x^2}+\frac {1}{18} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 79, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {453, 195, 239} \begin {gather*} -\frac {1}{6} x \left (x^3-1\right )^{2/3}-\frac {1}{6} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(x*(-1 + x^3)^(2/3)) + (-1 + x^3)^(5/3)/(2*x^2) + ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(3*Sqrt[3]
) - Log[-x + (-1 + x^3)^(1/3)]/6

Rule 195

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 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 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^3} \, dx &=\frac {\left (-1+x^3\right )^{5/3}}{2 x^2}-\frac {1}{2} \int \left (-1+x^3\right )^{2/3} \, dx\\ &=-\frac {1}{6} x \left (-1+x^3\right )^{2/3}+\frac {\left (-1+x^3\right )^{5/3}}{2 x^2}+\frac {1}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=-\frac {1}{6} x \left (-1+x^3\right )^{2/3}+\frac {\left (-1+x^3\right )^{5/3}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 50, normalized size = 0.49 \begin {gather*} \frac {\left (x^3-1\right )^{2/3} \left (-\frac {x^3 \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};x^3\right )}{\left (1-x^3\right )^{2/3}}+x^3-1\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.20, size = 103, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-3+2 x^3\right )}{6 x^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{18} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.98, size = 112, normalized size = 1.09 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - x^{2} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (2 \, x^{3} - 3\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{18 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(2*sqrt(3)*x^2*arctan(-(25382*sqrt(3)*(x^3 - 1)^(1/3)*x^2 - 13720*sqrt(3)*(x^3 - 1)^(2/3)*x + sqrt(3)*(58
31*x^3 - 7200))/(58653*x^3 - 8000)) - x^2*log(-3*(x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x + 1) + 3*(2*x^3 - 3
)*(x^3 - 1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.30, size = 56, normalized size = 0.54

method	result	size
risch	\(\frac {2 x^{6}-5 x^{3}+3}{6 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{3 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\)	\(56\)
meijerg	\(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} x \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2}}\)	\(63\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (2 x^{3}-3\right )}{6 x^{2}}+\frac {4 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \ln \left (182271728 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x [phone] \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}-1508552373 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1497160390 x^{3}-1458173824 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )[phone]\right )}{9}+\frac {\ln \left (182271728 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}-775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-821419388 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-1458173824 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-1129625404 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )[phone]\right )}{9}-\frac {4 \ln \left (182271728 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}-775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-821419388 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-1458173824 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-1129625404 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )[phone]\right ) \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )}{9}\)	\(457\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 106, normalized size = 1.03 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} + \frac {1}{18} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 2.26, size = 70, normalized size = 0.68 \begin {gather*} - \frac {x e^{- \frac {i \pi }{3}} \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^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*exp(-I*pi/3)*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), x**3)/(3*gamma(4/3)) + exp(2*I*pi/3)*gamma(-2/3)*hyper((
-2/3, -2/3), (1/3,), x**3)/(3*x**2*gamma(1/3))

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