3.17.9 \(\int \frac {(1-x^3)^{2/3} (-1+x^3)}{x^6 (-1+2 x^3)} \, dx\) [1609]

Optimal. Leaf size=110 \[ \frac {\left (-2-3 x^3\right ) \left (1-x^3\right )^{2/3}}{10 x^5}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \]

[Out]

1/10*(-3*x^3-2)*(-x^3+1)^(2/3)/x^5+1/3*arctan(3^(1/2)*x/(x+2*(-x^3+1)^(1/3)))*3^(1/2)-1/3*ln(-x+(-x^3+1)^(1/3)
)+1/6*ln(x^2+x*(-x^3+1)^(1/3)+(-x^3+1)^(2/3))

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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {21, 485, 597, 12, 384} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (2 x^3-1\right )-\frac {1}{2} \log \left (x-\sqrt [3]{1-x^3}\right )-\frac {\left (1-x^3\right )^{2/3}}{5 x^5}-\frac {3 \left (1-x^3\right )^{2/3}}{10 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - x^3)^(2/3)*(-1 + x^3))/(x^6*(-1 + 2*x^3)),x]

[Out]

-1/5*(1 - x^3)^(2/3)/x^5 - (3*(1 - x^3)^(2/3))/(10*x^2) + ArcTan[(1 + (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3]
+ Log[-1 + 2*x^3]/6 - Log[x - (1 - x^3)^(1/3)]/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 485

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[c*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)
*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx &=-\int \frac {\left (1-x^3\right )^{5/3}}{x^6 \left (-1+2 x^3\right )} \, dx\\ &=-\frac {\left (1-2 x^3\right )^{5/3} \, _2F_1\left (-\frac {5}{3},-\frac {5}{3};-\frac {2}{3};-\frac {x^3}{1-2 x^3}\right )}{5 x^5}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 110, normalized size = 1.00 \begin {gather*} -\frac {\left (1-x^3\right )^{2/3} \left (2+3 x^3\right )}{10 x^5}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \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]

Integrate[((1 - x^3)^(2/3)*(-1 + x^3))/(x^6*(-1 + 2*x^3)),x]

[Out]

-1/10*((1 - x^3)^(2/3)*(2 + 3*x^3))/x^5 + ArcTan[(Sqrt[3]*x)/(x + 2*(1 - x^3)^(1/3))]/Sqrt[3] - Log[-x + (1 -
x^3)^(1/3)]/3 + Log[x^2 + x*(1 - x^3)^(1/3) + (1 - x^3)^(2/3)]/6

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.84, size = 445, normalized size = 4.05

method	result	size
risch	\(\frac {3 x^{6}-x^{3}-2}{10 x^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}}+\frac {\ln \left (\frac {9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-2 \left (-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{2 x^{3}-1}\right )}{3}-\ln \left (\frac {9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-2 \left (-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{2 x^{3}-1}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+\RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\left (-x^{3}+1\right )^{\frac {2}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{2 x^{3}-1}\right )\)	\(445\)
trager	\(-\frac {\left (3 x^{3}+2\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}+\frac {\ln \left (\frac {374952960 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-10252800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -10252800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+527889 \left (-x^{3}+1\right )^{\frac {2}{3}} x +527889 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-64078 x^{3}-2999623680 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-9154}{2 x^{3}-1}\right )}{3}-32 \ln \left (\frac {374952960 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-10252800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -10252800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+527889 \left (-x^{3}+1\right )^{\frac {2}{3}} x +527889 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-64078 x^{3}-2999623680 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-9154}{2 x^{3}-1}\right ) \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+32 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (\frac {374952960 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x [phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-34273056 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+421089 \left (-x^{3}+1\right )^{\frac {2}{3}} x +421089 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+252248 x^{3}-2999623680 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-220717}{2 x^{3}-1}\right )\)	\(513\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(3*x^6-x^3-2)/x^5/(-x^3+1)^(1/3)+1/3*ln((9*RootOf(9*_Z^2-3*_Z+1)^2*x^3+3*RootOf(9*_Z^2-3*_Z+1)*(-x^3+1)^(
2/3)*x+3*RootOf(9*_Z^2-3*_Z+1)*(-x^3+1)^(1/3)*x^2-6*RootOf(9*_Z^2-3*_Z+1)*x^3-2*(-x^3+1)^(2/3)*x-2*(-x^3+1)^(1
/3)*x^2+x^3+3*RootOf(9*_Z^2-3*_Z+1)-1)/(2*x^3-1))-ln((9*RootOf(9*_Z^2-3*_Z+1)^2*x^3+3*RootOf(9*_Z^2-3*_Z+1)*(-
x^3+1)^(2/3)*x+3*RootOf(9*_Z^2-3*_Z+1)*(-x^3+1)^(1/3)*x^2-6*RootOf(9*_Z^2-3*_Z+1)*x^3-2*(-x^3+1)^(2/3)*x-2*(-x
^3+1)^(1/3)*x^2+x^3+3*RootOf(9*_Z^2-3*_Z+1)-1)/(2*x^3-1))*RootOf(9*_Z^2-3*_Z+1)+RootOf(9*_Z^2-3*_Z+1)*ln(-(-9*
RootOf(9*_Z^2-3*_Z+1)^2*x^3+3*RootOf(9*_Z^2-3*_Z+1)*(-x^3+1)^(2/3)*x+3*RootOf(9*_Z^2-3*_Z+1)*(-x^3+1)^(1/3)*x^
2+(-x^3+1)^(2/3)*x+(-x^3+1)^(1/3)*x^2+3*RootOf(9*_Z^2-3*_Z+1))/(2*x^3-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.73, size = 136, normalized size = 1.24 \begin {gather*} \frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {4 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (x^{3} - 1\right )}}{7 \, x^{3} + 1}\right ) - 5 \, x^{5} \log \left (\frac {2 \, x^{3} - 3 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - 1}{2 \, x^{3} - 1}\right ) - 3 \, {\left (3 \, x^{3} + 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(10*sqrt(3)*x^5*arctan(-(4*sqrt(3)*(-x^3 + 1)^(1/3)*x^2 - 2*sqrt(3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*(x^3 - 1
))/(7*x^3 + 1)) - 5*x^5*log((2*x^3 - 3*(-x^3 + 1)^(1/3)*x^2 + 3*(-x^3 + 1)^(2/3)*x - 1)/(2*x^3 - 1)) - 3*(3*x^
3 + 2)*(-x^3 + 1)^(2/3))/x^5

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \cdot \left (2 x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(2/3)*(x - 1)*(x**2 + x + 1)/(x**6*(2*x**3 - 1)), x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (1-x^3\right )}^{5/3}}{x^6\,\left (2\,x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1 - x^3)^(5/3)/(x^6*(2*x^3 - 1)),x)

[Out]

-int((1 - x^3)^(5/3)/(x^6*(2*x^3 - 1)), x)

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