3.17.73 \(\int \frac {(-1+x^3)^{2/3} (1-x^3+x^6)}{x^6 (-2-x^3+2 x^6)} \, dx\)

Optimal. Leaf size=112 \[ \frac {1}{12} \text {RootSum}\left [2 \text {$\#$1}^6-5 \text {$\#$1}^3+1\& ,\frac {-17 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+17 \text {$\#$1}^3 \log (x)+3 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )-3 \log (x)}{4 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ]+\frac {\left (x^3-1\right )^{2/3} \left (4-19 x^3\right )}{40 x^5} \]

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Rubi [B]  time = 0.61, antiderivative size = 227, normalized size of antiderivative = 2.03, number of steps used = 11, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6728, 264, 277, 239, 430, 429} \begin {gather*} \frac {\left (51+19 \sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {4 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (51-19 \sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {4 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {3}{8} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3-1\right )^{2/3}}{8 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(-1 + x^3)^(2/3))/(8*x^2) - (-1 + x^3)^(5/3)/(10*x^5) + ((51 + 19*Sqrt[17])*x*(-1 + x^3)^(2/3)*AppellF1[1/
3, -2/3, 1, 4/3, x^3, (4*x^3)/(1 - Sqrt[17])])/(34*(1 - Sqrt[17])*(1 - x^3)^(2/3)) + ((51 - 19*Sqrt[17])*x*(-1
+ x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, (4*x^3)/(1 + Sqrt[17])])/(34*(1 + Sqrt[17])*(1 - x^3)^(2/3)) +
(Sqrt[3]*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/4 - (3*Log[-x + (-1 + x^3)^(1/3)])/8

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 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

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

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (-2-x^3+2 x^6\right )} \, dx &=\int \left (-\frac {\left (-1+x^3\right )^{2/3}}{2 x^6}+\frac {3 \left (-1+x^3\right )^{2/3}}{4 x^3}+\frac {\left (11-6 x^3\right ) \left (-1+x^3\right )^{2/3}}{4 \left (-2-x^3+2 x^6\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (11-6 x^3\right ) \left (-1+x^3\right )^{2/3}}{-2-x^3+2 x^6} \, dx-\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx+\frac {3}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \int \left (\frac {\left (-6+\frac {38}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{-1-\sqrt {17}+4 x^3}+\frac {\left (-6-\frac {38}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{-1+\sqrt {17}+4 x^3}\right ) \, dx+\frac {3}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\frac {3}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{34} \left (-51+19 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1-\sqrt {17}+4 x^3} \, dx-\frac {1}{34} \left (51+19 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+\sqrt {17}+4 x^3} \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\frac {3}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {\left (\left (-51+19 \sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-1-\sqrt {17}+4 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}-\frac {\left (\left (51+19 \sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-1+\sqrt {17}+4 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\left (51+19 \sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {4 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (51-19 \sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {4 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\frac {3}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [B]  time = 0.58, size = 354, normalized size = 3.16 \begin {gather*} \frac {-2 \sqrt [3]{80295+19471 \sqrt {17}} \log \left (\sqrt [3]{5-\sqrt {17}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )+2 \sqrt [3]{80295-19471 \sqrt {17}} \log \left (\sqrt [3]{5+\sqrt {17}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )+\sqrt [3]{80295+19471 \sqrt {17}} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{10-2 \sqrt {17}} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (5-\sqrt {17}\right )^{2/3}\right )\right )-\sqrt [3]{80295-19471 \sqrt {17}} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (5+\sqrt {17}\right )^{2/3}\right )\right )}{48 \sqrt {17}}+\left (x^3-1\right )^{2/3} \left (\frac {1}{10 x^5}-\frac {19}{40 x^2}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1/(10*x^5) - 19/(40*x^2))*(-1 + x^3)^(2/3) + (-2*(80295 + 19471*Sqrt[17])^(1/3)*Log[(5 - Sqrt[17])^(1/3) - (2
^(1/3)*x)/(-1 + x^3)^(1/3)] + 2*(80295 - 19471*Sqrt[17])^(1/3)*Log[(5 + Sqrt[17])^(1/3) - (2^(1/3)*x)/(-1 + x^
3)^(1/3)] + (80295 + 19471*Sqrt[17])^(1/3)*(2*Sqrt[3]*ArcTan[(1 + ((2*(5 + Sqrt[17]))^(1/3)*x)/(-1 + x^3)^(1/3
))/Sqrt[3]] + Log[(5 - Sqrt[17])^(2/3) + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) + ((10 - 2*Sqrt[17])^(1/3)*x)/(-1 + x^
3)^(1/3)]) - (80295 - 19471*Sqrt[17])^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(2/(5 + Sqrt[17]))^(1/3)*x)/(-1 + x^3)^(
1/3))/Sqrt[3]] + Log[(5 + Sqrt[17])^(2/3) + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) + ((2*(5 + Sqrt[17]))^(1/3)*x)/(-1
+ x^3)^(1/3)]))/(48*Sqrt[17])

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IntegrateAlgebraic [A]  time = 0.00, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (4-19 x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+17 \log (x) \text {$\#$1}^3-17 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4 - 19*x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[1 - 5*#1^3 + 2*#1^6 & , (-3*Log[x] + 3*Log[(-1 + x^3)^(1/3)
- x*#1] + 17*Log[x]*#1^3 - 17*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/12

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 346.81, size = 7068, normalized size = 63.11

method	result	size
risch	\(\text {Expression too large to display}\)	\(7068\)
trager	\(\text {Expression too large to display}\)	\(10410\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^6-x^3+1\right )}{x^6\,\left (-2\,x^6+x^3+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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