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

Optimal. Leaf size=100 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3-1\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+\log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^4-\text {$\#$1}}\& \right ]-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]

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Rubi [A]  time = 0.51, antiderivative size = 172, normalized size of antiderivative = 1.72, number of steps used = 8, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6728, 277, 239, 429} \begin {gather*} \frac {\left (5+3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1-\sqrt {5}}\right )}{5 \left (1-\sqrt {5}\right )}+\frac {\left (5-3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1+\sqrt {5}}\right )}{5 \left (1+\sqrt {5}\right )}-\frac {1}{2} \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 )}{\sqrt {3}}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 + ((5 + 3*Sqrt[5])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, (2*x^3)/(1 - Sqrt[5])])/(5*(1
- Sqrt[5])) + ((5 - 3*Sqrt[5])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, (2*x^3)/(1 + Sqrt[5])])/(5*(1 + Sqrt[5])) +
ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-x + (1 + x^3)^(1/3)]/2

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 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 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 ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6}\right ) \, dx\\ &=\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (-1+\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3}+\frac {\left (-1-\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3}\right ) \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{5} \left (-5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3} \, dx-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (5+3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1-\sqrt {5}}\right )}{5 \left (1-\sqrt {5}\right )}+\frac {\left (5-3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1+\sqrt {5}}\right )}{5 \left (1+\sqrt {5}\right )}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [B]  time = 0.18, size = 342, normalized size = 3.42 \begin {gather*} -\frac {\left (x^3+1\right )^{2/3}}{2 x^2}-\frac {-2 \left (\sqrt {5}-1\right )^{4/3} \log \left (\sqrt [3]{\sqrt {5}-1}-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}\right )+2 \left (1+\sqrt {5}\right )^{4/3} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{1+\sqrt {5}}\right )+\left (\sqrt {5}-1\right )^{4/3} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {2}{\sqrt {5}-1}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2 \left (\sqrt {5}-1\right )} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (\sqrt {5}-1\right )^{2/3}\right )\right )+\left (1+\sqrt {5}\right )^{4/3} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} x}{\sqrt [3]{x^3+1}}-1}{\sqrt {3}}\right )-\log \left (-\frac {\sqrt [3]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (1+\sqrt {5}\right )^{2/3}\right )\right )}{12 \sqrt [3]{2} \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 - (-2*(-1 + Sqrt[5])^(4/3)*Log[(-1 + Sqrt[5])^(1/3) - (2^(1/3)*x)/(1 + x^3)^(1/3)] +
2*(1 + Sqrt[5])^(4/3)*Log[(1 + Sqrt[5])^(1/3) + (2^(1/3)*x)/(1 + x^3)^(1/3)] + (-1 + Sqrt[5])^(4/3)*(2*Sqrt[3]
*ArcTan[(1 + (2*(2/(-1 + Sqrt[5]))^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]] + Log[(-1 + Sqrt[5])^(2/3) + (2^(2/3)*x^
2)/(1 + x^3)^(2/3) + ((2*(-1 + Sqrt[5]))^(1/3)*x)/(1 + x^3)^(1/3)]) + (1 + Sqrt[5])^(4/3)*(2*Sqrt[3]*ArcTan[(-
1 + (2*(2/(1 + Sqrt[5]))^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]] - Log[(1 + Sqrt[5])^(2/3) + (2^(2/3)*x^2)/(1 + x^3
)^(2/3) - ((2*(1 + Sqrt[5]))^(1/3)*x)/(1 + x^3)^(1/3)]))/(12*2^(1/3)*Sqrt[5])

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IntegrateAlgebraic [A]  time = 0.00, size = 100, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \text {RootSum}\left [-1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 176.62, size = 11328, normalized size = 113.28

method	result	size
risch	\(\text {Expression too large to display}\)	\(11328\)
trager	\(\text {Expression too large to display}\)	\(15352\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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