3.17.71 \(\int \frac {(-1+x^3)^{2/3} (4+x^6)}{x^6 (4+2 x^3+x^6)} \, dx\)

Optimal. Leaf size=112 \[ \frac {\left (x^3-1\right )^{2/3} \left (9 x^3-4\right )}{20 x^5}-\frac {1}{12} \text {RootSum}\left [4 \text {$\#$1}^6-10 \text {$\#$1}^3+7\& ,\frac {-6 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+6 \text {$\#$1}^3 \log (x)+7 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )-7 \log (x)}{4 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ] \]

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Rubi [C]  time = 0.52, antiderivative size = 239, normalized size of antiderivative = 2.13, number of steps used = 11, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6728, 264, 277, 239, 430, 429} \begin {gather*} \frac {\left (\sqrt {3}+3 i\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 {x^3}{1-i \sqrt {3}}\right )}{12 \left (\sqrt {3}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (-\sqrt {3}+3 i\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 {x^3}{1+i \sqrt {3}}\right )}{12 \left (-\sqrt {3}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {1}{4} \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 )}{2 \sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{4 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1 + x^3)^(2/3)/(4*x^2) + (-1 + x^3)^(5/3)/(5*x^5) + ((3*I + Sqrt[3])*x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3,
1, 4/3, x^3, -(x^3/(1 - I*Sqrt[3]))])/(12*(I + Sqrt[3])*(1 - x^3)^(2/3)) + ((3*I - Sqrt[3])*x*(-1 + x^3)^(2/3)
*AppellF1[1/3, -2/3, 1, 4/3, x^3, -(x^3/(1 + I*Sqrt[3]))])/(12*(I - Sqrt[3])*(1 - x^3)^(2/3)) - ArcTan[(1 + (2
*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Log[-x + (-1 + x^3)^(1/3)]/4

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 (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^3\right )^{2/3}}{x^6}-\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{2 \left (4+2 x^3+x^6\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{4+2 x^3+x^6} \, dx+\int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{2} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3}\right ) \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx+\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {\left (\left (3-i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx}{6 \left (1-x^3\right )^{2/3}}+\frac {\left (\left (3+i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx}{6 \left (1-x^3\right )^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (3 i+\sqrt {3}\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 {x^3}{1-i \sqrt {3}}\right )}{12 \left (i+\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (3 i-\sqrt {3}\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 {x^3}{1+i \sqrt {3}}\right )}{12 \left (i-\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.31, size = 641, normalized size = 5.72 \begin {gather*} \left (x^3-1\right )^{2/3} \left (\frac {9}{20 x^2}-\frac {1}{5 x^5}\right )+\frac {1}{72} \left (\frac {2 \sqrt [3]{\frac {\sqrt {3}-i}{\sqrt {3}-2 i}} \left (5 \sqrt {3}-3 i\right ) \log \left (\sqrt [3]{-\sqrt {3}+i}-\frac {\sqrt [3]{-\sqrt {3}+2 i} x}{\sqrt [3]{x^3-1}}\right )}{\sqrt {3}-i}+\frac {2 \left (5 \sqrt {3}+3 i\right ) \log \left (\sqrt [3]{\sqrt {3}+i}-\frac {\sqrt [3]{\sqrt {3}+2 i} x}{\sqrt [3]{x^3-1}}\right )}{\left (\sqrt {3}+i\right )^{2/3} \sqrt [3]{\sqrt {3}+2 i}}-\frac {\left (5 \sqrt {3}+3 i\right ) \left (2 \sqrt {3} \sqrt [3]{\frac {1+3 i \sqrt {3}}{\sqrt {3}+i}} \tan ^{-1}\left (\frac {1+\frac {2 \left (\sqrt {3}+2 i\right )^{2/3} x}{\sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\sqrt [3]{\sqrt {3}+2 i} \log \left (\frac {\sqrt [3]{1+3 i \sqrt {3}} x}{\sqrt [3]{x^3-1}}+\frac {\left (\sqrt {3}+2 i\right )^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (\sqrt {3}+i\right )^{2/3}\right )\right )}{\left (1+3 i \sqrt {3}\right )^{2/3}}-\frac {\left (5 \sqrt {3}-3 i\right ) \left (2 \sqrt {-3 \left (-\sqrt {3}+i\right )^{2/3}} \tanh ^{-1}\left (\frac {\sqrt [3]{-\sqrt {3}+i} \sqrt [3]{x^3-1}+2 \sqrt [3]{-\sqrt {3}+2 i} x}{\sqrt {-3 \left (-\sqrt {3}+i\right )^{2/3}} \sqrt [3]{x^3-1}}\right )+\sqrt [3]{-\sqrt {3}+i} \log \left (-\frac {\sqrt [3]{-\sqrt {3}+i} \sqrt [3]{-\sqrt {3}+2 i} x}{\sqrt [3]{x^3-1}}-\frac {\left (-\sqrt {3}+2 i\right )^{2/3} x^2}{\left (x^3-1\right )^{2/3}}-\left (-\sqrt {3}+i\right )^{2/3}\right )\right )}{\sqrt [3]{-\sqrt {3}+2 i} \left (\sqrt {3}-i\right )}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1/5*1/x^5 + 9/(20*x^2))*(-1 + x^3)^(2/3) + ((2*((-I + Sqrt[3])/(-2*I + Sqrt[3]))^(1/3)*(-3*I + 5*Sqrt[3])*Lo
g[(I - Sqrt[3])^(1/3) - ((2*I - Sqrt[3])^(1/3)*x)/(-1 + x^3)^(1/3)])/(-I + Sqrt[3]) - ((-3*I + 5*Sqrt[3])*(2*S
qrt[-3*(I - Sqrt[3])^(2/3)]*ArcTanh[(2*(2*I - Sqrt[3])^(1/3)*x + (I - Sqrt[3])^(1/3)*(-1 + x^3)^(1/3))/(Sqrt[-
3*(I - Sqrt[3])^(2/3)]*(-1 + x^3)^(1/3))] + (I - Sqrt[3])^(1/3)*Log[-(I - Sqrt[3])^(2/3) - ((2*I - Sqrt[3])^(2
/3)*x^2)/(-1 + x^3)^(2/3) - ((I - Sqrt[3])^(1/3)*(2*I - Sqrt[3])^(1/3)*x)/(-1 + x^3)^(1/3)]))/((2*I - Sqrt[3])
^(1/3)*(-I + Sqrt[3])) - ((3*I + 5*Sqrt[3])*(2*Sqrt[3]*((1 + (3*I)*Sqrt[3])/(I + Sqrt[3]))^(1/3)*ArcTan[(1 + (
2*(2*I + Sqrt[3])^(2/3)*x)/((1 + (3*I)*Sqrt[3])^(1/3)*(-1 + x^3)^(1/3)))/Sqrt[3]] + (2*I + Sqrt[3])^(1/3)*Log[
(I + Sqrt[3])^(2/3) + ((2*I + Sqrt[3])^(2/3)*x^2)/(-1 + x^3)^(2/3) + ((1 + (3*I)*Sqrt[3])^(1/3)*x)/(-1 + x^3)^
(1/3)]))/(1 + (3*I)*Sqrt[3])^(2/3) + (2*(3*I + 5*Sqrt[3])*Log[(I + Sqrt[3])^(1/3) - ((2*I + Sqrt[3])^(1/3)*x)/
(-1 + x^3)^(1/3)])/((I + Sqrt[3])^(2/3)*(2*I + Sqrt[3])^(1/3)))/72

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IntegrateAlgebraic [A]  time = 0.00, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-4+9 x^3\right )}{20 x^5}-\frac {1}{12} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-7 \log (x)+7 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x^3)^(2/3)*(-4 + 9*x^3))/(20*x^5) - RootSum[7 - 10*#1^3 + 4*#1^6 & , (-7*Log[x] + 7*Log[(-1 + x^3)^(1/3
) - x*#1] + 6*Log[x]*#1^3 - 6*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+4)/x^6/(x^6+2*x^3+4),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} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\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+4)/x^6/(x^6+2*x^3+4),x, algorithm="giac")

[Out]

integrate((x^6 + 4)*(x^3 - 1)^(2/3)/((x^6 + 2*x^3 + 4)*x^6), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}+4\right )}{x^{6} \left (x^{6}+2 x^{3}+4\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^6+4)/x^6/(x^6+2*x^3+4),x)

[Out]

int((x^3-1)^(2/3)*(x^6+4)/x^6/(x^6+2*x^3+4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\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+4)/x^6/(x^6+2*x^3+4),x, algorithm="maxima")

[Out]

integrate((x^6 + 4)*(x^3 - 1)^(2/3)/((x^6 + 2*x^3 + 4)*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+4\right )}{x^6\,\left (x^6+2\,x^3+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(x^6 + 4))/(x^6*(2*x^3 + x^6 + 4)),x)

[Out]

int(((x^3 - 1)^(2/3)*(x^6 + 4))/(x^6*(2*x^3 + x^6 + 4)), 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+4)/x**6/(x**6+2*x**3+4),x)

[Out]

Timed out

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