3.13.85 \(\int \frac {1+x^2}{(-1+x+x^2) \sqrt [3]{-x^2+x^4}} \, dx\)

Optimal. Leaf size=93 \[ -\log \left (\sqrt [3]{x^4-x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^4-x^2} x+\left (x^4-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4-x^2}-x}\right ) \]

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Rubi [C]  time = 1.25, antiderivative size = 293, normalized size of antiderivative = 3.15, number of steps used = 23, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2056, 6728, 365, 364, 959, 466, 430, 429, 465, 511, 510} \begin {gather*} \frac {3 \sqrt [3]{1-x^2} x^2 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{2 \left (1-\sqrt {5}\right ) \sqrt [3]{x^4-x^2}}+\frac {3 \sqrt [3]{1-x^2} x^2 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{2 \left (1+\sqrt {5}\right ) \sqrt [3]{x^4-x^2}}-\frac {3 \sqrt [3]{1-x^2} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{x^4-x^2}}-\frac {3 \sqrt [3]{1-x^2} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{x^4-x^2}}+\frac {3 \sqrt [3]{1-x^2} x \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{x^4-x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*x*(1 - x^2)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (4*x^2)/(1 - Sqrt[5])^2, x^2])/(-x^2 + x^4)^(1/3) - (3*x*(1 -
x^2)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (4*x^2)/(1 + Sqrt[5])^2, x^2])/(-x^2 + x^4)^(1/3) + (3*x^2*(1 - x^2)^(1
/3)*AppellF1[2/3, 1, 1/3, 5/3, (4*x^2)/(1 - Sqrt[5])^2, x^2])/(2*(1 - Sqrt[5])*(-x^2 + x^4)^(1/3)) + (3*x^2*(1
- x^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (4*x^2)/(1 + Sqrt[5])^2, x^2])/(2*(1 + Sqrt[5])*(-x^2 + x^4)^(1/3)) +
(3*x*(1 - x^2)^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, x^2])/(-x^2 + x^4)^(1/3)

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

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

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 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 510

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

Rule 511

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

Rule 959

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[(d*(g*x)^n)/x^n, In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[(e*(g*x)^n)/x^n, Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

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 {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1+x^2}{x^{2/3} \sqrt [3]{-1+x^2} \left (-1+x+x^2\right )} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \left (\frac {1}{x^{2/3} \sqrt [3]{-1+x^2}}+\frac {2-x}{x^{2/3} \sqrt [3]{-1+x^2} \left (-1+x+x^2\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {2-x}{x^{2/3} \sqrt [3]{-1+x^2} \left (-1+x+x^2\right )} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1-x^2}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1-x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \left (\frac {-1+\sqrt {5}}{x^{2/3} \left (1-\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^2}}+\frac {-1-\sqrt {5}}{x^{2/3} \left (1+\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^2}}\right ) \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \left (1+\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \left (1-\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (2 \left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {\sqrt [3]{x}}{\left (\left (1+\sqrt {5}\right )^2-4 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}-\frac {\left (2 \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {\sqrt [3]{x}}{\left (\left (1-\sqrt {5}\right )^2-4 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \left (\left (1-\sqrt {5}\right )^2-4 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{x^{2/3} \left (\left (1+\sqrt {5}\right )^2-4 x^2\right ) \sqrt [3]{-1+x^2}} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (6 \left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (\left (1+\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (6 \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (\left (1-\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (3 \left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{1-x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (3 \left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+\sqrt {5}\right )^2-4 x^6\right ) \sqrt [3]{1-x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (3 \left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1+\sqrt {5}\right )^2-4 x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (3 \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1-\sqrt {5}\right )^2-4 x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (3 \left (-1-\sqrt {5}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1+\sqrt {5}\right )^2-4 x^3\right ) \sqrt [3]{1-x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {\left (3 \left (-1+\sqrt {5}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1-\sqrt {5}\right )^2-4 x^3\right ) \sqrt [3]{1-x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{-x^2+x^4}}-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {3 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};\frac {4 x^2}{\left (1-\sqrt {5}\right )^2},x^2\right )}{2 \left (1-\sqrt {5}\right ) \sqrt [3]{-x^2+x^4}}+\frac {3 x^2 \sqrt [3]{1-x^2} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};\frac {4 x^2}{\left (1+\sqrt {5}\right )^2},x^2\right )}{2 \left (1+\sqrt {5}\right ) \sqrt [3]{-x^2+x^4}}+\frac {3 x \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\sqrt [3]{-x^2+x^4}}\\ \end {align*}

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Mathematica [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.35, size = 93, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x^2+x^4}}\right )-\log \left (x+\sqrt [3]{-x^2+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x^2+x^4}+\left (-x^2+x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.42, size = 130, normalized size = 1.40 \begin {gather*} -\sqrt {3} \arctan \left (-\frac {128537192 \, \sqrt {3} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (1454911 \, x^{3} - 69864736 \, x^{2} - 1454911 \, x\right )} - 14102102 \, \sqrt {3} {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}}}{226981 \, x^{3} + 171879616 \, x^{2} - 226981 \, x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{3} + x^{2} + 3 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x^{2} - x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^2+x-1)/(x^4-x^2)^(1/3),x, algorithm="fricas")

[Out]

-sqrt(3)*arctan(-(128537192*sqrt(3)*(x^4 - x^2)^(1/3)*x + sqrt(3)*(1454911*x^3 - 69864736*x^2 - 1454911*x) - 1
4102102*sqrt(3)*(x^4 - x^2)^(2/3))/(226981*x^3 + 171879616*x^2 - 226981*x)) - 1/2*log((x^3 + x^2 + 3*(x^4 - x^
2)^(1/3)*x - x + 3*(x^4 - x^2)^(2/3))/(x^3 + x^2 - x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^2+x-1)/(x^4-x^2)^(1/3),x, algorithm="giac")

[Out]

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

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maple [C]  time = 2.52, size = 581, normalized size = 6.25

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}-741 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-2222 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x -494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x +1604 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+2220 x^{3}+1980 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-1980 x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}+2222 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -740 x^{2}-2220 x}{x \left (x^{2}+x -1\right )}\right )}{2}-\frac {\ln \left (-\frac {-494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+741 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-246 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -1360 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+248 x^{3}-4692 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+4692 x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}+246 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +496 x^{2}-248 x}{x \left (x^{2}+x -1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )}{2}+\ln \left (-\frac {-494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+741 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-246 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-1356 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +494 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -1360 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+248 x^{3}-4692 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+4692 x \left (x^{4}-x^{2}\right )^{\frac {1}{3}}+246 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +496 x^{2}-248 x}{x \left (x^{2}+x -1\right )}\right )\)	\(581\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(x^2+x-1)/(x^4-x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^2-2*_Z+4)*ln((494*RootOf(_Z^2-2*_Z+4)^2*x^3-741*RootOf(_Z^2-2*_Z+4)^2*x^2-2222*RootOf(_Z^2-2*_Z+
4)*x^3+1356*RootOf(_Z^2-2*_Z+4)*(x^4-x^2)^(2/3)-1356*RootOf(_Z^2-2*_Z+4)*(x^4-x^2)^(1/3)*x-494*RootOf(_Z^2-2*_
Z+4)^2*x+1604*RootOf(_Z^2-2*_Z+4)*x^2+2220*x^3+1980*(x^4-x^2)^(2/3)-1980*x*(x^4-x^2)^(1/3)+2222*RootOf(_Z^2-2*
_Z+4)*x-740*x^2-2220*x)/x/(x^2+x-1))-1/2*ln(-(-494*RootOf(_Z^2-2*_Z+4)^2*x^3+741*RootOf(_Z^2-2*_Z+4)^2*x^2-246
*RootOf(_Z^2-2*_Z+4)*x^3+1356*RootOf(_Z^2-2*_Z+4)*(x^4-x^2)^(2/3)-1356*RootOf(_Z^2-2*_Z+4)*(x^4-x^2)^(1/3)*x+4
94*RootOf(_Z^2-2*_Z+4)^2*x-1360*RootOf(_Z^2-2*_Z+4)*x^2+248*x^3-4692*(x^4-x^2)^(2/3)+4692*x*(x^4-x^2)^(1/3)+24
6*RootOf(_Z^2-2*_Z+4)*x+496*x^2-248*x)/x/(x^2+x-1))*RootOf(_Z^2-2*_Z+4)+ln(-(-494*RootOf(_Z^2-2*_Z+4)^2*x^3+74
1*RootOf(_Z^2-2*_Z+4)^2*x^2-246*RootOf(_Z^2-2*_Z+4)*x^3+1356*RootOf(_Z^2-2*_Z+4)*(x^4-x^2)^(2/3)-1356*RootOf(_
Z^2-2*_Z+4)*(x^4-x^2)^(1/3)*x+494*RootOf(_Z^2-2*_Z+4)^2*x-1360*RootOf(_Z^2-2*_Z+4)*x^2+248*x^3-4692*(x^4-x^2)^
(2/3)+4692*x*(x^4-x^2)^(1/3)+246*RootOf(_Z^2-2*_Z+4)*x+496*x^2-248*x)/x/(x^2+x-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^2+x-1)/(x^4-x^2)^(1/3),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2+1}{{\left (x^4-x^2\right )}^{1/3}\,\left (x^2+x-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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