3.13.2 \(\int \frac {1+x^4}{(-1-x^2+x^4) \sqrt [3]{-x+x^5}} \, dx\)

Optimal. Leaf size=88 \[ \frac {1}{2} \log \left (\sqrt [3]{x^5-x}-x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5-x}+x}\right )-\frac {1}{4} \log \left (\sqrt [3]{x^5-x} x+\left (x^5-x\right )^{2/3}+x^2\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 1.16, antiderivative size = 305, normalized size of antiderivative = 3.47, number of steps used = 22, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2056, 6715, 6728, 246, 245, 1438, 430, 429, 465, 511, 510} \begin {gather*} -\frac {3 \sqrt [3]{1-x^4} x F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{2 \sqrt [3]{x^5-x}}-\frac {3 \sqrt [3]{1-x^4} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {2 x^4}{3+\sqrt {5}},x^4\right )}{2 \sqrt [3]{x^5-x}}-\frac {3 \left (1-\sqrt {5}\right ) \sqrt [3]{1-x^4} x^3 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{8 \left (3-\sqrt {5}\right ) \sqrt [3]{x^5-x}}-\frac {3 \left (1+\sqrt {5}\right ) \sqrt [3]{1-x^4} x^3 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,\frac {2 x^4}{3+\sqrt {5}}\right )}{8 \left (3+\sqrt {5}\right ) \sqrt [3]{x^5-x}}+\frac {3 \sqrt [3]{1-x^4} x \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{x^5-x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(1 + x^4)/((-1 - x^2 + x^4)*(-x + x^5)^(1/3)),x]

[Out]

(-3*x*(1 - x^4)^(1/3)*AppellF1[1/6, 1/3, 1, 7/6, x^4, (2*x^4)/(3 - Sqrt[5])])/(2*(-x + x^5)^(1/3)) - (3*x*(1 -
x^4)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (2*x^4)/(3 + Sqrt[5]), x^4])/(2*(-x + x^5)^(1/3)) - (3*(1 - Sqrt[5])*x^
3*(1 - x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^4, (2*x^4)/(3 - Sqrt[5])])/(8*(3 - Sqrt[5])*(-x + x^5)^(1/3)) -
(3*(1 + Sqrt[5])*x^3*(1 - x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^4, (2*x^4)/(3 + Sqrt[5])])/(8*(3 + Sqrt[5])
*(-x + x^5)^(1/3)) + (3*x*(1 - x^4)^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, x^4])/(2*(-x + x^5)^(1/3))

Rule 245

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || 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 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 1438

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2
*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 0]

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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {1+x^4}{\sqrt [3]{x} \sqrt [3]{-1+x^4} \left (-1-x^2+x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^6}{\sqrt [3]{-1+x^6} \left (-1-x^3+x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{-1+x^6}}+\frac {2+x^3}{\sqrt [3]{-1+x^6} \left (-1-x^3+x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {2+x^3}{\sqrt [3]{-1+x^6} \left (-1-x^3+x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1+\sqrt {5}}{\left (-1-\sqrt {5}+2 x^3\right ) \sqrt [3]{-1+x^6}}+\frac {1-\sqrt {5}}{\left (-1+\sqrt {5}+2 x^3\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+\sqrt {5}+2 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-\sqrt {5}+2 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1-\sqrt {5}}{2 \sqrt [3]{-1+x^6} \left (-3+\sqrt {5}+2 x^6\right )}+\frac {x^3}{\sqrt [3]{-1+x^6} \left (-3+\sqrt {5}+2 x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-\sqrt {5}}{2 \left (3+\sqrt {5}-2 x^6\right ) \sqrt [3]{-1+x^6}}+\frac {x^3}{\sqrt [3]{-1+x^6} \left (-3-\sqrt {5}+2 x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^6} \left (-3+\sqrt {5}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^6} \left (-3+\sqrt {5}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^6} \left (-3-\sqrt {5}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3+\sqrt {5}-2 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right )^2 \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6} \left (-3+\sqrt {5}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3+\sqrt {5}-2 x^6\right ) \sqrt [3]{1-x^6}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (-3+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (-3-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=-\frac {3 x \sqrt [3]{1-x^4} F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{2 \sqrt [3]{-x+x^5}}-\frac {3 x \sqrt [3]{1-x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {2 x^4}{3+\sqrt {5}},x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (-3+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (-3-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=-\frac {3 x \sqrt [3]{1-x^4} F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{2 \sqrt [3]{-x+x^5}}-\frac {3 x \sqrt [3]{1-x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {2 x^4}{3+\sqrt {5}},x^4\right )}{2 \sqrt [3]{-x+x^5}}-\frac {3 \left (1-\sqrt {5}\right ) x^3 \sqrt [3]{1-x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{8 \left (3-\sqrt {5}\right ) \sqrt [3]{-x+x^5}}-\frac {3 \left (1+\sqrt {5}\right ) x^3 \sqrt [3]{1-x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,\frac {2 x^4}{3+\sqrt {5}}\right )}{8 \left (3+\sqrt {5}\right ) \sqrt [3]{-x+x^5}}+\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 1.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 + x^4)/((-1 - x^2 + x^4)*(-x + x^5)^(1/3)),x]

[Out]

Integrate[(1 + x^4)/((-1 - x^2 + x^4)*(-x + x^5)^(1/3)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.50, size = 88, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^5}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-x+x^5}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^4)/((-1 - x^2 + x^4)*(-x + x^5)^(1/3)),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.25, size = 109, normalized size = 1.24 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} - 1\right )} - 2 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}}}{x^{4} + 8 \, x^{2} - 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} - x^{2} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} - 1}{x^{4} - x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^4-x^2-1)/(x^5-x)^(1/3),x, algorithm="fricas")

[Out]

-1/2*sqrt(3)*arctan(-(4*sqrt(3)*(x^5 - x)^(1/3)*x + sqrt(3)*(x^4 - 1) - 2*sqrt(3)*(x^5 - x)^(2/3))/(x^4 + 8*x^
2 - 1)) + 1/4*log((x^4 - x^2 + 3*(x^5 - x)^(1/3)*x - 3*(x^5 - x)^(2/3) - 1)/(x^4 - x^2 - 1))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} - x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^4-x^2-1)/(x^5-x)^(1/3),x, algorithm="giac")

[Out]

integrate((x^4 + 1)/((x^5 - x)^(1/3)*(x^4 - x^2 - 1)), x)

________________________________________________________________________________________

maple [C]  time = 5.42, size = 601, normalized size = 6.83

method	result	size
trager	\(-\frac {\ln \left (-\frac {8793584675980112 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}+19508929144777782 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-32975942534925420 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-3267606844103087 x^{4}+41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}+41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x +9556171565521246 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+2090065390295748 \left (x^{5}-x \right )^{\frac {2}{3}}+2090065390295748 x \left (x^{5}-x \right )^{\frac {1}{3}}-8793584675980112 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-687917230337492 x^{2}-19508929144777782 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+3267606844103087}{x^{4}-x^{2}-1}\right )}{2}-\ln \left (-\frac {8793584675980112 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}+19508929144777782 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-32975942534925420 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-3267606844103087 x^{4}+41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}+41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x +9556171565521246 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+2090065390295748 \left (x^{5}-x \right )^{\frac {2}{3}}+2090065390295748 x \left (x^{5}-x \right )^{\frac {1}{3}}-8793584675980112 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-687917230337492 x^{2}-19508929144777782 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+3267606844103087}{x^{4}-x^{2}-1}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (\frac {-8793584675980112 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}+10715344468797670 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}+32975942534925420 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+10823675247496950 x^{4}+41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}+41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x +42532114100446666 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+18488074429590093 \left (x^{5}-x \right )^{\frac {2}{3}}+18488074429590093 x \left (x^{5}-x \right )^{\frac {1}{3}}+8793584675980112 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+13709988646829470 x^{2}-10715344468797670 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-10823675247496950}{x^{4}-x^{2}-1}\right )\)	\(601\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)/(x^4-x^2-1)/(x^5-x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(-(8793584675980112*RootOf(4*_Z^2+2*_Z+1)^2*x^4+19508929144777782*RootOf(4*_Z^2+2*_Z+1)*x^4-32975942534
925420*RootOf(4*_Z^2+2*_Z+1)^2*x^2-3267606844103087*x^4+41156279639771682*RootOf(4*_Z^2+2*_Z+1)*(x^5-x)^(2/3)+
41156279639771682*RootOf(4*_Z^2+2*_Z+1)*(x^5-x)^(1/3)*x+9556171565521246*RootOf(4*_Z^2+2*_Z+1)*x^2+20900653902
95748*(x^5-x)^(2/3)+2090065390295748*x*(x^5-x)^(1/3)-8793584675980112*RootOf(4*_Z^2+2*_Z+1)^2-687917230337492*
x^2-19508929144777782*RootOf(4*_Z^2+2*_Z+1)+3267606844103087)/(x^4-x^2-1))-ln(-(8793584675980112*RootOf(4*_Z^2
+2*_Z+1)^2*x^4+19508929144777782*RootOf(4*_Z^2+2*_Z+1)*x^4-32975942534925420*RootOf(4*_Z^2+2*_Z+1)^2*x^2-32676
06844103087*x^4+41156279639771682*RootOf(4*_Z^2+2*_Z+1)*(x^5-x)^(2/3)+41156279639771682*RootOf(4*_Z^2+2*_Z+1)*
(x^5-x)^(1/3)*x+9556171565521246*RootOf(4*_Z^2+2*_Z+1)*x^2+2090065390295748*(x^5-x)^(2/3)+2090065390295748*x*(
x^5-x)^(1/3)-8793584675980112*RootOf(4*_Z^2+2*_Z+1)^2-687917230337492*x^2-19508929144777782*RootOf(4*_Z^2+2*_Z
+1)+3267606844103087)/(x^4-x^2-1))*RootOf(4*_Z^2+2*_Z+1)+RootOf(4*_Z^2+2*_Z+1)*ln((-8793584675980112*RootOf(4*
_Z^2+2*_Z+1)^2*x^4+10715344468797670*RootOf(4*_Z^2+2*_Z+1)*x^4+32975942534925420*RootOf(4*_Z^2+2*_Z+1)^2*x^2+1
0823675247496950*x^4+41156279639771682*RootOf(4*_Z^2+2*_Z+1)*(x^5-x)^(2/3)+41156279639771682*RootOf(4*_Z^2+2*_
Z+1)*(x^5-x)^(1/3)*x+42532114100446666*RootOf(4*_Z^2+2*_Z+1)*x^2+18488074429590093*(x^5-x)^(2/3)[phone]
0093*x*(x^5-x)^(1/3)+8793584675980112*RootOf(4*_Z^2+2*_Z+1)^2+13709988646829470*x^2-10715344468797670*RootOf(4
*_Z^2+2*_Z+1)-10823675247496950)/(x^4-x^2-1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} - x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^4-x^2-1)/(x^5-x)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)/((x^5 - x)^(1/3)*(x^4 - x^2 - 1)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x^4+1}{{\left (x^5-x\right )}^{1/3}\,\left (-x^4+x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 + 1)/((x^5 - x)^(1/3)*(x^2 - x^4 + 1)),x)

[Out]

int(-(x^4 + 1)/((x^5 - x)^(1/3)*(x^2 - x^4 + 1)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)/(x**4-x**2-1)/(x**5-x)**(1/3),x)

[Out]

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

________________________________________________________________________________________