3.19.17 \(\int \frac {1-3 x^3+3 x^6}{x^6 (-1+2 x^3) \sqrt [4]{-x+x^4}} \, dx\)

Optimal. Leaf size=123 \[ -\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4-x}}{\sqrt {x^4-x}-x^2}\right )-\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {x^4-x}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{x^4-x}}\right )+\frac {4 \left (x^4-x\right )^{3/4} \left (x^3-1\right )}{21 x^6} \]

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Rubi [B]  time = 0.82, antiderivative size = 338, normalized size of antiderivative = 2.75, number of steps used = 20, number of rules used = 14, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2056, 6725, 271, 264, 466, 465, 494, 461, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\sqrt [4]{x} \sqrt [4]{x^3-1} \log \left (\frac {x^{3/2}}{\sqrt {x^3-1}}-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{x^3-1}}+1\right )}{3 \sqrt {2} \sqrt [4]{x^4-x}}-\frac {\sqrt [4]{x} \sqrt [4]{x^3-1} \log \left (\frac {x^{3/2}}{\sqrt {x^3-1}}+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{x^3-1}}+1\right )}{3 \sqrt {2} \sqrt [4]{x^4-x}}+\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{x^3-1} \tan ^{-1}\left (1-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{3 \sqrt [4]{x^4-x}}-\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{x^3-1} \tan ^{-1}\left (\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{x^3-1}}+1\right )}{3 \sqrt [4]{x^4-x}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{x^4-x}}+\frac {1-x^3}{7 x^5 \sqrt [4]{x^4-x}}-\frac {1-x^3}{7 x^2 \sqrt [4]{x^4-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - x^3)/(7*x^5*(-x + x^4)^(1/4)) - (1 - x^3)/(7*x^2*(-x + x^4)^(1/4)) + (1 - x^3)^2/(21*x^5*(-x + x^4)^(1/4)
) + (Sqrt[2]*x^(1/4)*(-1 + x^3)^(1/4)*ArcTan[1 - (Sqrt[2]*x^(3/4))/(-1 + x^3)^(1/4)])/(3*(-x + x^4)^(1/4)) - (
Sqrt[2]*x^(1/4)*(-1 + x^3)^(1/4)*ArcTan[1 + (Sqrt[2]*x^(3/4))/(-1 + x^3)^(1/4)])/(3*(-x + x^4)^(1/4)) + (x^(1/
4)*(-1 + x^3)^(1/4)*Log[1 + x^(3/2)/Sqrt[-1 + x^3] - (Sqrt[2]*x^(3/4))/(-1 + x^3)^(1/4)])/(3*Sqrt[2]*(-x + x^4
)^(1/4)) - (x^(1/4)*(-1 + x^3)^(1/4)*Log[1 + x^(3/2)/Sqrt[-1 + x^3] + (Sqrt[2]*x^(3/4))/(-1 + x^3)^(1/4)])/(3*
Sqrt[2]*(-x + x^4)^(1/4))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

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 271

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

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

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 494

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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 6725

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

Rubi steps

\begin {align*} \int \frac {1-3 x^3+3 x^6}{x^6 \left (-1+2 x^3\right ) \sqrt [4]{-x+x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1-3 x^3+3 x^6}{x^{25/4} \sqrt [4]{-1+x^3} \left (-1+2 x^3\right )} \, dx}{\sqrt [4]{-x+x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \left (-\frac {3}{4 x^{25/4} \sqrt [4]{-1+x^3}}+\frac {3}{2 x^{13/4} \sqrt [4]{-1+x^3}}+\frac {1}{4 x^{25/4} \sqrt [4]{-1+x^3} \left (-1+2 x^3\right )}\right ) \, dx}{\sqrt [4]{-x+x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{25/4} \sqrt [4]{-1+x^3} \left (-1+2 x^3\right )} \, dx}{4 \sqrt [4]{-x+x^4}}-\frac {\left (3 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{25/4} \sqrt [4]{-1+x^3}} \, dx}{4 \sqrt [4]{-x+x^4}}+\frac {\left (3 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{-1+x^3}} \, dx}{2 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {2 \left (1-x^3\right )}{3 x^2 \sqrt [4]{-x+x^4}}-\frac {\left (3 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{-1+x^3}} \, dx}{7 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{22} \sqrt [4]{-1+x^{12}} \left (-1+2 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {10 \left (1-x^3\right )}{21 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{-1+x^4} \left (-1+2 x^4\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {10 \left (1-x^3\right )}{21 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^2}{x^8 \left (-1-x^4\right )} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {10 \left (1-x^3\right )}{21 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{x^8}+\frac {3}{x^4}-\frac {4}{1+x^4}\right ) \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}-\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}-\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {\sqrt [4]{x} \sqrt [4]{-1+x^3} \log \left (1+\frac {x^{3/2}}{\sqrt {-1+x^3}}-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}-\frac {\sqrt [4]{x} \sqrt [4]{-1+x^3} \log \left (1+\frac {x^{3/2}}{\sqrt {-1+x^3}}+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}-\frac {\left (\sqrt {2} \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}+\frac {\left (\sqrt {2} \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {1-x^3}{7 x^5 \sqrt [4]{-x+x^4}}-\frac {1-x^3}{7 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (1-x^3\right )^2}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{-1+x^3} \tan ^{-1}\left (1-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}-\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{-1+x^3} \tan ^{-1}\left (1+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}+\frac {\sqrt [4]{x} \sqrt [4]{-1+x^3} \log \left (1+\frac {x^{3/2}}{\sqrt {-1+x^3}}-\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}-\frac {\sqrt [4]{x} \sqrt [4]{-1+x^3} \log \left (1+\frac {x^{3/2}}{\sqrt {-1+x^3}}+\frac {\sqrt {2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{-x+x^4}}\\ \end {align*}

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Mathematica [C]  time = 1.37, size = 192, normalized size = 1.56 \begin {gather*} \frac {16 \left (1-2 x^3\right )^2 x^3 \, _3F_2\left (\frac {5}{4},2,2;1,\frac {9}{4};\frac {x^3}{1-x^3}\right )-8 x^3 \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};\frac {x^3}{1-x^3}\right )+192 x^9 \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};\frac {x^3}{1-x^3}\right )-80 x^6 \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};\frac {x^3}{1-x^3}\right )-5 \left (128 x^9-144 x^6+13 x^3+3\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {x^3}{1-x^3}\right )+150 x^9-345 x^6+240 x^3-45}{315 x^4 \left (x \left (x^3-1\right )\right )^{5/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-45 + 240*x^3 - 345*x^6 + 150*x^9 - 5*(3 + 13*x^3 - 144*x^6 + 128*x^9)*Hypergeometric2F1[1/4, 1, 5/4, x^3/(1
- x^3)] - 8*x^3*Hypergeometric2F1[5/4, 2, 9/4, x^3/(1 - x^3)] - 80*x^6*Hypergeometric2F1[5/4, 2, 9/4, x^3/(1 -
x^3)] + 192*x^9*Hypergeometric2F1[5/4, 2, 9/4, x^3/(1 - x^3)] + 16*x^3*(1 - 2*x^3)^2*HypergeometricPFQ[{5/4,
2, 2}, {1, 9/4}, x^3/(1 - x^3)])/(315*x^4*(x*(-1 + x^3))^(5/4))

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IntegrateAlgebraic [A]  time = 0.69, size = 123, normalized size = 1.00 \begin {gather*} \frac {4 \left (-1+x^3\right ) \left (-x+x^4\right )^{3/4}}{21 x^6}-\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x+x^4}}{-x^2+\sqrt {-x+x^4}}\right )-\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-x+x^4}}{\sqrt {2}}}{x \sqrt [4]{-x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-1 + x^3)*(-x + x^4)^(3/4))/(21*x^6) - (Sqrt[2]*ArcTan[(Sqrt[2]*x*(-x + x^4)^(1/4))/(-x^2 + Sqrt[-x + x^4]
)])/3 - (Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[-x + x^4]/Sqrt[2])/(x*(-x + x^4)^(1/4))])/3

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fricas [B]  time = 2.07, size = 479, normalized size = 3.89 \begin {gather*} \frac {28 \, \sqrt {2} x^{6} \arctan \left (-\frac {\sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} + {\left (2 \, x^{4} - \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} - 2 \, x\right )} \sqrt {\frac {2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{4} - x} x + 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1}{2 \, x^{3} - 1}}}{2 \, {\left (x^{4} - x\right )}}\right ) + 28 \, \sqrt {2} x^{6} \arctan \left (-\frac {\sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} - {\left (2 \, x^{4} + \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} - 2 \, x\right )} \sqrt {\frac {2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{4} - x} x - 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1}{2 \, x^{3} - 1}}}{2 \, {\left (x^{4} - x\right )}}\right ) - 7 \, \sqrt {2} x^{6} \log \left (\frac {2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{4} - x} x + 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1}{2 \, x^{3} - 1}\right ) + 7 \, \sqrt {2} x^{6} \log \left (\frac {2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{4} - x} x - 2 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1}{2 \, x^{3} - 1}\right ) + 16 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )}}{84 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-3*x^3+1)/x^6/(2*x^3-1)/(x^4-x)^(1/4),x, algorithm="fricas")

[Out]

1/84*(28*sqrt(2)*x^6*arctan(-1/2*(sqrt(2)*(x^4 - x)^(3/4)*x - sqrt(2)*(x^4 - x)^(1/4)*(x^3 - 1) + (2*x^4 - sqr
t(2)*(x^4 - x)^(3/4)*x - sqrt(2)*(x^4 - x)^(1/4)*(x^3 - 1) - 2*x)*sqrt((2*x^3 + 2*sqrt(2)*(x^4 - x)^(1/4)*x^2
+ 4*sqrt(x^4 - x)*x + 2*sqrt(2)*(x^4 - x)^(3/4) - 1)/(2*x^3 - 1)))/(x^4 - x)) + 28*sqrt(2)*x^6*arctan(-1/2*(sq
rt(2)*(x^4 - x)^(3/4)*x - sqrt(2)*(x^4 - x)^(1/4)*(x^3 - 1) - (2*x^4 + sqrt(2)*(x^4 - x)^(3/4)*x + sqrt(2)*(x^
4 - x)^(1/4)*(x^3 - 1) - 2*x)*sqrt((2*x^3 - 2*sqrt(2)*(x^4 - x)^(1/4)*x^2 + 4*sqrt(x^4 - x)*x - 2*sqrt(2)*(x^4
- x)^(3/4) - 1)/(2*x^3 - 1)))/(x^4 - x)) - 7*sqrt(2)*x^6*log((2*x^3 + 2*sqrt(2)*(x^4 - x)^(1/4)*x^2 + 4*sqrt(
x^4 - x)*x + 2*sqrt(2)*(x^4 - x)^(3/4) - 1)/(2*x^3 - 1)) + 7*sqrt(2)*x^6*log((2*x^3 - 2*sqrt(2)*(x^4 - x)^(1/4
)*x^2 + 4*sqrt(x^4 - x)*x - 2*sqrt(2)*(x^4 - x)^(3/4) - 1)/(2*x^3 - 1)) + 16*(x^4 - x)^(3/4)*(x^3 - 1))/x^6

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giac [A]  time = 0.24, size = 125, normalized size = 1.02 \begin {gather*} -\frac {4}{21} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} - \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{3} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (\sqrt {2} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-3*x^3+1)/x^6/(2*x^3-1)/(x^4-x)^(1/4),x, algorithm="giac")

[Out]

-4/21*(-1/x^3 + 1)^(7/4) - 1/3*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(-1/x^3 + 1)^(1/4))) - 1/3*sqrt(2)*arct
an(-1/2*sqrt(2)*(sqrt(2) - 2*(-1/x^3 + 1)^(1/4))) + 1/6*sqrt(2)*log(sqrt(2)*(-1/x^3 + 1)^(1/4) + sqrt(-1/x^3 +
1) + 1) - 1/6*sqrt(2)*log(-sqrt(2)*(-1/x^3 + 1)^(1/4) + sqrt(-1/x^3 + 1) + 1)

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maple [C]  time = 7.33, size = 181, normalized size = 1.47

method	result	size
trager	\(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{21 x^{6}}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (x^{4}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{2 x^{3}-1}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x +2 \left (x^{4}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{3}-1}\right )}{3}\)	\(181\)
risch	\(\frac {\frac {4}{21} x^{6}-\frac {8}{21} x^{3}+\frac {4}{21}}{x^{5} \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (x^{4}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{2 x^{3}-1}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x +2 \left (x^{4}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{3}-1}\right )}{3}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/21*(x^3-1)*(x^4-x)^(3/4)/x^6+1/3*RootOf(_Z^4+1)*ln((2*(x^4-x)^(1/2)*RootOf(_Z^4+1)^3*x-2*(x^4-x)^(1/4)*RootO
f(_Z^4+1)^2*x^2+2*(x^4-x)^(3/4)+RootOf(_Z^4+1))/(2*x^3-1))+1/3*RootOf(_Z^4+1)^3*ln((2*(x^4-x)^(1/4)*RootOf(_Z^
4+1)^2*x^2+2*(x^4-x)^(1/2)*RootOf(_Z^4+1)*x+2*(x^4-x)^(3/4)+RootOf(_Z^4+1)^3)/(2*x^3-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-3*x^3+1)/x^6/(2*x^3-1)/(x^4-x)^(1/4),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x^6-3\,x^3+1}{x^6\,{\left (x^4-x\right )}^{1/4}\,\left (2\,x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{6} - 3 x^{3} + 1}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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