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

Optimal. Leaf size=61 \[ \frac {\sqrt [3]{(x-1)^2} \text {RootSum}\left [\text {$\#$1}^6+2 \text {$\#$1}^3+3\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x-1}-\text {$\#$1}\right )}{\text {$\#$1}^3+1}\& \right ]}{2 (x-1)^{2/3}} \]

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Rubi [C]  time = 0.45, antiderivative size = 421, normalized size of antiderivative = 6.90, number of steps used = 13, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {970, 712, 50, 56, 617, 204, 31} \begin {gather*} \frac {i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \log \left (\sqrt {2}+i x\right )}{4 \sqrt {2} (x-1)^{2/3}}-\frac {i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \log \left (x+i \sqrt {2}\right )}{4 \sqrt {2} (x-1)^{2/3}}-\frac {3 i \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \log \left (\sqrt [3]{2 x-2}+\sqrt [3]{2 \left (1-i \sqrt {2}\right )}\right )}{4 \sqrt {2} (x-1)^{2/3}}+\frac {3 i \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \log \left (\sqrt [3]{2 x-2}+\sqrt [3]{2 \left (1+i \sqrt {2}\right )}\right )}{4 \sqrt {2} (x-1)^{2/3}}-\frac {i \sqrt {\frac {3}{2}} \left (1-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-i \sqrt {2}}}}{\sqrt {3}}\right )}{2 (x-1)^{2/3}}+\frac {i \sqrt {\frac {3}{2}} \left (1+i \sqrt {2}\right )^{2/3} \sqrt [3]{x^2-2 x+1} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+i \sqrt {2}}}}{\sqrt {3}}\right )}{2 (x-1)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(b*c - a*d)/b]

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 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 712

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2
), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[m]

Rule 970

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[(a + b*x + c*x^2)^F
racPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q, x], x] /; Free
Q[{a, b, c, d, f, p, q}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]

Rubi steps

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

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Mathematica [C]  time = 0.08, size = 75, normalized size = 1.23 \begin {gather*} \frac {3 i \sqrt [3]{(x-1)^2} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {i (x-1)}{i+\sqrt {2}}\right )-\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {i (x-1)}{-i+\sqrt {2}}\right )\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((3*I)/4)*((-1 + x)^2)^(1/3)*(-Hypergeometric2F1[2/3, 1, 5/3, (I*(-1 + x))/(-I + Sqrt[2])] + Hypergeometric2F
1[2/3, 1, 5/3, ((-I)*(-1 + x))/(I + Sqrt[2])]))/Sqrt[2]

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IntegrateAlgebraic [A]  time = 4.19, size = 61, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{(-1+x)^2} \text {RootSum}\left [3+2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+\text {$\#$1}^3}\&\right ]}{2 (-1+x)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.53, size = 2025, normalized size = 33.20

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*18^(1/6)*4^(2/3)*cos(2/3*arctan(2*sqrt(2) - 3))*log(-24*(18^(1/6)*4^(2/3)*sqrt(2)*(x^2 - 2*x + 1)^(1/3)*(x
- 1)*sin(2/3*arctan(2*sqrt(2) - 3)) + 2*18^(1/6)*4^(2/3)*(x^2 - 2*x + 1)^(1/3)*(x - 1)*cos(2/3*arctan(2*sqrt(
2) - 3)) - 18^(1/3)*4^(1/3)*(x^2 - 2*x + 1) - 6*(x^2 - 2*x + 1)^(2/3))/(x^2 - 2*x + 1)) - 1/2*18^(1/6)*4^(2/3)
*arctan(-1/72*(6*18^(5/6)*4^(1/3)*sqrt(2)*(x^2 - 2*x + 1)^(1/3)*cos(2/3*arctan(2*sqrt(2) - 3)) - sqrt(6)*(18^(
5/6)*4^(1/3)*sqrt(2)*(x - 1)*cos(2/3*arctan(2*sqrt(2) - 3)) + 2*18^(5/6)*4^(1/3)*(x - 1)*sin(2/3*arctan(2*sqrt
(2) - 3)))*sqrt(-(18^(1/6)*4^(2/3)*sqrt(2)*(x^2 - 2*x + 1)^(1/3)*(x - 1)*sin(2/3*arctan(2*sqrt(2) - 3)) + 2*18
^(1/6)*4^(2/3)*(x^2 - 2*x + 1)^(1/3)*(x - 1)*cos(2/3*arctan(2*sqrt(2) - 3)) - 18^(1/3)*4^(1/3)*(x^2 - 2*x + 1)
- 6*(x^2 - 2*x + 1)^(2/3))/(x^2 - 2*x + 1)) - 12*(18*(x - 1)*cos(2/3*arctan(2*sqrt(2) - 3)) - 18^(5/6)*4^(1/3
)*(x^2 - 2*x + 1)^(1/3))*sin(2/3*arctan(2*sqrt(2) - 3)) - 72*sqrt(2)*(x - 1))/(3*(x - 1)*cos(2/3*arctan(2*sqrt
(2) - 3))^2 - 2*x + 2))*sin(2/3*arctan(2*sqrt(2) - 3)) - 1/4*(18^(1/6)*4^(2/3)*sqrt(3)*cos(2/3*arctan(2*sqrt(2
) - 3)) - 18^(1/6)*4^(2/3)*sin(2/3*arctan(2*sqrt(2) - 3)))*arctan(1/36*(72*18^(5/6)*4^(1/3)*(x^2 - 2*x + 1)^(1
/3)*(2*sqrt(3) + sqrt(2))*cos(2/3*arctan(2*sqrt(2) - 3))^3 - 6*18^(5/6)*4^(1/3)*(x^2 - 2*x + 1)^(1/3)*(14*sqrt
(3) + 17*sqrt(2))*cos(2/3*arctan(2*sqrt(2) - 3)) + 432*(sqrt(3)*(x - 1) - 2*sqrt(2)*(x - 1))*cos(2/3*arctan(2*
sqrt(2) - 3))^2 - sqrt(3)*(12*18^(5/6)*4^(1/3)*(2*sqrt(3)*(x - 1) + sqrt(2)*(x - 1))*cos(2/3*arctan(2*sqrt(2)
- 3))^3 - 18^(5/6)*4^(1/3)*(14*sqrt(3)*(x - 1) + 17*sqrt(2)*(x - 1))*cos(2/3*arctan(2*sqrt(2) - 3)) - (12*18^(
5/6)*4^(1/3)*(sqrt(3)*sqrt(2)*(x - 1) - 2*x + 2)*cos(2/3*arctan(2*sqrt(2) - 3))^2 - 18^(5/6)*4^(1/3)*(sqrt(3)*
sqrt(2)*(x - 1) + 2*x - 2))*sin(2/3*arctan(2*sqrt(2) - 3)))*sqrt(-(18^(1/6)*4^(2/3)*(sqrt(3)*sqrt(2)*(x - 1) -
2*x + 2)*(x^2 - 2*x + 1)^(1/3)*cos(2/3*arctan(2*sqrt(2) - 3)) - 18^(1/6)*4^(2/3)*(x^2 - 2*x + 1)^(1/3)*(2*sqr
t(3)*(x - 1) + sqrt(2)*(x - 1))*sin(2/3*arctan(2*sqrt(2) - 3)) - 2*18^(1/3)*4^(1/3)*(x^2 - 2*x + 1) - 12*(x^2
- 2*x + 1)^(2/3))/(x^2 - 2*x + 1)) - 6*(12*18^(5/6)*4^(1/3)*(x^2 - 2*x + 1)^(1/3)*(sqrt(3)*sqrt(2) - 2)*cos(2/
3*arctan(2*sqrt(2) - 3))^2 - 864*(x - 1)*cos(2/3*arctan(2*sqrt(2) - 3))^3 - 18^(5/6)*4^(1/3)*(x^2 - 2*x + 1)^(
1/3)*(sqrt(3)*sqrt(2) + 2) + 72*(2*sqrt(3)*sqrt(2)*(x - 1) + 5*x - 5)*cos(2/3*arctan(2*sqrt(2) - 3)))*sin(2/3*
arctan(2*sqrt(2) - 3)) + 108*sqrt(3)*(x - 1) + 144*sqrt(2)*(x - 1))/(144*(x - 1)*cos(2/3*arctan(2*sqrt(2) - 3)
)^4 - 120*(x - 1)*cos(2/3*arctan(2*sqrt(2) - 3))^2 + x - 1)) - 1/4*(18^(1/6)*4^(2/3)*sqrt(3)*cos(2/3*arctan(2*
sqrt(2) - 3)) + 18^(1/6)*4^(2/3)*sin(2/3*arctan(2*sqrt(2) - 3)))*arctan(1/36*(72*18^(5/6)*4^(1/3)*(x^2 - 2*x +
1)^(1/3)*(2*sqrt(3) - sqrt(2))*cos(2/3*arctan(2*sqrt(2) - 3))^3 - 6*18^(5/6)*4^(1/3)*(x^2 - 2*x + 1)^(1/3)*(1
4*sqrt(3) - 17*sqrt(2))*cos(2/3*arctan(2*sqrt(2) - 3)) + 432*(sqrt(3)*(x - 1) + 2*sqrt(2)*(x - 1))*cos(2/3*arc
tan(2*sqrt(2) - 3))^2 - sqrt(3)*(12*18^(5/6)*4^(1/3)*(2*sqrt(3)*(x - 1) - sqrt(2)*(x - 1))*cos(2/3*arctan(2*sq
rt(2) - 3))^3 - 18^(5/6)*4^(1/3)*(14*sqrt(3)*(x - 1) - 17*sqrt(2)*(x - 1))*cos(2/3*arctan(2*sqrt(2) - 3)) - (1
2*18^(5/6)*4^(1/3)*(sqrt(3)*sqrt(2)*(x - 1) + 2*x - 2)*cos(2/3*arctan(2*sqrt(2) - 3))^2 - 18^(5/6)*4^(1/3)*(sq
rt(3)*sqrt(2)*(x - 1) - 2*x + 2))*sin(2/3*arctan(2*sqrt(2) - 3)))*sqrt((18^(1/6)*4^(2/3)*(sqrt(3)*sqrt(2)*(x -
1) + 2*x - 2)*(x^2 - 2*x + 1)^(1/3)*cos(2/3*arctan(2*sqrt(2) - 3)) - 18^(1/6)*4^(2/3)*(x^2 - 2*x + 1)^(1/3)*(
2*sqrt(3)*(x - 1) - sqrt(2)*(x - 1))*sin(2/3*arctan(2*sqrt(2) - 3)) + 2*18^(1/3)*4^(1/3)*(x^2 - 2*x + 1) + 12*
(x^2 - 2*x + 1)^(2/3))/(x^2 - 2*x + 1)) - 6*(12*18^(5/6)*4^(1/3)*(x^2 - 2*x + 1)^(1/3)*(sqrt(3)*sqrt(2) + 2)*c
os(2/3*arctan(2*sqrt(2) - 3))^2 + 864*(x - 1)*cos(2/3*arctan(2*sqrt(2) - 3))^3 - 18^(5/6)*4^(1/3)*(x^2 - 2*x +
1)^(1/3)*(sqrt(3)*sqrt(2) - 2) + 72*(2*sqrt(3)*sqrt(2)*(x - 1) - 5*x + 5)*cos(2/3*arctan(2*sqrt(2) - 3)))*sin
(2/3*arctan(2*sqrt(2) - 3)) + 108*sqrt(3)*(x - 1) - 144*sqrt(2)*(x - 1))/(144*(x - 1)*cos(2/3*arctan(2*sqrt(2)
- 3))^4 - 120*(x - 1)*cos(2/3*arctan(2*sqrt(2) - 3))^2 + x - 1)) + 1/16*(18^(1/6)*4^(2/3)*sqrt(3)*sin(2/3*arc
tan(2*sqrt(2) - 3)) - 18^(1/6)*4^(2/3)*cos(2/3*arctan(2*sqrt(2) - 3)))*log(48*(18^(1/6)*4^(2/3)*(sqrt(3)*sqrt(
2)*(x - 1) + 2*x - 2)*(x^2 - 2*x + 1)^(1/3)*cos(2/3*arctan(2*sqrt(2) - 3)) - 18^(1/6)*4^(2/3)*(x^2 - 2*x + 1)^
(1/3)*(2*sqrt(3)*(x - 1) - sqrt(2)*(x - 1))*sin(2/3*arctan(2*sqrt(2) - 3)) + 2*18^(1/3)*4^(1/3)*(x^2 - 2*x + 1
) + 12*(x^2 - 2*x + 1)^(2/3))/(x^2 - 2*x + 1)) - 1/16*(18^(1/6)*4^(2/3)*sqrt(3)*sin(2/3*arctan(2*sqrt(2) - 3))
+ 18^(1/6)*4^(2/3)*cos(2/3*arctan(2*sqrt(2) - 3)))*log(-48*(18^(1/6)*4^(2/3)*(sqrt(3)*sqrt(2)*(x - 1) - 2*x +
2)*(x^2 - 2*x + 1)^(1/3)*cos(2/3*arctan(2*sqrt(2) - 3)) - 18^(1/6)*4^(2/3)*(x^2 - 2*x + 1)^(1/3)*(2*sqrt(3)*(
x - 1) + sqrt(2)*(x - 1))*sin(2/3*arctan(2*sqrt(2) - 3)) - 2*18^(1/3)*4^(1/3)*(x^2 - 2*x + 1) - 12*(x^2 - 2*x
+ 1)^(2/3))/(x^2 - 2*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 17.82, size = 15639, normalized size = 256.38

method	result	size
trager	\(\text {Expression too large to display}\)	\(15639\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x^2-2\,x+1\right )}^{1/3}}{x^2+2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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