∫ (1+x2) 3√_________−1−x2 +x4 +x6 _____________________________________

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

Optimal. Leaf size=330 \[ \frac {\left (x^2-1\right )^{2/3} \left (x^2+1\right )^{4/3} \left (\frac {1}{12} \sqrt [3]{x^2-1} \left (3 \left (x^2+1\right )^{5/3}-2 \left (x^2+1\right )^{2/3}\right )+\frac {1}{2} \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}+\frac {1}{18} \log \left (\sqrt [3]{x^2-1}-\sqrt [3]{x^2+1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^2-1}+\sqrt [3]{x^2+1}\right )+\frac {1}{4} \log \left (\left (x^2-1\right )^{2/3}-\sqrt [3]{x^2+1} \sqrt [3]{x^2-1}+\left (x^2+1\right )^{2/3}\right )-\frac {1}{36} \log \left (\left (x^2-1\right )^{2/3}+\sqrt [3]{x^2+1} \sqrt [3]{x^2-1}+\left (x^2+1\right )^{2/3}\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2+1}}{2 \sqrt [3]{x^2-1}-\sqrt [3]{x^2+1}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2+1}}{2 \sqrt [3]{x^2-1}+\sqrt [3]{x^2+1}}\right )}{6 \sqrt {3}}\right )}{\left (\left (x^2-1\right ) \left (x^2+1\right )^2\right )^{2/3}} \]

________________________________________________________________________________________

Rubi [A] time = 0.55, antiderivative size = 488, normalized size of antiderivative = 1.48, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6719, 101, 157, 50, 59, 105, 91} \begin {gather*} \frac {1}{4} \sqrt [3]{x^6+x^4-x^2-1} \left (x^2+1\right )+\frac {1}{3} \sqrt [3]{x^6+x^4-x^2-1}+\frac {\sqrt [3]{x^6+x^4-x^2-1} \log \left (x^2\right )}{4 \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}}+\frac {\sqrt [3]{x^6+x^4-x^2-1} \log \left (x^2-1\right )}{36 \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}}-\frac {3 \sqrt [3]{x^6+x^4-x^2-1} \log \left (-\sqrt [3]{x^2-1}-\sqrt [3]{x^2+1}\right )}{4 \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}}+\frac {\sqrt [3]{x^6+x^4-x^2-1} \log \left (\frac {\sqrt [3]{x^2+1}}{\sqrt [3]{x^2-1}}-1\right )}{12 \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}}-\frac {\sqrt {3} \sqrt [3]{x^6+x^4-x^2-1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3} \sqrt [3]{x^2-1}}\right )}{2 \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}}+\frac {\sqrt {3} \sqrt [3]{x^6+x^4-x^2-1} \tan ^{-1}\left (\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3} \sqrt [3]{x^2-1}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}}-\frac {4 \sqrt [3]{x^6+x^4-x^2-1} \tan ^{-1}\left (\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3} \sqrt [3]{x^2-1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

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

^2])/(4*(-1 + x^2)^(1/3)*(1 + x^2)^(2/3)) + ((-1 - x^2 + x^4 + x^6)^(1/3)*Log[-1 + x^2])/(36*(-1 + x^2)^(1/3)*

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

3)*(1 + x^2)^(2/3)) + ((-1 - x^2 + x^4 + x^6)^(1/3)*Log[-1 + (1 + x^2)^(1/3)/(-1 + x^2)^(1/3)])/(12*(-1 + x^2)

^(1/3)*(1 + x^2)^(2/3))

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 59

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt

[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x

)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[

b*c - a*d, 0] && PosQ[d/b]

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[

(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/

3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*

x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +

b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -

1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))

*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ

ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a

+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]

/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su

mSimplerQ[m, -1] || !SumSimplerQ[n, -1])))

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]

:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x

), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F

racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !

FreeQ[v, x] && !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1+x) \sqrt [3]{(-1+x) (1+x)^2}}{x} \, dx,x,x^2\right )\\ &=\frac {\sqrt [3]{\left (-1+x^2\right ) \left (1+x^2\right )^2} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x} (1+x)^{5/3}}{x} \, dx,x,x^2\right )}{2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}\\ &=\frac {1}{4} \left (1+x^2\right ) \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )}-\frac {\sqrt [3]{\left (-1+x^2\right ) \left (1+x^2\right )^2} \operatorname {Subst}\left (\int \frac {\left (2-\frac {4 x}{3}\right ) (1+x)^{2/3}}{(-1+x)^{2/3} x} \, dx,x,x^2\right )}{4 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}\\ &=\frac {1}{4} \left (1+x^2\right ) \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )}+\frac {\sqrt [3]{\left (-1+x^2\right ) \left (1+x^2\right )^2} \operatorname {Subst}\left (\int \frac {(1+x)^{2/3}}{(-1+x)^{2/3}} \, dx,x,x^2\right )}{3 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{\left (-1+x^2\right ) \left (1+x^2\right )^2} \operatorname {Subst}\left (\int \frac {(1+x)^{2/3}}{(-1+x)^{2/3} x} \, dx,x,x^2\right )}{2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}\\ &=\frac {1}{3} \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )}+\frac {1}{4} \left (1+x^2\right ) \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )}+\frac {\left (4 \sqrt [3]{\left (-1+x^2\right ) \left (1+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} \sqrt [3]{1+x}} \, dx,x,x^2\right )}{9 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{\left (-1+x^2\right ) \left (1+x^2\right )^2} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} \sqrt [3]{1+x}} \, dx,x,x^2\right )}{2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{\left (-1+x^2\right ) \left (1+x^2\right )^2} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x \sqrt [3]{1+x}} \, dx,x,x^2\right )}{2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}\\ &=\frac {1}{3} \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )}+\frac {1}{4} \left (1+x^2\right ) \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )}-\frac {\sqrt {3} \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x^2}}{\sqrt {3} \sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}-\frac {4 \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^2}}{\sqrt {3} \sqrt [3]{-1+x^2}}\right )}{3 \sqrt {3} \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}+\frac {\sqrt {3} \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^2}}{\sqrt {3} \sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}+\frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )} \log (x)}{2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}+\frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )} \log \left (1-x^2\right )}{36 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}-\frac {3 \sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )} \log \left (\sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}+\frac {\sqrt [3]{-\left (\left (1-x^2\right ) \left (1+x^2\right )^2\right )} \log \left (1-\frac {\sqrt [3]{1+x^2}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] time = 0.11, size = 188, normalized size = 0.57 \begin {gather*} -\frac {3 \sqrt [3]{\left (x^2-1\right ) \left (x^2+1\right )^2} \left (2^{2/3} x^2 \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {1}{2} \left (1-x^2\right )\right )-4\ 2^{2/3} \left (x^2+1\right ) \, _2F_1\left (-\frac {5}{3},\frac {1}{3};\frac {4}{3};\frac {1}{2} \left (1-x^2\right )\right )+2\ 2^{2/3} \left (x^2+1\right ) \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};\frac {1}{2} \left (1-x^2\right )\right )+2^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {1}{2} \left (1-x^2\right )\right )+2 \left (x^2+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {1-x^2}{x^2+1}\right )\right )}{4 \left (x^2+1\right )^{5/3}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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

________________________________________________________________________________________

IntegrateAlgebraic [A] time = 7.85, size = 308, normalized size = 0.93 \begin {gather*} \frac {\left (-1+x^2\right )^{2/3} \left (1+x^2\right )^{4/3} \left (\frac {1}{12} \sqrt [3]{-1+x^2} \left (4 \left (1+x^2\right )^{2/3}+3 \left (1+x^2\right )^{5/3}\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1+x^2}}{2 \sqrt [3]{-1+x^2}-\sqrt [3]{1+x^2}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1+x^2}}{2 \sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}}\right )}{6 \sqrt {3}}+\frac {1}{18} \log \left (\sqrt [3]{-1+x^2}-\sqrt [3]{1+x^2}\right )-\frac {1}{2} \log \left (\sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}\right )+\frac {1}{4} \log \left (\left (-1+x^2\right )^{2/3}-\sqrt [3]{-1+x^2} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-\frac {1}{36} \log \left (\left (-1+x^2\right )^{2/3}+\sqrt [3]{-1+x^2} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{\left (\left (-1+x^2\right ) \left (1+x^2\right )^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-1 + x^2)^(2/3)*(1 + x^2)^(4/3)*(((-1 + x^2)^(1/3)*(4*(1 + x^2)^(2/3) + 3*(1 + x^2)^(5/3)))/12 + (Sqrt[3]*Ar

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

2*(-1 + x^2)^(1/3) + (1 + x^2)^(1/3))]/(6*Sqrt[3]) + Log[(-1 + x^2)^(1/3) - (1 + x^2)^(1/3)]/18 - Log[(-1 + x^

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

Log[(-1 + x^2)^(2/3) + (-1 + x^2)^(1/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/36))/((-1 + x^2)*(1 + x^2)^2)^(2/3

)

________________________________________________________________________________________

fricas [A] time = 0.49, size = 305, normalized size = 0.92 \begin {gather*} -\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x^{2} + 1\right )} + 2 \, \sqrt {3} {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x^{2} + 1\right )}}\right ) - \frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (x^{2} + 1\right )} - 2 \, \sqrt {3} {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x^{2} + 1\right )}}\right ) + \frac {1}{12} \, {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} {\left (3 \, x^{2} + 7\right )} - \frac {1}{36} \, \log \left (\frac {x^{4} + 2 \, x^{2} + {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {2}{3}} + 1}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + 2 \, x^{2} - {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {2}{3}} + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} + 1}{x^{2} + 1}\right ) + \frac {1}{18} \, \log \left (-\frac {x^{2} - {\left (x^{6} + x^{4} - x^{2} - 1\right )}^{\frac {1}{3}} + 1}{x^{2} + 1}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*sqrt(3)*arctan(1/3*(sqrt(3)*(x^2 + 1) + 2*sqrt(3)*(x^6 + x^4 - x^2 - 1)^(1/3))/(x^2 + 1)) - 1/2*sqrt(3)*

arctan(-1/3*(sqrt(3)*(x^2 + 1) - 2*sqrt(3)*(x^6 + x^4 - x^2 - 1)^(1/3))/(x^2 + 1)) + 1/12*(x^6 + x^4 - x^2 - 1

)^(1/3)*(3*x^2 + 7) - 1/36*log((x^4 + 2*x^2 + (x^6 + x^4 - x^2 - 1)^(1/3)*(x^2 + 1) + (x^6 + x^4 - x^2 - 1)^(2

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

- 1)^(2/3) + 1)/(x^4 + 2*x^2 + 1)) - 1/2*log((x^2 + (x^6 + x^4 - x^2 - 1)^(1/3) + 1)/(x^2 + 1)) + 1/18*log(-(x

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C] time = 62.24, size = 4208, normalized size = 12.75


method	result	size

trager	\(\text {Expression too large to display}\)	\(4208\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/4*x^2+7/12)*(x^6+x^4-x^2-1)^(1/3)-1/18*ln((-351253861284635030780891080042612082507770230553507917611312046

4327428721518261731293967309748584316363213250560*x^20-6746389289678077923428764810604167136947931869021226834

2862504291743786612860389474681240476732543769808465113280*x^10+5267927477412893796534426126586625020127634112

1710369582430706980165796549239620007340698525251375371432562462080*x^6-14842283170367790226271482519764753425

385942246466594065004761265496645063655774074048616341442669901500132596640*x^2+255088153384240881467577442639

19025400673839142208728799803283638455703435896773308395878117326105007475243014720*x^8-3168543061057415052959

185013903006086433610993967145387041722593555876414455706773748203553112968312965706401920*x^4-372930353101016

36179328899559738961728311492960880432101889227389127684085673615291145454107760869705383124782720*x^12+203088

62540464149811959881796093130698045026694957500163400751522254901546072938198037121655858979748076366572000*x^

16+22730162900801230121941388830036650311102910397751848332888733870571072380089197719303826839498903328463519

7440*x^22-1059507377446528661097367015220731620387863392610333886876182763131520720760440819004416990516178813

74822771744*RootOf(_Z^2-3*_Z+9)^2-7445856372466336854641029537910891590528702926351860620242168685456838651089

235799706964314361158444717196964000*x^18+36845794808838101650418331878779754402173337753205441679778798531429

467975539636026732604490663769881326936977280*x^14-17282348318610904882172123607676019444082077123692993665388

51705805363716660761007508541756785266377307461120*RootOf(_Z^2-3*_Z+9)^2*x^20+22413433231852528924281438529511

3591319229581186654058705830711949163533042428320093257366022070686015644160256*RootOf(_Z^2-3*_Z+9)*x^20-22782

0312028461981261179993675765526859625653654076442414344854731483758892476378477927296930316902381704726944*Roo

tOf(_Z^2-3*_Z+9)^2*x^18+45627758038111765342039665210746209773117152376295781693811487126394616418872590548015

73531089260096766833508965*RootOf(_Z^2-3*_Z+9)*x^18-1573715884936107199680217320341158701038249527282289790607

11916734377234896010179391669030890348444781341186968*RootOf(_Z^2-3*_Z+9)^2*x^2-579510745548257823836084292557

8031416094181580439765880762242397840431141994976475846878104799239434550402137691*(x^6+x^4-x^2-1)^(1/3)+51364

9750555857432060310710939014156141821862591649149294203833248919221585602339311128396436564803925869700744*Roo

tOf(_Z^2-3*_Z+9)^2*x^4+203316396759215427589291903362411473873030866042098620118441748160359338453805014940476

9453388726186368301897929*RootOf(_Z^2-3*_Z+9)-9193010031607746136970126518396493949441476076167971661739921327

538899396363574019691055378089041752706561426748*RootOf(_Z^2-3*_Z+9)*x^4+3683695096475897610351694027889566062

517145439831607612428056055950525372377410259872484493372890760959350593469*RootOf(_Z^2-3*_Z+9)*x^2-1180299776

69068878547173266180289406813797199347038718923725511276532167697353926644133555470000415498520655336*RootOf(_

Z^2-3*_Z+9)^2*x^16+1038672995877658212545741789320720128963634579406193290393786127901474384966249198077337251

380107829074391296785*RootOf(_Z^2-3*_Z+9)*x^16+832533794574288042612432172954213552978476256754426867711197977

643644510653542809975329944871627044231262816952*RootOf(_Z^2-3*_Z+9)^2*x^14-1720237045283778738691605087778876

7541976970849258256958722581676491156787566332925254352152845292057983719786748*RootOf(_Z^2-3*_Z+9)*x^14+53926

0172409068885261871916627336836655698491788345631350709633707926541940978967859535590747972466777838171672*Roo

tOf(_Z^2-3*_Z+9)^2*x^12-77887330739580928928682898222464434242962160427431749272126052320372328453397126945422

81310564018108708145607948*RootOf(_Z^2-3*_Z+9)*x^12-1194998862699794078156671494935157472747352890409401624360

016957722442857317469154624237588332876603846718564904*RootOf(_Z^2-3*_Z+9)^2*x^10+2549119546370260155999623489

9141786761834368971317393902234023486044813784606843889084575979060865302180044306118*RootOf(_Z^2-3*_Z+9)*x^10

-8272233952031325554994723878161030422245664286563730808685077105945032154199338565182577416838489528157350025

20*RootOf(_Z^2-3*_Z+9)^2*x^8+136861621221390381225914985204037028081655528288661181296354891559420656575911736

33969807503783177933294272154478*RootOf(_Z^2-3*_Z+9)*x^8+73456896118100969308109084202779023523606183156182279

6823050584688256958145456684088253229521481298561282151336*RootOf(_Z^2-3*_Z+9)^2*x^6-1628418080755271749983004

8715981395933118256313635826613749627795103496084858602642441238512835822887986622489068*RootOf(_Z^2-3*_Z+9)*x

^6-25072479123763394737123846633509643312696895491569102060402186521998320801196304875120845192087844059798365

7984*x^22*RootOf(_Z^2-3*_Z+9)+13065584982779970369933265350152861272227587878637674099348283923255326923544899

530573190982718362854541492224*x^22*RootOf(_Z^2-3*_Z+9)^2+2270417180916391380237886019934924798346243397301727

40449889393984220482919452380920352532710315558059996020736*(x^6+x^4-x^2-1)^(1/3)*x^20+22727365084391655782725

5777004777768623816992948384615601004339519725071999838827996968815411775871398717685760*(x^6+x^4-x^2-1)^(2/3)

*x^18-35893346819109567581070829062481282978764471441451172828234414919035192264058413524957063931077577468800

09913856*(x^6+x^4-x^2-1)^(1/3)*x^18-36647090914105407322157800048567653317793965479580061829480652437170596367

06830377742475135227919173014792139520*(x^6+x^4-x^2-1)^(2/3)*x^16-61472334529129268256195866528773903873346736

09385824426667424170516899309120612733209452613695717423677031662545*(x^6+x^4-x^2-1)^(1/3)*x^16-48184867432710

61974263639633130894090461168742385084074699886773854105299329059450210146369494839832931248474255*(x^6+x^4-x^

2-1)^(2/3)*x^14+2082957903166189729528617311649396742765077311529127183510040145078355675649893982464482390148

7630494097112371406*(x^6+x^4-x^2-1)^(1/3)*x^14+208055484821232896272448445837167731839902075660028207283130982

29206653099801707602401705096857911760459384830455*(x^6+x^4-x^2-1)^(2/3)*x^12+27098769649345548976229687230714

951473640888369965349112690332094547887985365438509506156300704367364362269922894*(x^6+x^4-x^2-1)^(1/3)*x^12+1

76838241395077408398094620206071902650217287658583653474699013256580715745253848542993089149924454348707299014

65*(x^6+x^4-x^2-1)^(2/3)*x^10-38255142578164215758552935749842846956040922957619237530263974387037684411357540

032584351701707403040842096631498*(x^6+x^4-x^2-1)^(1/3)*x^10-3551577494567396982502551958986652608661040738802

9555745331198509949954455824911764672973180213590964705817034665*(x^6+x^4-x^2-1)^(2/3)*x^8-4312922873194525056

8021143202653314723112540166556051758911376711925651031495820697092037040034913162764345335620*(x^6+x^4-x^2-1)

^(1/3)*x^8-220105265669186429170020498174313974872938815015351885132311294946927364789558417646863051194476344

67743985967045*(x^6+x^4-x^2-1)^(2/3)*x^6+276910401523576479278826547240465068007843378567596291327062576736480

57362886310612733855330017327442395755181674*(x^6+x^4-x^2-1)^(1/3)*x^6+222823925879139855827604514083840597040

37745422407710560621249442771675314467701575124931799611647558420598052205*(x^6+x^4-x^2-1)^(2/3)*x^4+277413238

64347090190833569827267607173069112640050765276863909889524618708019498461407008289710688747826153894946*(x^6+

x^4-x^2-1)^(1/3)*x^4+33681107160086945536486599885619985432111060617970800638471726610804630147108011933383077

276787142000206588928*RootOf(_Z^2-3*_Z+9)^2*(x^6+x^4-x^2-1)^(2/3)+88677937984847774001113105757464888383816086

37112713031415736606456487180242048422204040528504096897795083152955*(x^6+x^4-x^2-1)^(2/3)*x^2-397598482420532

510559844300662311382756412651044467429429562386258261287328984805904109018271807865483226514432*(x^6+x^4-x^2-

1)^(2/3)*RootOf(_Z^2-3*_Z+9)-721302436008879117410795709338161842502591836140119750441727155386688485628025294

15171670677513534202100154400*RootOf(_Z^2-3*_Z+9)^2*(x^6+x^4-x^2-1)^(1/3)-668057633250085003342333630558218437

4514510876921901091056155147716664787928889606613471772094295499514120305006*(x^6+x^4-x^2-1)^(1/3)*x^2+1635147

324220587738836563140853892378904647724289366932739933592854296317110664721844999581824966705367527298249*(x^6

+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)-457427480850334710292890837557174899450134827388811312772599996419508607

27670581720337832012634565135618336504*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^14+1923797092891334288450

6573809228139046023320189768249610068068913082958677232334363430695268039669105680144128*(x^6+x^4-x^2-1)^(2/3)

*RootOf(_Z^2-3*_Z+9)^2*x^12-3041103426439053025441827065238676749010489546396441367776218610730647623467769462

53523531848729946126776294898*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^14-149157063266712040080066004268410

6825335078769909688724317547969474419636217055218705357480570631772129383813032*(x^6+x^4-x^2-1)^(2/3)*RootOf(_

Z^2-3*_Z+9)*x^12+747799115677575031501549410875188643157838665964203388908716264240803336462876330925066352910

568048461972969584*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^12+648840380994234664285584158472594508158918

429635132209773469110953601770221005959327794291970247505778306297088*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9

)^2*x^10-15147174575419323292539859764030875922455256056524610415349021012661946284368396286157419920882612168

688363652522*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^12-12804150466163519737353663731265294410006995443448

674910589953355586731283142440352126234253684887550371699959096*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^10

+2480413459772589943962627760746765578174561875159630237205747849547369192444496238789700256340461850926401161

52*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^10+3831944557941108781583723853123927491841630626844883524170

9691427254323566226476623568657519495491020770318848*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^8-243873331

0903095643597973091805002578015302273695093364598729364478906406784984364723173077048313642610449601146*(x^6+x

^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^10+124824026227337614112756456463265781445047733424279645003323282596431

9404182957366277641820669509201945651436336*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^8-96206651478783806611

7525012798857360300465954096752313501398264039449246339858374959464685544135500769994145888*(x^6+x^4-x^2-1)^(1

/3)*RootOf(_Z^2-3*_Z+9)^2*x^8-70140394313590875369586808874997044492033117048295714912579278625885397520747964

3477680020321297409350492717568*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^6+199396702412570816827209817703

89592483335446582124467453151852407810974666175972723497203806125322509796223599372*(x^6+x^4-x^2-1)^(1/3)*Root

Of(_Z^2-3*_Z+9)*x^8+140001057234262978435122758505254667658343797007486418545007034112783632735746705875224980

39521909735445222598608*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^6-3340976054658606157498393142376375731377

44842067561004733464049538657639168698635892248283622500014406074946952*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z

+9)^2*x^6-8173420053417203011659147782801081958515917178984688502955389986264342550163950794935769159039465445

8397076736*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^4+458933679207564821442895894171391320948296537793172

4801502653438967698799571228280103847070662815581473679798282*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^6+26

9157993749983118700707779572149683501270616575429323110041419665900310642775519987912531757517554234285505528*

(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^4+4924904574957588163656895615376140406117029626986572613239256875

74929088077001582289880842562287652092262132592*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^4+24878697007875

9110852029031505060660915484707027364420942256957370537632319332317841545391955027452627838740736*(x^6+x^4-x^2

-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^2-10517220107999747891133254365826085392465137874007611211863642551578254598

342611139762478455670657687479892737222*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^4-502397042758452746544225

9737664117896590240365592567662303457406123464615563450593296528006047296031051132361752*(x^6+x^4-x^2-1)^(2/3)

*RootOf(_Z^2-3*_Z+9)*x^2+1376503229029056267062045845246195254987279139312194332844492352068198940921479616417

24008197485359787558578888*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^2-21496024601613906797285220416233582

64492309108600025711629350553581579254925044027602054501631263986490444953758*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^

2-3*_Z+9)*x^2+130367059920718411259749002049446364637381696289325915572374011764230779367182777224969981288657

54496248250368*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^20+1306247629788044332636014187286522410698179776

4289466574027950680368032278983438508787696206805789311661768704*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x

^18-2506381542655095596393633708994717587015007001665757729776892169794864610514831833269798733593206155231039

32416*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^20-250715465182935366240519095903233521631231584572646398028

060865491321324078278665685851967593140719969344487424*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^18-61059384

93178187470330659875568845164358162766305410673152263972675326681179896769875437691101374606010260224*(x^6+x^4

-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^18-104333616173271150316046289429068949855264259022291554799880686061697

96564204732074846373572597383653334261760*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^16+3080045980832416450

28425214076195615712833644553166844569018832684982219427988839399259115517041224525926675200*(x^6+x^4-x^2-1)^(

1/3)*RootOf(_Z^2-3*_Z+9)*x^18+39122078575486808818967792977115592608014439590930186748955767683260806780953564

9078064715007941302148057001728*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^16-2193841439405873642816009861547

01600613487046706594042923160666405764420814886815424575356874776829345893900896*(x^6+x^4-x^2-1)^(1/3)*RootOf(

_Z^2-3*_Z+9)^2*x^16-210470049208246211088188909166334662343528162797300881827277697675050707417568384587643754

600992249924881769472*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^14+434511054865740955534163890535082851916

8939009835091447364537273073546955418264944829029937613850487299545048219*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*

_Z+9)*x^16+410307734862461336855349778352858430805384861052480310326797534643242817790963141450944555263799651

6349239085312*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^14-3953139815812989501011593885186751609572610974780

255817076000799217627642942799293398006013886194030281821204795*(x^6+x^4-x^2-1)^(2/3)-184322212252269081683878

3816715012737729315460920965580731006106095910839347379047415854650788900250821225188640)/x^18/(x^2+1))+1/54*R

ootOf(_Z^2-3*_Z+9)*ln((341799187506635442187396355672915082000754870020462719042899221613317326905246323902787

7821270084783203403816960*x^20+2088955396301134975249584996784389865855271945340288410716144460388179311157012

425720771087195304958955601396160*x^10-12493127802820157771114270981473357329614478176576219428968550092950767

952070926262998238023556083371253950992000*x^6+623293793728736446827604687966874943028036700049561751236173466

0164868876406996996868731838303971833387926397600*x^2-70763818877222587930629541422358033730526247473591512047

945873886560891135326464310745816054040674869417029440320*x^8+312688378712642879790225575464810098939886899899

16760823530383827771228334539401534971334459441823959483118394240*x^4+6679440235583516918488813736488382673538

3143317464906723705481179901027183673351870886966208954949057573440042880*x^12-2676594183497126053665257901440

9090078832822909038280414915248886891446761961138282805954736308911703196903162080*x^16+4874788015998344696778

09456214392359170510086459477468769097978351090032407865030264514830644335607369726689280*x^22-538078034944597

987996558178700277852042979524632310330419955194556095789456104546843470851991672534671310880736*RootOf(_Z^2-3

*_Z+9)^2-50175351211053661365563078929160825362129688651078953576324094932603739911174559517498009103483521557

20540564320*x^18+869972534702858624285180546776262546153503232022610577832056180593277469875216320194907056847

7380861313289041280*x^14-8776958254805119060110259161117954174497748435458699042808731426889686557316849244279

179185270237307115809280*RootOf(_Z^2-3*_Z+9)^2*x^20-9203809971991761228464917706969061545165112753519962362034

49915212643647791708375938560866338313374184079495424*RootOf(_Z^2-3*_Z+9)*x^20-1157000965035060573316301943082

787153889448989841451999571719387348206534686044927239491061370739831000266109536*RootOf(_Z^2-3*_Z+9)^2*x^18+5

55182657476091263593919583768601857547049456503664729468225064619846836224898190328439194374776010819523284957

3*RootOf(_Z^2-3*_Z+9)*x^18-79922232629310325314258222797655124046346118571299063789962437274693123035385394029

4984802791729118342237279592*RootOf(_Z^2-3*_Z+9)^2*x^2-3022767916953086887160114381695044839951235463863386434

85627558861012233294693679356682268642107637328284856347*(x^6+x^4-x^2-1)^(1/3)+2608605228356017337502286237125

208489142075803624974788093425192775008769534808189713663926488863601242731631736*RootOf(_Z^2-3*_Z+9)^2*x^4+30

32990698251717967602998964153277382282808747344261998867281029906577139790570774129897125767443706775159477257

*RootOf(_Z^2-3*_Z+9)-18075006737028269833796713567739788112345573818365277821700771103715884509786567513414781

675259955327423325122596*RootOf(_Z^2-3*_Z+9)*x^4+1133934351899277294355613205230968642112214914815048171125342

085692437375553714056074566787933854815486018233365*RootOf(_Z^2-3*_Z+9)*x^2-5994232772761656464186819802992616

35495957935550825753466774195905306189623557193885736786802214246181538965784*RootOf(_Z^2-3*_Z+9)^2*x^16+96065

63674527538422455081662249548601466517284377837271927692556369026802527900179080254830406410483644463718137*Ro

otOf(_Z^2-3*_Z+9)*x^16+422807955607756616775091642110605087632245375140463396096650872650771578648999043765596

7283921834149395428962888*RootOf(_Z^2-3*_Z+9)^2*x^14-176046775181244520066960228481803353702198861857370116150

29175012204481488523092229311385841056204545727694349652*RootOf(_Z^2-3*_Z+9)*x^14+2738669499339101448689154506

166537053018929325609929326628874120822444913392325096556187976051369683465530190568*RootOf(_Z^2-3*_Z+9)^2*x^1

2-284357533420092759653787198606590418261235016731997736077289700575471289474078930812593299413050837275657777

73892*RootOf(_Z^2-3*_Z+9)*x^12-6068883081797944172847557739426762694321985636798596145878467467983735731760683

850815814566060149930692488514776*RootOf(_Z^2-3*_Z+9)^2*x^10+2120988041016950766710969208207485105497388631830

9092892248591126081425442268765616638272851663800504321354812686*RootOf(_Z^2-3*_Z+9)*x^10-42011103313300340871

52949518677120105536812274830602835591780094750517640072254032631735362395547698511130155880*RootOf(_Z^2-3*_Z+

9)^2*x^8+34792431562270503099636261831074738258333936034094169569678590653012186389773526834663639630078522697

986076960358*RootOf(_Z^2-3*_Z+9)*x^8+3730558480075521439611130300682531194948589606786416109117189069446746219

317731509727575715111328923680785789784*RootOf(_Z^2-3*_Z+9)^2*x^6-99160979897363024968820191960735727842052447

60091618413624292676721802559731923835843732816419351201884792371060*RootOf(_Z^2-3*_Z+9)*x^6-37402097015590552

6154041822356101969033789553432937154562343006233913904710616693580993822520835221157601411072*x^22*RootOf(_Z^

2-3*_Z+9)+6635446286253633750753599515148701231461009994769520946709452908305586821076143463137715335498437499

5943161856*x^22*RootOf(_Z^2-3*_Z+9)^2-800543217434298664475840532560852419350875689228926619670934831333672316

11235187179337340336356703372941721600*(x^6+x^4-x^2-1)^(1/3)*x^20-40731045802396555095706023384913577930704158

0890949379316926700830926609724146222481497821506394184704359858176*(x^6+x^4-x^2-1)^(2/3)*x^18-534348626550573

788386384870703054263786130155563577182674656641503475708555065314582712059074175882426756436736*(x^6+x^4-x^2-

1)^(1/3)*x^18-258494698870188050293118787602945960840870119335016297979928483067476360235606601918189835235747

1719450629490176*(x^6+x^4-x^2-1)^(2/3)*x^16+965195764294458837269192022060755952021766074312939377420896838950

511736289707250933773618739078035123999170799*(x^6+x^4-x^2-1)^(1/3)*x^16+5596514859728562231603153534957846872

608623624013617298658539986367722867641719332029396497010219466792532856433*(x^6+x^4-x^2-1)^(2/3)*x^14+3831514

642946625708532341797148866896507534576332699731280465926527676930783918811760749774679475181528366445638*(x^6

+x^4-x^2-1)^(1/3)*x^14+165039783224821485108034236199475059593991811379816688018510669410011408192456329555565

08912558373466022354688511*(x^6+x^4-x^2-1)^(2/3)*x^12-22810774989810070656010604830086598598181821134876981676

22781970246948309075404433952031625726431973121771180258*(x^6+x^4-x^2-1)^(1/3)*x^12-14889063830034706393681271

746935342391568991697771469496338736742519706342912882568129245218329989664239613302031*(x^6+x^4-x^2-1)^(2/3)*

x^10-851315583046073078297694967444331903782651276090020139929976454864550734869269647105789369428418198999194

3906434*(x^6+x^4-x^2-1)^(1/3)*x^10-314261791486391416113002907491873991689932286288851268780561128092117073311

80001376227929800529603939681929856025*(x^6+x^4-x^2-1)^(2/3)*x^8+157758051955307471690863205775013600863290824

2201943782281653612422136763972389033496184533856691490742595543452*(x^6+x^4-x^2-1)^(1/3)*x^8+1367715511513707

1830271964935395268805926277066364122708138845662812667564900853206582068816226418054437247370667*(x^6+x^4-x^2

-1)^(2/3)*x^6+755323280484815748662797970278826170998312051872056878602940034886383153730289797580693704465186

7253260060083826*(x^6+x^4-x^2-1)^(1/3)*x^6+2235425876435638666781325144664841137827512472602537656468538860300

5585416881272563544450170570648330997880878165*(x^6+x^4-x^2-1)^(2/3)*x^4+1221958895437634311658862706635155399

75263591362571560676997355461127852461643953307689514537571254819155811954*(x^6+x^4-x^2-1)^(1/3)*x^4+336811071

60086945536486599885619985432111060617970800638471726610804630147108011933383077276787142000206588928*RootOf(_

Z^2-3*_Z+9)^2*(x^6+x^4-x^2-1)^(2/3)-39650347535599729985472073537003189031497500964187101670143229955790679755

74312497111634962042544121707765265677*(x^6+x^4-x^2-1)^(2/3)*x^2+195511839460010837340924701348591470163746287

336642625598732026593433506446336734303810554611085013481986980864*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)+4

39332554284273702458217982887536332718498552661739375481846531094873008996897524604641271192796741217266183200

*RootOf(_Z^2-3*_Z+9)^2*(x^6+x^4-x^2-1)^(1/3)-23356794298119272592179325787505973899964674995089646880314002906

50076832096648064677484633544540095506973418294*(x^6+x^4-x^2-1)^(1/3)*x^2+397314091505374570189054980395286254

458453984652667908676023059379518141319225329889520542655482318592035033097*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-

3*_Z+9)+278610986916623173515132435975089606922140707529315237716917816281213877466061415279682076779952695214

163614712*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^14+192379709289133428845065738092281390460233201897682

49610068068913082958677232334363430695268039669105680144128*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^12-7

91625421555893569872865559802374376648505360923501305201600059803067980008084259921500102687363861241504997745

0*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^14+1376142807093640343493620599828737991058938848771079226657139

555995921884153661212524773308962393757495302948264*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^12-45547121315

71200184411855008580509982084822476598188768165710958896173014890522923788743801353239560336567267952*(x^6+x^4

-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^12+648840380994234664285584158472594508158918429635132209773469110953601

770221005959327794291970247505778306297088*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^10+687461052529780932

2330317514795445384514353971032060100326504304561633058938384927639378362556633704941530738310*(x^6+x^4-x^2-1)

^(1/3)*RootOf(_Z^2-3*_Z+9)*x^12+891110818019811175164015878042972736105348486563788165194913868986512066181640

4596159468501863402515701862176568*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^10-1510775961041631862689389329

465137527946145988260201586623419953054399366205289216701853632746967472443281296056*(x^6+x^4-x^2-1)^(1/3)*Roo

tOf(_Z^2-3*_Z+9)^2*x^10+38319445579411087815837238531239274918416306268448835241709691427254323566226476623568

657519495491020770318848*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^8+1802816611377807782410793489990039124

2637780846534906341119975676373303660135063199270588209045508652888463113790*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2

-3*_Z+9)*x^10-147815693574984266802258799582009346396097517185348946148349097452784534558031622601905376578648

2148070273349424*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^8+58597769566926419077793169170810219128816727113

00508368913847384388890838382397981039179309148382168133479892064*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*

x^8-7014039431359087536958680887499704449203311704829571491257927862588539752074796434776800203212974093504927

17568*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^6-60891208371263563990168113887551995785089026073397647942

75278041309803243668162168965252970732581952455640335316*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^8-9791682

064610845321337067318025644096312392677850898959745946693725239422329792726656417917594125279342266293200*(x^6

+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^6+20349294146535951764629225221990292422435903415452993497706871355465

56575944255010343138189337534874225158328456*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^6-81734200534172030

116591477828010819585159171789846885029553899862643425501639507949357691590394654458397076736*(x^6+x^4-x^2-1)^

(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^4-163374277778109166859325037506588360464054515030917907744145394464716617663009

64476284111195552547146692284886782*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^6+2212472094550490619988410873

95915234009684414163651987067281979509960242367061527708233617784850372516096954888*(x^6+x^4-x^2-1)^(2/3)*Root

Of(_Z^2-3*_Z+9)*x^4-299967225744373898832696341921957936354609148515248296922206656789991934998590835837453536

3268201437643351467376*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^4+248786970078759110852029031505060660915

484707027364420942256957370537632319332317841545391955027452627838740736*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_

Z+9)^2*x^2+122938019003613001713343566651571115478367913020316559935000680324967241151556366819091252092123768

6598122834922*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^4+35312486071119728003300855486337539310973321234283

81136649915661900238821647456686247255654317131315284099917336*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^2-8

38403767129994295338628604766566272664982622214229658151045426059270616671383165856459118295662646321142341064

*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^2+5155021047598983110608945003067618852662635234076980971325202

366370493128266889918032373303510875851497476194858*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^2-794042700679

23552803544451567777643318438460306914072987224947053779879259468915295311553028280751052891029504*(x^6+x^4-x^

2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^20+130624762978804433263601418728652241069817977642894665740279506803680322

78983438508787696206805789311661768704*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^18+2016423554827731357806

28521320336012734988443230815746816757246695868710615410785413308142568000997294085832704*(x^6+x^4-x^2-1)^(1/3

)*RootOf(_Z^2-3*_Z+9)*x^20+17234060739565270628235824466604217698934079798690959858389316140911313040437803463

3125790352305984099373875200*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^18+3719019125117193994065720362726810

8541460812577686072774892164533043667733142459187354038695226157744103471872*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2

-3*_Z+9)^2*x^18-1043336161732711503160462894290689498552642590222915547998806860616979656420473207484637357259

7383653334261760*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^16+10665565047372685371360640449075319048413587

88683064643930347211784016092225646716293547035210269985242995441664*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)

*x^18-32862061605090539800005015611371455616698584049592693460962926519558928842430725662898647357235700022805

1431168*(x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^16+1336230012755711247195522206968991900445244409273207481

467340295613252535023391194066631691078626448101065467488*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)^2*x^16-210

470049208246211088188909166334662343528162797300881827277697675050707417568384587643754600992249924881769472*(

x^6+x^4-x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)^2*x^14-26177646524512535592248406950786170407313031648107964309490281

62551417764474629784070470272628304024448534217637*(x^6+x^4-x^2-1)^(1/3)*RootOf(_Z^2-3*_Z+9)*x^16-284025705337

5136102024364328530576333992679633740997812304309160382123933404221106983583025032043016799948468480*(x^6+x^4-

x^2-1)^(2/3)*RootOf(_Z^2-3*_Z+9)*x^14-484280529863380452286274738820310588895284938235192089961844241849516983

3605781603709885373210533348729641447739*(x^6+x^4-x^2-1)^(2/3)-39530368316805668701178701052861463890065773141

19127891237155131727319914441958611279358308600713493593977683680)/x^18/(x^2+1))

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]