3.30.16 \(\int \frac {(1+x^2) \sqrt [3]{-1-x^2+x^4+x^6}}{x} \, dx\) [2916]

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

[Out]

(x^2-1)^(2/3)*(x^2+1)^(4/3)*(1/2*(x^2-1)^(1/3)*(x^2+1)^(2/3)+1/12*(x^2-1)^(1/3)*(-2*(x^2+1)^(2/3)+3*(x^2+1)^(5
/3))+1/2*3^(1/2)*arctan(3^(1/2)*(x^2+1)^(1/3)/(2*(x^2-1)^(1/3)-(x^2+1)^(1/3)))+1/18*arctan(3^(1/2)*(x^2+1)^(1/
3)/(2*(x^2-1)^(1/3)+(x^2+1)^(1/3)))*3^(1/2)+1/18*ln((x^2-1)^(1/3)-(x^2+1)^(1/3))-1/2*ln((x^2-1)^(1/3)+(x^2+1)^
(1/3))+1/4*ln((x^2-1)^(2/3)-(x^2-1)^(1/3)*(x^2+1)^(1/3)+(x^2+1)^(2/3))-1/36*ln((x^2-1)^(2/3)+(x^2-1)^(1/3)*(x^
2+1)^(1/3)+(x^2+1)^(2/3)))/((x^2-1)*(x^2+1)^2)^(2/3)

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Rubi [A]
time = 0.38, antiderivative size = 412, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6851, 103, 161, 93, 81, 61} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{x^6+x^4-x^2-1} \text {ArcTan}\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]{x^6+x^4-x^2-1} \text {ArcTan}\left (\frac {2 \sqrt [3]{x^2+1}}{\sqrt {3} \sqrt [3]{x^2-1}}+\frac {1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [3]{x^2-1} \left (x^2+1\right )^{2/3}}+\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}} \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))
+ ((-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))])/(6*Sqrt[3
]*(-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 61

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt
[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*
((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] /; FreeQ[{a, b, c, d}, x] && Ne
Q[b*c - a*d, 0] && PosQ[d/b]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 93

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 103

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 161

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbol]
:> Dist[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/f^(m + n + 2)), Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x
], x] + Dist[1/f^(m + n + 2), Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 2)*(c + d*x)^(m + n
+ 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 6851

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

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Mathematica [A]
time = 0.93, size = 306, normalized size = 0.93 \begin {gather*} \frac {\left (-1+x^2\right )^{2/3} \left (1+x^2\right )^{4/3} \left (21 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}+9 x^2 \sqrt [3]{-1+x^2} \left (1+x^2\right )^{2/3}+18 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{1+x^2}}{2 \sqrt [3]{-1+x^2}-\sqrt [3]{1+x^2}}\right )+2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{1+x^2}}{2 \sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}}\right )+2 \log \left (\sqrt [3]{-1+x^2}-\sqrt [3]{1+x^2}\right )-18 \log \left (\sqrt [3]{-1+x^2}+\sqrt [3]{1+x^2}\right )+9 \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 )-\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 )}{36 \left (\left (-1+x^2\right ) \left (1+x^2\right )^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((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)*(21*(-1 + x^2)^(1/3)*(1 + x^2)^(2/3) + 9*x^2*(-1 + x^2)^(1/3)*(1 + x^2)^(2/3
) + 18*Sqrt[3]*ArcTan[(Sqrt[3]*(1 + x^2)^(1/3))/(2*(-1 + x^2)^(1/3) - (1 + x^2)^(1/3))] + 2*Sqrt[3]*ArcTan[(Sq
rt[3]*(1 + x^2)^(1/3))/(2*(-1 + x^2)^(1/3) + (1 + x^2)^(1/3))] + 2*Log[(-1 + x^2)^(1/3) - (1 + x^2)^(1/3)] - 1
8*Log[(-1 + x^2)^(1/3) + (1 + x^2)^(1/3)] + 9*Log[(-1 + x^2)^(2/3) - (-1 + x^2)^(1/3)*(1 + x^2)^(1/3) + (1 + x
^2)^(2/3)] - 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))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 71.96, size = 6693, normalized size = 20.28

method	result	size
trager	\(\text {Expression too large to display}\)	\(6693\)

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]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)

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Fricas [A]
time = 0.51, 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))

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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 ) \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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)

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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)

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