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

Optimal. Leaf size=167 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}-x}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{2 \sqrt [3]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^4+x^2}}\right )}{\sqrt [3]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^4+x^2\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^4+x^2}}\right )}{2 \sqrt [3]{2}} \]

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Rubi [C]  time = 0.10, antiderivative size = 42, normalized size of antiderivative = 0.25, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2056, 466, 429} \begin {gather*} -\frac {3 x \sqrt [3]{x^2+1} F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};x^2,-x^2\right )}{\sqrt [3]{x^4+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
 -q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 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 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]

Rubi steps

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

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Mathematica [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.52, size = 167, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}-x}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{2 \sqrt [3]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^4+x^2}}\right )}{\sqrt [3]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^4+x^2\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^4+x^2}}\right )}{2 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.11, size = 232, normalized size = 1.39 \begin {gather*} -\frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{5} + 8 \, x^{4} - 2 \, x^{3} + 8 \, x^{2} + x\right )} + 8 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 2 \, x^{2} + x\right )} + 8 \cdot 2^{\frac {1}{6}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} - 2 \, x + 1\right )}\right )}}{6 \, {\left (x^{5} - 8 \, x^{4} - 2 \, x^{3} - 8 \, x^{2} + x\right )}}\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left (-\frac {2^{\frac {2}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} + 4 \cdot 2^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + 2 \, x^{2} + x}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \log \left (\frac {2 \cdot 2^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{3} + 2 \, x^{2} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}{x^{3} + 2 \, x^{2} + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^5 + 8*x^4 - 2*x^3 + 8*x^2 + x) + 8*sqrt(2)*(x^4 +
x^2)^(1/3)*(x^3 + 2*x^2 + x) + 8*2^(1/6)*(x^4 + x^2)^(2/3)*(x^2 - 2*x + 1))/(x^5 - 8*x^4 - 2*x^3 - 8*x^2 + x))
 + 1/4*2^(2/3)*log(-(2^(2/3)*(x^3 - 2*x^2 + x) + 4*2^(1/3)*(x^4 + x^2)^(1/3)*x - 4*(x^4 + x^2)^(2/3))/(x^3 + 2
*x^2 + x)) - 1/8*2^(2/3)*log((2*2^(2/3)*(x^4 + x^2)^(2/3) + 2^(1/3)*(x^3 + 2*x^2 + x) + 4*(x^4 + x^2)^(1/3)*x)
/(x^3 + 2*x^2 + x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 25.52, size = 2591, normalized size = 15.51 \begin {gather*} \text {Expression too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*ln((1106640125761716*x^2*(x^4+x^2)^(2/3)+1721440195629336*
(x^4+x^2)^(2/3)*x+1106640125761716*(x^4+x^2)^(2/3)+41772375754200*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z
^2)^2*RootOf(_Z^3-4)^2*x^2-36525817072836*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2
+122536959111840*RootOf(_Z^3-4)^4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(2/3)*x^2-214431
5288715600*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^4+206541191229100*RootOf(RootOf(_Z^3-4)^2+_Z*Root
Of(_Z^3-4)+_Z^2)*x^3-103270595614550*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^5-2144315288715600*Root
Of(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^2-103270595614550*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2
)*x+1874991943072248*RootOf(_Z^3-4)*x^4-180599873304578*RootOf(_Z^3-4)*x^3+1874991943072248*RootOf(_Z^3-4)*x^2
+90299936652289*RootOf(_Z^3-4)*x-46413750838000*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-
4)^2*x+40584241192040*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x+90299936652289*RootOf
(_Z^3-4)*x^5+92827501676000*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^3-81168482384
080*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^3-46413750838000*RootOf(RootOf(_Z^3-4)^
2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^5+40584241192040*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2
)*RootOf(_Z^3-4)^3*x^5+41772375754200*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^4-3
6525817072836*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^4+122536959111840*RootOf(_Z^3
-4)^4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(2/3)-2085103979818662*RootOf(_Z^3-4)^2*(x^4
+x^2)^(1/3)*x^3+797336343950328*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(2/
3)+219484629454596*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x^2-2085103979818662*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x-30
6342397779600*RootOf(_Z^3-4)^4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(2/3)*x-21948462945
4596*RootOf(_Z^3-4)^4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*x^3-107343584948232*Root
Of(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*x^3+548711573636490*RootOf(_Z^3
-4)^4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*x^2+268358962370580*RootOf(_Z^3-4)^3*Roo
tOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*x^2-219484629454596*RootOf(_Z^3-4)^4*RootOf(Roo
tOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*x-107343584948232*RootOf(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)
^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*x+797336343950328*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+_Z*Roo
tOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(2/3)*x^2+242958643915260*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_
Z^3-4)^2*(x^4+x^2)^(2/3)*x-1019764057008204*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^
4+x^2)^(1/3)*x^3+107343584948232*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3
)*x^2-1019764057008204*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*(x^4+x^2)^(1/3)*x)/(1+x)
^2/(-1+x)^2/x)+1/4*RootOf(_Z^3-4)*ln((-122113904250156*x^2*(x^4+x^2)^(2/3)-4151872744505304*(x^4+x^2)^(2/3)*x-
122113904250156*(x^4+x^2)^(2/3)+36525817072836*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4
)^2*x^2-41772375754200*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2+122536959111840*Ro
otOf(_Z^3-4)^4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(2/3)*x^2+2021095211363592*RootOf(R
ootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^4+144074056231742*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^
3-72037028115871*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^5+2021095211363592*RootOf(RootOf(_Z^3-4)^2+
_Z*RootOf(_Z^3-4)+_Z^2)*x^2-72037028115871*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x-2311404791732400*
RootOf(_Z^3-4)*x^4-164768815474900*RootOf(_Z^3-4)*x^3-2311404791732400*RootOf(_Z^3-4)*x^2+82384407737450*RootO
f(_Z^3-4)*x-40584241192040*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+46413750838000
*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x+82384407737450*RootOf(_Z^3-4)*x^5+81168482
384080*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^3-92827501676000*RootOf(RootOf(_Z^
3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^3-40584241192040*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_
Z^2)^2*RootOf(_Z^3-4)^2*x^5+46413750838000*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^
5+36525817072836*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^4-41772375754200*RootOf(
RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^4+122536959111840*RootOf(_Z^3-4)^4*RootOf(RootOf(_
Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(2/3)+1207165462000278*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x^3+18295
9328944392*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(2/3)+1975361665091364*R
ootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x^2+1207165462000278*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x-306342397779600*RootOf
(_Z^3-4)^4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(2/3)*x-219484629454596*RootOf(_Z^3-4)^
4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*x^3-112141044506364*RootOf(_Z^3-4)^3*RootOf(
RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*x^3+548711573636490*RootOf(_Z^3-4)^4*RootOf(RootOf(
_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*x^2+280352611265910*RootOf(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)^2
+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*x^2-219484629454596*RootOf(_Z^3-4)^4*RootOf(RootOf(_Z^3-4)^2+_Z*Roo
tOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*x-112141044506364*RootOf(_Z^3-4)^3*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)
+_Z^2)^2*(x^4+x^2)^(1/3)*x+182959328944392*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x
^4+x^2)^(2/3)*x^2-2693697826152060*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^
(2/3)*x+616775744785002*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*x^3+100
9269400557276*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*x^2+6167757447850
02*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*(x^4+x^2)^(1/3)*x)/(1+x)^2/(-1+x)^2/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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