∫ 1+x2 ___________________________(−1+x2 ) 3√___

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

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 = 0.53, size = 167, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^2+x^4}}\right )}{\sqrt [3]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^2+x^4\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^2+x^4}}\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 = 1.75, 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 = 27.20, size = 3893, normalized size = 23.31


method	result	size

trager	\(\text {Expression too large to display}\)	\(3893\)

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,method=_RETURNVERBOSE)

[Out]

-1/4*ln((103270595614550*RootOf(_Z^3-4)*x^5+4019307231787848*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x

^2+193570532266839*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x+2144315288715600*RootOf(_Z^3-4)*x^4-20654

1191229100*RootOf(_Z^3-4)*x^3+2144315288715600*RootOf(_Z^3-4)*x^2+103270595614550*RootOf(_Z^3-4)*x+11066401257

61716*(x^4+x^2)^(2/3)+4019307231787848*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^4-387141064533678*Roo

tOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^3+193570532266839*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^

2)*x^5+86997992030040*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+46413750838000*Root

Of(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x-78298192827036*RootOf(RootOf(_Z^3-4)^2+_Z*RootO

f(_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^4-173995984060080*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^3-9282750167

6000*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^3+112141044506364*RootOf(RootOf(_Z^3-4

)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^4*x^3-78298192827036*RootOf(RootOf(_Z^3-4)^2+_Z*Roo

tOf(_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^2-548711573636490*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*RootOf(_Z^3-4

)^3*x^2-280352611265910*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^4*x^2+2

19484629454596*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^3*(x^4+x^2)^(1/3)*x+7973363439

50328*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(2/3)*RootOf(_Z^3-4)^2*x^2+112141044506364*Roo

tOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^4*(x^4+x^2)^(1/3)*x+242958643915260*(x^4+x^2)^(2/3

)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2*x+2085103979818662*RootOf(RootOf(_Z^3-4)^2+

_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*x^3-219484629454596*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_

Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*x^2+2085103979818662*(x^4+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+_Z*Ro

otOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*x+122536959111840*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2

)^(2/3)*RootOf(_Z^3-4)^4*x^2-306342397779600*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^

4*(x^4+x^2)^(2/3)*x+219484629454596*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*RootOf(_

Z^3-4)^3*x^3+86997992030040*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^5+46413750838

000*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^5+1065339922810458*(x^4+x^2)^(1/3)*Root

Of(_Z^3-4)^2*x+122536959111840*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^4*(x^4+x^2)^(2

/3)+1065339922810458*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x^3+797336343950328*(x^4+x^2)^(2/3)*RootOf(RootOf(_Z^3-4

)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2-112141044506364*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x^2+172144019562

9336*(x^4+x^2)^(2/3)*x+1106640125761716*(x^4+x^2)^(2/3)*x^2)/(-1+x)^2/(1+x)^2/x)*RootOf(RootOf(_Z^3-4)^2+_Z*Ro

otOf(_Z^3-4)+_Z^2)-1/4*ln((103270595614550*RootOf(_Z^3-4)*x^5+4019307231787848*RootOf(RootOf(_Z^3-4)^2+_Z*Root

Of(_Z^3-4)+_Z^2)*x^2+193570532266839*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x+2144315288715600*RootOf

(_Z^3-4)*x^4-206541191229100*RootOf(_Z^3-4)*x^3+2144315288715600*RootOf(_Z^3-4)*x^2+103270595614550*RootOf(_Z^

3-4)*x+1106640125761716*(x^4+x^2)^(2/3)+4019307231787848*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^4-3

87141064533678*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^3+193570532266839*RootOf(RootOf(_Z^3-4)^2+_Z*

RootOf(_Z^3-4)+_Z^2)*x^5+86997992030040*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+4

6413750838000*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x-78298192827036*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^4-173995984060080*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+112141044506364*Ro

otOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^4*x^3-78298192827036*RootOf(RootO

f(_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^2-548711573636490*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(

1/3)*RootOf(_Z^3-4)^3*x^2-280352611265910*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*Root

Of(_Z^3-4)^4*x^2+219484629454596*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^3*(x^4+x^2)^

(1/3)*x+797336343950328*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(2/3)*RootOf(_Z^3-4)^2*x^2+1

12141044506364*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^4*(x^4+x^2)^(1/3)*x+242958643915

260*(x^4+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2*x+2085103979818662*RootOf

(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*x^3-219484629454596*RootOf(RootOf(_Z^

3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*x^2+2085103979818662*(x^4+x^2)^(1/3)*RootOf(Root

Of(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*x+122536959111840*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4

)+_Z^2)^2*(x^4+x^2)^(2/3)*RootOf(_Z^3-4)^4*x^2-306342397779600*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)

^2*RootOf(_Z^3-4)^4*(x^4+x^2)^(2/3)*x+219484629454596*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x

^2)^(1/3)*RootOf(_Z^3-4)^3*x^3+86997992030040*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+1065339922810458*(x

^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x+122536959111840*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3

-4)^4*(x^4+x^2)^(2/3)+1065339922810458*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x^3+797336343950328*(x^4+x^2)^(2/3)*Ro

otOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2-112141044506364*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^

2*x^2+1721440195629336*(x^4+x^2)^(2/3)*x+1106640125761716*(x^4+x^2)^(2/3)*x^2)/(-1+x)^2/(1+x)^2/x)*RootOf(_Z^3

-4)+1/4*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*ln((72037028115871*RootOf(_Z^3-4)*x^5-4332500003095992

*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^2+154421435853321*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)

+_Z^2)*x-2021095211363592*RootOf(_Z^3-4)*x^4-144074056231742*RootOf(_Z^3-4)*x^3-2021095211363592*RootOf(_Z^3-4

)*x^2+72037028115871*RootOf(_Z^3-4)*x-122113904250156*(x^4+x^2)^(2/3)-4332500003095992*RootOf(RootOf(_Z^3-4)^2

+_Z*RootOf(_Z^3-4)+_Z^2)*x^4-308842871706642*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^3+1544214358533

21*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*x^5+86997992030040*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

-78298192827036*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^4-36525817072836*RootOf(R

ootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^4-173995984060080*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf

(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^3-81168482384080*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z

^3-4)^3*x^3+107343584948232*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^4*x

^3-78298192827036*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^2-548711573636490*RootOf(RootOf(_Z^3-4)^2+_Z*Root

Of(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^3*x^2-268358962370580*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3

-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^4*x^2+219484629454596*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)

^2*RootOf(_Z^3-4)^3*(x^4+x^2)^(1/3)*x+182959328944392*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2

)^(2/3)*RootOf(_Z^3-4)^2*x^2+107343584948232*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^4*

(x^4+x^2)^(1/3)*x-2693697826152060*(x^4+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3

-4)^2*x-1207165462000278*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*x^3-19

75361665091364*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*x^2-120716546200

0278*(x^4+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*x+122536959111840*RootOf(R

ootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(2/3)*RootOf(_Z^3-4)^4*x^2-306342397779600*RootOf(RootOf(_

Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^4*(x^4+x^2)^(2/3)*x+219484629454596*RootOf(RootOf(_Z^3-4)^2+

_Z*RootOf(_Z^3-4)+_Z^2)^2*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^3*x^3+86997992030040*RootOf(RootOf(_Z^3-4)^2+_Z*RootO

f(_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^5-590389717215276*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x+122536959111840*RootOf(RootOf(_Z^3-4)^2+_Z*Roo

tOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^4*(x^4+x^2)^(2/3)-590389717215276*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x^3+1829

59328944392*(x^4+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2-966092264534088*(

x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x^2-4151872744505304*(x^4+x^2)^(2/3)*x-122113904250156*(x^4+x^2)^(2/3)*x^2)/(-

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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