∫ x2 ___________________________(1+x4 ) 4√____

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

Optimal. Leaf size=452 \[ -\frac {1}{16} \sqrt {\sqrt {2}-1} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} \sqrt [4]{x^6-x^2} x-\sqrt {2} \sqrt {x^6-x^2}\right )+\frac {1}{16} \sqrt {\sqrt {2}-1} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} \sqrt [4]{x^6-x^2} x+\sqrt {4-2 \sqrt {2}} \sqrt {x^6-x^2}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x}{2^{3/4} \sqrt [4]{x^6-x^2}-\sqrt {2+\sqrt {2}} x}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x}{2^{3/4} \sqrt [4]{x^6-x^2}+\sqrt {2+\sqrt {2}} x}\right )+\frac {1}{8} \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{x^6-x^2}}{\sqrt {2} \sqrt {x^6-x^2}-2 x^2}\right )-\frac {1}{8} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {x^6-x^2}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{x^6-x^2}}\right ) \]

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Rubi [C] time = 0.16, antiderivative size = 50, normalized size of antiderivative = 0.11, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2042, 466, 511, 510} \begin {gather*} \frac {2 x^3 \sqrt [4]{1-x^4} F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};x^4,-x^4\right )}{5 \sqrt [4]{x^6-x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*x^3*(1 - x^4)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^4, -x^4])/(5*(-x^2 + x^6)^(1/4))

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 510

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

*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free

Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a

, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

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

rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x

] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && !(IntegerQ[

p] || GtQ[a, 0])

Rule 2042

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol]

:> Dist[(e^IntPart[m]*(e*x)^FracPart[m]*(a*x^j + b*x^(j + n))^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a

+ b*x^n)^FracPart[p]), Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n,

p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])

Rubi steps

\begin {align*} \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{3/2}}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx}{\sqrt [4]{-x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1+x^8} \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{1-x^8} \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^6}}\\ &=\frac {2 x^3 \sqrt [4]{1-x^4} F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};x^4,-x^4\right )}{5 \sqrt [4]{-x^2+x^6}}\\ \end {align*}

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Mathematica [C] time = 0.02, size = 50, normalized size = 0.11 \begin {gather*} \frac {2 x^3 \sqrt [4]{1-x^4} F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};x^4,-x^4\right )}{5 \sqrt [4]{x^2 \left (x^4-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*x^3*(1 - x^4)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^4, -x^4])/(5*(x^2*(-1 + x^4))^(1/4))

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IntegrateAlgebraic [C] time = 0.00, size = 153, normalized size = 0.34 \begin {gather*} -\frac {1}{4} \sqrt {\frac {1}{2}-\frac {i}{2}} \tan ^{-1}\left (\frac {\sqrt {-1-i} x}{\sqrt [4]{-x^2+x^6}}\right )-\frac {1}{4} \sqrt {\frac {1}{2}+\frac {i}{2}} \tan ^{-1}\left (\frac {\sqrt {-1+i} x}{\sqrt [4]{-x^2+x^6}}\right )+\frac {1}{4} \sqrt {-\frac {1}{2}-\frac {i}{2}} \tan ^{-1}\left (\frac {\sqrt {1-i} x}{\sqrt [4]{-x^2+x^6}}\right )+\frac {1}{4} \sqrt {-\frac {1}{2}+\frac {i}{2}} \tan ^{-1}\left (\frac {\sqrt {1+i} x}{\sqrt [4]{-x^2+x^6}}\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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

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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C] time = 113.30, size = 2931, normalized size = 6.48


method	result	size

trager	\(\text {Expression too large to display}\)	\(2931\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*RootOf(32*_Z^4-8*_Z^2+1)*ln(-(-230912*RootOf(32*_Z^4-8*_Z^2+1)^5*x^5+865920*RootOf(32*_Z^4-8*_Z^2+1)^5*x^

3+27672*RootOf(32*_Z^4-8*_Z^2+1)^3*x^5+208080*(x^6-x^2)^(1/2)*RootOf(32*_Z^4-8*_Z^2+1)^3*x+230912*RootOf(32*_Z

^4-8*_Z^2+1)^5*x-364448*RootOf(32*_Z^4-8*_Z^2+1)^3*x^3+5789*RootOf(32*_Z^4-8*_Z^2+1)*x^5+73984*(x^6-x^2)^(3/4)

*RootOf(32*_Z^4-8*_Z^2+1)^2-43928*(x^6-x^2)^(1/4)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^2-36992*(x^6-x^2)^(1/2)*RootOf(

32*_Z^4-8*_Z^2+1)*x-27672*RootOf(32*_Z^4-8*_Z^2+1)^3*x+38042*RootOf(32*_Z^4-8*_Z^2+1)*x^3-5491*(x^6-x^2)^(3/4)

-7514*(x^6-x^2)^(1/4)*x^2-5789*RootOf(32*_Z^4-8*_Z^2+1)*x)/(8*RootOf(32*_Z^4-8*_Z^2+1)^2*x^2-32*RootOf(32*_Z^4

-8*_Z^2+1)^2-5*x^2+3)^2/x)-1/8*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*ln((230912*RootOf(_Z^2+4*RootOf(32*

_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^4*x^5-865920*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(3

2*_Z^4-8*_Z^2+1)^4*x^3-87784*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^5+208080

*(x^6-x^2)^(1/2)*RootOf(32*_Z^4-8*_Z^2+1)^2*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x-230912*RootOf(_Z^2+4

*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^4*x+68512*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*

RootOf(32*_Z^4-8*_Z^2+1)^2*x^3+1725*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x^5+147968*(x^6-x^2)^(3/4)*Roo

tOf(32*_Z^4-8*_Z^2+1)^2-87856*(x^6-x^2)^(1/4)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^2-15028*(x^6-x^2)^(1/2)*RootOf(_Z^2

+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x+87784*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^

2*x-1050*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x^3-26010*(x^6-x^2)^(3/4)+36992*(x^6-x^2)^(1/4)*x^2-1725*

RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x)/(8*RootOf(32*_Z^4-8*_Z^2+1)^2*x^2-32*RootOf(32*_Z^4-8*_Z^2+1)^2

+3*x^2+5)^2/x)-ln(-(20168550173184*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^4*x^5-

75632063149440*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^4*x^3-5372907862360*RootOf

(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^5-22189554285584*(x^6-x^2)^(1/2)*RootOf(32*

_Z^4-8*_Z^2+1)^2*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x-20168550173184*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_

Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^4*x-2619997860096*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_

Z^4-8*_Z^2+1)^2*x^3-673887370171*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x^5+661540638128*(x^6-x^2)^(3/4)*

RootOf(32*_Z^4-8*_Z^2+1)^2+22189554285584*(x^6-x^2)^(1/4)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^2+165385159532*(x^6-x^2

)^(1/2)*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x+5372907862360*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1

)*RootOf(32*_Z^4-8*_Z^2+1)^2*x+410192312278*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x^3+2608309126166*(x^6

-x^2)^(3/4)-165385159532*(x^6-x^2)^(1/4)*x^2+673887370171*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x)/(8*Ro

otOf(32*_Z^4-8*_Z^2+1)^2*x^2-32*RootOf(32*_Z^4-8*_Z^2+1)^2+3*x^2+5)^2/x)*RootOf(32*_Z^4-8*_Z^2+1)^2*RootOf(_Z^

2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)+1/8*ln(-(20168550173184*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(3

2*_Z^4-8*_Z^2+1)^4*x^5-75632063149440*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^4*x

^3-5372907862360*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^5-22189554285584*(x^

6-x^2)^(1/2)*RootOf(32*_Z^4-8*_Z^2+1)^2*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x-20168550173184*RootOf(_Z

^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^4*x-2619997860096*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_

Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^3-673887370171*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x^5+661540

638128*(x^6-x^2)^(3/4)*RootOf(32*_Z^4-8*_Z^2+1)^2+22189554285584*(x^6-x^2)^(1/4)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^

2+165385159532*(x^6-x^2)^(1/2)*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)*x+5372907862360*RootOf(_Z^2+4*RootO

f(32*_Z^4-8*_Z^2+1)^2-1)*RootOf(32*_Z^4-8*_Z^2+1)^2*x+410192312278*RootOf(_Z^2+4*RootOf(32*_Z^4-8*_Z^2+1)^2-1)

*x^3+2608309126166*(x^6-x^2)^(3/4)-165385159532*(x^6-x^2)^(1/4)*x^2+673887370171*RootOf(_Z^2+4*RootOf(32*_Z^4-

8*_Z^2+1)^2-1)*x)/(8*RootOf(32*_Z^4-8*_Z^2+1)^2*x^2-32*RootOf(32*_Z^4-8*_Z^2+1)^2+3*x^2+5)^2/x)*RootOf(_Z^2+4*

RootOf(32*_Z^4-8*_Z^2+1)^2-1)+2*RootOf(32*_Z^4-8*_Z^2+1)^3*ln(-(-192512*RootOf(32*_Z^4-8*_Z^2+1)^5*x^5+721920*

RootOf(32*_Z^4-8*_Z^2+1)^5*x^3-25856*RootOf(32*_Z^4-8*_Z^2+1)^3*x^5-87856*(x^6-x^2)^(1/2)*RootOf(32*_Z^4-8*_Z^

2+1)^3*x+192512*RootOf(32*_Z^4-8*_Z^2+1)^5*x-120368*RootOf(32*_Z^4-8*_Z^2+1)^3*x^3-525*RootOf(32*_Z^4-8*_Z^2+1

)*x^5+73984*(x^6-x^2)^(3/4)*RootOf(32*_Z^4-8*_Z^2+1)^2+43928*(x^6-x^2)^(1/4)*RootOf(32*_Z^4-8*_Z^2+1)^2*x^2-15

028*(x^6-x^2)^(1/2)*RootOf(32*_Z^4-8*_Z^2+1)*x+25856*RootOf(32*_Z^4-8*_Z^2+1)^3*x-3450*RootOf(32*_Z^4-8*_Z^2+1

)*x^3-5491*(x^6-x^2)^(3/4)+7514*(x^6-x^2)^(1/4)*x^2+525*RootOf(32*_Z^4-8*_Z^2+1)*x)/(8*RootOf(32*_Z^4-8*_Z^2+1

)^2*x^2-32*RootOf(32*_Z^4-8*_Z^2+1)^2-5*x^2+3)^2/x)-1/4*RootOf(32*_Z^4-8*_Z^2+1)*ln(-(-192512*RootOf(32*_Z^4-8

*_Z^2+1)^5*x^5+721920*RootOf(32*_Z^4-8*_Z^2+1)^5*x^3-25856*RootOf(32*_Z^4-8*_Z^2+1)^3*x^5-87856*(x^6-x^2)^(1/2

)*RootOf(32*_Z^4-8*_Z^2+1)^3*x+192512*RootOf(32*_Z^4-8*_Z^2+1)^5*x-120368*RootOf(32*_Z^4-8*_Z^2+1)^3*x^3-525*R

ootOf(32*_Z^4-8*_Z^2+1)*x^5+73984*(x^6-x^2)^(3/4)*RootOf(32*_Z^4-8*_Z^2+1)^2+43928*(x^6-x^2)^(1/4)*RootOf(32*_

Z^4-8*_Z^2+1)^2*x^2-15028*(x^6-x^2)^(1/2)*RootOf(32*_Z^4-8*_Z^2+1)*x+25856*RootOf(32*_Z^4-8*_Z^2+1)^3*x-3450*R

ootOf(32*_Z^4-8*_Z^2+1)*x^3-5491*(x^6-x^2)^(3/4)+7514*(x^6-x^2)^(1/4)*x^2+525*RootOf(32*_Z^4-8*_Z^2+1)*x)/(8*R

ootOf(32*_Z^4-8*_Z^2+1)^2*x^2-32*RootOf(32*_Z^4-8*_Z^2+1)^2-5*x^2+3)^2/x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/((x**2*(x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(x**4 + 1)), x)

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