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

Optimal. Leaf size=162 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{4 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{4 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{4\ 2^{3/4}} \]

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Rubi [C]  time = 0.13, antiderivative size = 46, normalized size of antiderivative = 0.28, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2042, 466, 510} \begin {gather*} -\frac {2 x^3 \sqrt [4]{x^4+1} F_1\left (\frac {5}{8};1,\frac {1}{4};\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, 1/4, 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 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}}{\left (-1+x^4\right ) \sqrt [4]{1+x^4}} \, 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}{\left (-1+x^8\right ) \sqrt [4]{1+x^8}} \, 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};1,\frac {1}{4};\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 = 44, normalized size = 0.27 \begin {gather*} -\frac {2 x \left (x^6+x^2\right )^{3/4} F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,x^4\right )}{5 \left (x^4+1\right )^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 162, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{4\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 14.00, size = 712, normalized size = 4.40 \begin {gather*} \frac {1}{8} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{2 \, {\left (x^{5} + x\right )}}\right ) - \frac {1}{32} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + \frac {1}{32} \cdot 2^{\frac {3}{4}} \log \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{5} + 2 \, x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + 2 \, x^{3} + x}} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{5} - 2 \, x^{3} + x\right )}}\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{5} + 2 \, x^{3} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x - 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + 2 \, x^{3} + x}} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{5} - 2 \, x^{3} + x\right )}}\right ) + \frac {1}{32} \cdot 2^{\frac {1}{4}} \log \left (\frac {2 \, {\left (x^{5} + 2 \, x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + 2 \, x^{3} + x}\right ) - \frac {1}{32} \cdot 2^{\frac {1}{4}} \log \left (\frac {2 \, {\left (x^{5} + 2 \, x^{3} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x - 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + 2 \, x^{3} + x}\right ) \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]

1/8*2^(3/4)*arctan(1/2*2^(3/4)*(x^6 + x^2)^(1/4)*(x^4 + 1)/(x^5 + x)) - 1/32*2^(3/4)*log((4*2^(1/4)*(x^6 + x^2
)^(1/4)*x^2 + 2*2^(3/4)*(x^6 + x^2)^(3/4) + sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)
) + 1/32*2^(3/4)*log(-(4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 +
x) - 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) + 1/8*2^(1/4)*arctan(1/2*(4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(
2)*(2*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 - sqrt(2)*(x^5 + 2*x^3 + x) - 4*sqrt(x^6 + x^2)*x + 2*2^(1/4)*(x^6 + x^2)^
(3/4))*sqrt((x^5 + 2*x^3 + 4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^6 + x^2)*x + 2*2^(3/4)*(x^6 + x^
2)^(3/4) + x)/(x^5 + 2*x^3 + x)) + 2*2^(3/4)*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) + 1/8*2^(1/4)*arctan(1/2*(4
*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(2*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt
(x^6 + x^2)*x + 2*2^(1/4)*(x^6 + x^2)^(3/4))*sqrt((x^5 + 2*x^3 - 4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(2)*s
qrt(x^6 + x^2)*x - 2*2^(3/4)*(x^6 + x^2)^(3/4) + x)/(x^5 + 2*x^3 + x)) + 2*2^(3/4)*(x^6 + x^2)^(3/4))/(x^5 - 2
*x^3 + x)) + 1/32*2^(1/4)*log(2*(x^5 + 2*x^3 + 4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^6 + x^2)*x +
2*2^(3/4)*(x^6 + x^2)^(3/4) + x)/(x^5 + 2*x^3 + x)) - 1/32*2^(1/4)*log(2*(x^5 + 2*x^3 - 4*2^(1/4)*(x^6 + x^2)
^(1/4)*x^2 + 4*sqrt(2)*sqrt(x^6 + x^2)*x - 2*2^(3/4)*(x^6 + x^2)^(3/4) + x)/(x^5 + 2*x^3 + x))

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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 = 21.64, size = 639, normalized size = 3.94

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+8\right ) x}{x \left (x^{2}+1\right )^{2}}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x}{x \left (x^{2}+1\right )^{2}}\right )}{16}-\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{3}}{32}-\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{32}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{5}+2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+8 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}-8 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x -8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x +16 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{32}\)	\(639\)

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]

1/16*RootOf(_Z^4+8)*ln(-(-RootOf(_Z^4+8)^3*(x^6+x^2)^(1/2)*x+RootOf(_Z^4+8)*x^5-2*(x^6+x^2)^(1/4)*RootOf(_Z^4+
8)^2*x^2-2*RootOf(_Z^4+8)*x^3+4*(x^6+x^2)^(3/4)+RootOf(_Z^4+8)*x)/x/(x^2+1)^2)-1/16*RootOf(_Z^2+RootOf(_Z^4+8)
^2)*ln(-(-RootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)^2*(x^6+x^2)^(1/2)*x-RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^5+2
*(x^6+x^2)^(1/4)*RootOf(_Z^4+8)^2*x^2+2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^3+4*(x^6+x^2)^(3/4)-RootOf(_Z^2+RootOf
(_Z^4+8)^2)*x)/x/(x^2+1)^2)-1/32*ln(-(RootOf(_Z^4+8)^2*x^2-2*RootOf(_Z^4+8)*(x^6+x^2)^(1/4)*x+2*(x^6+x^2)^(1/2
))/x/(-1+x)/(1+x))*RootOf(_Z^4+8)^3-1/32*ln(-(RootOf(_Z^4+8)^2*x^2-2*RootOf(_Z^4+8)*(x^6+x^2)^(1/4)*x+2*(x^6+x
^2)^(1/2))/x/(-1+x)/(1+x))*RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)+1/32*RootOf(_Z^4+8)^2*RootOf(_Z^2+Ro
otOf(_Z^4+8)^2)*ln((RootOf(_Z^4+8)^3*x^5-RootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)^2*x^5+2*RootOf(_Z^4+8)^3
*x^3-2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)^2*x^3+8*(x^6+x^2)^(1/4)*RootOf(_Z^4+8)*RootOf(_Z^2+RootOf(
_Z^4+8)^2)*x^2-8*(x^6+x^2)^(1/2)*RootOf(_Z^4+8)*x-8*RootOf(_Z^2+RootOf(_Z^4+8)^2)*(x^6+x^2)^(1/2)*x+RootOf(_Z^
4+8)^3*x-RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x+16*(x^6+x^2)^(3/4))/(-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}}{{\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 = -0.01 \begin {gather*} \int \frac {x^2}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/((x^2 + x^6)^(1/4)*(x^4 - 1)), 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^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 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**4 + 1))**(1/4)*(x - 1)*(x + 1)*(x**2 + 1)), x)

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