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

Optimal. Leaf size=62 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\& ,\frac {\text {$\#$1}^3 \log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-\text {$\#$1}^3 \log (x)}{2 \text {$\#$1}^4-3}\& \right ] \]

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Rubi [C]  time = 0.51, antiderivative size = 477, normalized size of antiderivative = 7.69, number of steps used = 21, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2056, 1269, 1428, 408, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} -\frac {\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \left (\sqrt {3}+i\right ) \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\left (-\sqrt {3}+i\right ) \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \left (\sqrt {3}+i\right ) \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}}-\frac {\left (-\sqrt {3}+i\right ) \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{\left (\sqrt {3}+i\right ) \sqrt [4]{x^4+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 377

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

Rule 408

Int[((a_) + (b_.)*(x_)^4)^(p_)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[(a + b*x^4)^(p - 1), x], x] -
Dist[(b*c - a*d)/d, Int[(a + b*x^4)^(p - 1)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0
] && (EqQ[p, 3/4] || EqQ[p, 5/4])

Rule 1269

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With
[{k = Denominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + (e*x^(2*k))/f^2)^q*(a + (b*x^(2*k))/f^k + (c
*x^(4*k))/f^4)^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && Fra
ctionQ[m] && IntegerQ[p]

Rule 1428

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -
4*a*c, 2]}, Dist[(2*c)/r, Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[(2*c)/r, Int[(d + e*x^n)^q/(b +
r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] &&  !IntegerQ[q]

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+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {\left (1+x^2\right )^{3/4}}{\sqrt {x} \left (1-x^2+x^4\right )} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{1-x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (4 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{-1-i \sqrt {3}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}+\frac {\left (4 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^{3/4}}{-1+i \sqrt {3}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (2 i \left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}-\frac {\left (2 i \left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-1-i \sqrt {3}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (2 i \left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (-3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}-\frac {\left (2 i \left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (-3-i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i-\sqrt {3}}-\sqrt {3 i-\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i-\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i-\sqrt {3}}+\sqrt {3 i-\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i-\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i+\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}-\frac {\left (\left (-3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3 \left (i+\sqrt {3}\right )} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i-\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i+\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i-\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^2}}\right )}{\left (i+\sqrt {3}\right ) \sqrt [4]{x^2+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 72, normalized size = 1.16 \begin {gather*} -\frac {\left (x^4+x^2\right )^{3/4} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\&,\frac {\text {$\#$1}^3 \log \left (\sqrt [4]{\frac {1}{x^2}+1}-\text {$\#$1}\right )}{2 \text {$\#$1}^4-3}\&\right ]}{2 \left (\frac {1}{x^2}+1\right )^{3/4} x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((x^2 + x^4)^(3/4)*RootSum[3 - 3*#1^4 + #1^8 & , (Log[(1 + x^(-2))^(1/4) - #1]*#1^3)/(-3 + 2*#1^4) & ])/(
(1 + x^(-2))^(3/4)*x^3)

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IntegrateAlgebraic [A]  time = 0.20, size = 62, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*RootSum[3 - 3*#1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(x^2 + x^4)^(1/4) - x*#1]*#1^3)/(-3 + 2*#1^4) & ]

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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+1)/(x^4-x^2+1)/(x^4+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} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 43.74, size = 1629, normalized size = 26.27

method	result	size
trager	\(\text {Expression too large to display}\)	\(1629\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*ln(-(RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(
_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^8*x^3+4*RootOf(_Z^8+2187)^5*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*Root
Of(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*(x^4+x^2)^(1/2)*x-RootOf(_Z^2-RootOf(_Z^8+2187)*Ro
otOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^8*x-264*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^5
*(x^4+x^2)^(1/4)*x^2-558*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^4*x
^3+756*RootOf(_Z^8+2187)^4*(x^4+x^2)^(3/4)-4860*RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2
-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*(x^4+x^2)^(1/2)*x-144*RootOf(_Z^2-RootOf(_Z^8+2187)*RootO
f(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^4*x-7776*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)*(x^
4+x^2)^(1/4)*x^2+23085*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^3+67068*(x^4+x^2)^(3/
4)+8505*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x)/x/(x^2*RootOf(_Z^8+2187)^4-RootOf(_
Z^8+2187)^4+27*x^2+27))+1/6*RootOf(_Z^8+2187)*ln(-(RootOf(_Z^8+2187)^9*x^3-RootOf(_Z^8+2187)^9*x-4*(x^4+x^2)^(
1/2)*RootOf(_Z^8+2187)^7*x+264*RootOf(_Z^8+2187)^6*(x^4+x^2)^(1/4)*x^2+558*RootOf(_Z^8+2187)^5*x^3-756*RootOf(
_Z^8+2187)^4*(x^4+x^2)^(3/4)+144*RootOf(_Z^8+2187)^5*x-4860*(x^4+x^2)^(1/2)*RootOf(_Z^8+2187)^3*x-7776*RootOf(
_Z^8+2187)^2*(x^4+x^2)^(1/4)*x^2+23085*RootOf(_Z^8+2187)*x^3+67068*(x^4+x^2)^(3/4)+8505*RootOf(_Z^8+2187)*x)/(
x^2*RootOf(_Z^8+2187)^4-RootOf(_Z^8+2187)^4-27*x^2-27)/x)+1/6*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*ln(-(x^3*RootOf
(_Z^8+2187)^8*RootOf(_Z^2+RootOf(_Z^8+2187)^2)-RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^8*x+4*(x^4+x
^2)^(1/2)*RootOf(_Z^8+2187)^6*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x-264*RootOf(_Z^8+2187)^6*(x^4+x^2)^(1/4)*x^2+5
58*x^3*RootOf(_Z^8+2187)^4*RootOf(_Z^2+RootOf(_Z^8+2187)^2)-756*RootOf(_Z^8+2187)^4*(x^4+x^2)^(3/4)+144*RootOf
(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^4*x+4860*(x^4+x^2)^(1/2)*RootOf(_Z^8+2187)^2*RootOf(_Z^2+RootOf(_
Z^8+2187)^2)*x+7776*RootOf(_Z^8+2187)^2*(x^4+x^2)^(1/4)*x^2+23085*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x^3+67068*(
x^4+x^2)^(3/4)+8505*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x)/(x^2*RootOf(_Z^8+2187)^4-RootOf(_Z^8+2187)^4-27*x^2-27
)/x)-1/13122*RootOf(_Z^8+2187)^7*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+Ro
otOf(_Z^8+2187)^2))*ln(-(5*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_
Z^8+2187)^2))*RootOf(_Z^8+2187)^11*x^3-5*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^11*RootOf(_Z^2-Roo
tOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x-2322*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2-RootOf(_Z
^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^7*x^3-1188*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootO
f(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^7*x+972*RootOf(_Z^2-RootOf(_Z^8+2
187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^6*(x^4+x^2)^(1/2)*x+64152*RootOf(_Z^2+RootOf(_Z^8+218
7)^2)*RootOf(_Z^8+2187)^5*(x^4+x^2)^(1/4)*x^2-124659*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2-RootOf(_Z^8+
2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^3*x^3+183708*RootOf(_Z^8+2187)^4*(x^4+x^2)^(3/4)-459
27*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^
8+2187)^3*x-1180980*RootOf(_Z^2-RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^2*(x^4+x
^2)^(1/2)*x+1889568*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)*(x^4+x^2)^(1/4)*x^2+16297524*(x^4+x^2)^
(3/4))/x/(x^2*RootOf(_Z^8+2187)^4-RootOf(_Z^8+2187)^4+27*x^2+27))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 \, {\left (8 \, x^{5} - 7 \, {\left (x^{3} + x\right )} x^{2} + 9 \, x^{3} + x\right )}}{21 \, {\left (x^{\frac {9}{2}} - x^{\frac {5}{2}} + \sqrt {x}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} + \int -\frac {4 \, {\left (16 \, x^{4} - 8 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} x^{2} + 11 \, x^{2} - 5\right )}}{21 \, {\left (x^{\frac {17}{2}} - 2 \, x^{\frac {13}{2}} + 3 \, x^{\frac {9}{2}} - 2 \, x^{\frac {5}{2}} + \sqrt {x}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(8*x^5 - 7*(x^3 + x)*x^2 + 9*x^3 + x)/((x^(9/2) - x^(5/2) + sqrt(x))*(x^2 + 1)^(1/4)) + integrate(-4/21*(
16*x^4 - 8*(x^4 + 2*x^2 + 1)*x^2 + 11*x^2 - 5)/((x^(17/2) - 2*x^(13/2) + 3*x^(9/2) - 2*x^(5/2) + sqrt(x))*(x^2
+ 1)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2+1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-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)^(1/4)*(x^4 - x^2 + 1)),x)

[Out]

int((x^2 + 1)/((x^2 + x^4)^(1/4)*(x^4 - 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 [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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