3.17.48 \(\int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx\)

Optimal. Leaf size=111 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{x^{20}+3 x^{16}+2 x^{12}-2 x^8-3 x^4-1}}{\sqrt [4]{2} x \left (x^4+1\right )}\right )}{2 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{x^{20}+3 x^{16}+2 x^{12}-2 x^8-3 x^4-1}}{\sqrt [4]{2} x \left (x^4+1\right )}\right )}{2 \sqrt [4]{2}} \]

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Rubi [A]  time = 0.06, antiderivative size = 121, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6688, 6719, 377, 212, 206, 203} \begin {gather*} \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}}+\frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 3*x^4 - 2*x^8 + 2*x^12 + 3*x^16 + x^20)^(-1/4),x]

[Out]

((-1 + x^4)^(1/4)*(1 + x^4)*ArcTan[(2^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*2^(1/4)*(-((1 - x^4)*(1 + x^4)^4))^(1/4))
+ ((-1 + x^4)^(1/4)*(1 + x^4)*ArcTanh[(2^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*2^(1/4)*(-((1 - x^4)*(1 + x^4)^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 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 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 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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx &=\int \frac {1}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}} \, dx\\ &=\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}\\ &=\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}\\ &=\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}+\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}\\ &=\frac {\sqrt [4]{-1+x^4} \left (1+x^4\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (1+x^4\right )^4\right )}}+\frac {\sqrt [4]{-1+x^4} \left (1+x^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (1+x^4\right )^4\right )}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 75, normalized size = 0.68 \begin {gather*} \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )\right )}{2 \sqrt [4]{2} \sqrt [4]{\left (x^4-1\right ) \left (x^4+1\right )^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 3*x^4 - 2*x^8 + 2*x^12 + 3*x^16 + x^20)^(-1/4),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.42, size = 111, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt [4]{2} x \left (1+x^4\right )}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt [4]{2} x \left (1+x^4\right )}\right )}{2 \sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 - 3*x^4 - 2*x^8 + 2*x^12 + 3*x^16 + x^20)^(-1/4),x]

[Out]

-1/2*ArcTan[(-1 - 3*x^4 - 2*x^8 + 2*x^12 + 3*x^16 + x^20)^(1/4)/(2^(1/4)*x*(1 + x^4))]/2^(1/4) + ArcTanh[(-1 -
3*x^4 - 2*x^8 + 2*x^12 + 3*x^16 + x^20)^(1/4)/(2^(1/4)*x*(1 + x^4))]/(2*2^(1/4))

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fricas [B]  time = 7.35, size = 503, normalized size = 4.53 \begin {gather*} -\frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x + 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} + 2^{\frac {1}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )}\right )}}{2 \, {\left (x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1\right )}}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )} + 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} + 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )} - 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} - 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4),x, algorithm="fricas")

[Out]

-1/4*2^(3/4)*arctan(1/2*(4*2^(3/4)*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(1/4)*(x^11 + 2*x^7 + x^3) + 4
*2^(1/4)*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(3/4)*x + 2^(3/4)*(2*2^(3/4)*sqrt(x^20 + 3*x^16 + 2*x^12
- 2*x^8 - 3*x^4 - 1)*(x^6 + x^2) + 2^(1/4)*(3*x^16 + 8*x^12 + 6*x^8 - 1)))/(x^16 + 4*x^12 + 6*x^8 + 4*x^4 + 1
)) + 1/16*2^(3/4)*log((2^(3/4)*(3*x^16 + 8*x^12 + 6*x^8 - 1) + 4*sqrt(2)*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x
^4 - 1)^(1/4)*(x^11 + 2*x^7 + x^3) + 4*2^(1/4)*sqrt(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)*(x^6 + x^2) +
4*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(3/4)*x)/(x^16 + 4*x^12 + 6*x^8 + 4*x^4 + 1)) - 1/16*2^(3/4)*lo
g(-(2^(3/4)*(3*x^16 + 8*x^12 + 6*x^8 - 1) - 4*sqrt(2)*(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(1/4)*(x^11
+ 2*x^7 + x^3) + 4*2^(1/4)*sqrt(x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)*(x^6 + x^2) - 4*(x^20 + 3*x^16 +
2*x^12 - 2*x^8 - 3*x^4 - 1)^(3/4)*x)/(x^16 + 4*x^12 + 6*x^8 + 4*x^4 + 1))

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giac [A]  time = 0.35, size = 65, normalized size = 0.59 \begin {gather*} \frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4),x, algorithm="giac")

[Out]

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

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maple [C]  time = 2.30, size = 635, normalized size = 5.72

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-3 x^{16} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )-2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{11}-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{12}+\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{6}-4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{7}-6 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{8}+\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}-2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (x^{4}+1\right )^{4}}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {3 x^{16} \RootOf \left (\textit {\_Z}^{4}-8\right )+2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{11}+8 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{12}+\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{6}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{7}+6 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{8}+\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-8\right )}{\left (x^{4}+1\right )^{4}}\right )}{8}\)	\(635\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/8*RootOf(_Z^2+RootOf(_Z^4-8)^2)*ln((-3*x^16*RootOf(_Z^2+RootOf(_Z^4-8)^2)-2*(x^20+3*x^16+2*x^12-2*x^8-3*x^4
-1)^(1/4)*RootOf(_Z^4-8)^2*x^11-8*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^12+(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/2)*
RootOf(_Z^4-8)^2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^6-4*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2
*x^7-6*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^8+(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/2)*RootOf(_Z^4-8)^2*RootOf(_Z^2
+RootOf(_Z^4-8)^2)*x^2-2*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2*x^3+4*(x^20+3*x^16+2*x^12-2
*x^8-3*x^4-1)^(3/4)*x+RootOf(_Z^2+RootOf(_Z^4-8)^2))/(x^4+1)^4)+1/8*RootOf(_Z^4-8)*ln((3*x^16*RootOf(_Z^4-8)+2
*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2*x^11+8*RootOf(_Z^4-8)*x^12+(x^20+3*x^16+2*x^12-2*x^
8-3*x^4-1)^(1/2)*RootOf(_Z^4-8)^3*x^6+4*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4)*RootOf(_Z^4-8)^2*x^7+6*RootOf
(_Z^4-8)*x^8+(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/2)*RootOf(_Z^4-8)^3*x^2+2*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1
)^(1/4)*RootOf(_Z^4-8)^2*x^3+4*(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(3/4)*x-RootOf(_Z^4-8))/(x^4+1)^4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^20 + 3*x^16 + 2*x^12 - 2*x^8 - 3*x^4 - 1)^(-1/4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (x^{20}+3\,x^{16}+2\,x^{12}-2\,x^8-3\,x^4-1\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^12 - 2*x^8 - 3*x^4 + 3*x^16 + x^20 - 1)^(1/4),x)

[Out]

int(1/(2*x^12 - 2*x^8 - 3*x^4 + 3*x^16 + x^20 - 1)^(1/4), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{20} + 3 x^{16} + 2 x^{12} - 2 x^{8} - 3 x^{4} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**20+3*x**16+2*x**12-2*x**8-3*x**4-1)**(1/4),x)

[Out]

Integral((x**20 + 3*x**16 + 2*x**12 - 2*x**8 - 3*x**4 - 1)**(-1/4), x)

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