3.25.2 \(\int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} (-1-x^4+x^8)} \, dx\) [2402]

Optimal. Leaf size=193 \[ -\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right ) \]

[Out]

-1/4*(-2+2*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))+1/4*(2+2*5^(1/2))^(1/2)*arctan(1/2*
(2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))-1/4*(-2+2*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4)
)+1/4*(2+2*5^(1/2))^(1/2)*arctanh(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))

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Rubi [A]
time = 0.17, antiderivative size = 189, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {6860, 385, 218, 212, 209} \begin {gather*} -\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(((3 - Sqrt[5])/2)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(-1 + x^4)^(1/4)]) + (((3 + Sqrt[5])/2)^(1/4)
*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/2 - (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))
^(1/4)*x)/(-1 + x^4)^(1/4)])/2 + (((3 + Sqrt[5])/2)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)
])/2

Rule 209

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

Rule 212

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 385

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 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]
time = 0.50, size = 168, normalized size = 0.87 \begin {gather*} \frac {-\sqrt {-1+\sqrt {5}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {1+\sqrt {5}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {-1+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {1+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[-1 + Sqrt[5]]*ArcTan[(Sqrt[(-1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)]) + Sqrt[1 + Sqrt[5]]*ArcTan[(Sqrt[(1
+ Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[-1 + Sqrt[5]]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)]
+ Sqrt[1 + Sqrt[5]]*ArcTanh[(Sqrt[(1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)])/(2*Sqrt[2])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 10.28, size = 1623, normalized size = 8.41

method	result	size
trager	\(\text {Expression too large to display}\)	\(1623\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x,method=_RETURNVERBOSE)

[Out]

-RootOf(256*_Z^4+16*_Z^2-1)*ln(-(512*RootOf(256*_Z^4+16*_Z^2-1)^5*x^4+192*RootOf(256*_Z^4+16*_Z^2-1)^3*(x^4-1)
^(1/2)*x^2-128*x^4*RootOf(256*_Z^4+16*_Z^2-1)^3+48*RootOf(256*_Z^4+16*_Z^2-1)^2*(x^4-1)^(3/4)*x-64*(x^4-1)^(1/
4)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^3-4*RootOf(256*_Z^4+16*_Z^2-1)*(x^4-1)^(1/2)*x^2+8*x^4*RootOf(256*_Z^4+16*_Z
^2-1)-(x^4-1)^(3/4)*x+3*x^3*(x^4-1)^(1/4)+32*RootOf(256*_Z^4+16*_Z^2-1)^3-4*RootOf(256*_Z^4+16*_Z^2-1))/(4*Roo
tOf(256*_Z^4+16*_Z^2-1)*x^2-1)/(4*RootOf(256*_Z^4+16*_Z^2-1)*x^2+1))-1/4*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^
2-1)^2+1)*ln((-256*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)*RootOf(256*_Z^4+16*_Z^2-1)^4*x^4+96*(x^4-1)^
(1/2)*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^2-96*RootOf(_Z^2+16*RootOf
(256*_Z^4+16*_Z^2-1)^2+1)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^4+96*RootOf(256*_Z^4+16*_Z^2-1)^2*(x^4-1)^(3/4)*x-128
*(x^4-1)^(1/4)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^3+8*(x^4-1)^(1/2)*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)
*x^2-9*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)*x^4+8*(x^4-1)^(3/4)*x-14*x^3*(x^4-1)^(1/4)+16*RootOf(256
*_Z^4+16*_Z^2-1)^2*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)+3*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^
2+1))/(16*RootOf(256*_Z^4+16*_Z^2-1)^2*x^4+x^4+1))+4*RootOf(256*_Z^4+16*_Z^2-1)^2*RootOf(_Z^2+16*RootOf(256*_Z
^4+16*_Z^2-1)^2+1)*ln(-(1024*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)*RootOf(256*_Z^4+16*_Z^2-1)^4*x^4-2
24*(x^4-1)^(1/2)*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^2-176*RootOf(_Z
^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^4+128*RootOf(256*_Z^4+16*_Z^2-1)^2*(x^4-1
)^(3/4)*x+224*(x^4-1)^(1/4)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^3+8*(x^4-1)^(1/2)*RootOf(_Z^2+16*RootOf(256*_Z^4+16
*_Z^2-1)^2+1)*x^2+6*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)*x^4-6*(x^4-1)^(3/4)*x-8*x^3*(x^4-1)^(1/4)+6
4*RootOf(256*_Z^4+16*_Z^2-1)^2*RootOf(_Z^2+16*RootOf(256*_Z^4+16*_Z^2-1)^2+1)-3*RootOf(_Z^2+16*RootOf(256*_Z^4
+16*_Z^2-1)^2+1))/(4*RootOf(256*_Z^4+16*_Z^2-1)*x^2-1)/(4*RootOf(256*_Z^4+16*_Z^2-1)*x^2+1))+16*ln((2048*RootO
f(256*_Z^4+16*_Z^2-1)^5*x^4+448*RootOf(256*_Z^4+16*_Z^2-1)^3*(x^4-1)^(1/2)*x^2+608*x^4*RootOf(256*_Z^4+16*_Z^2
-1)^3+64*RootOf(256*_Z^4+16*_Z^2-1)^2*(x^4-1)^(3/4)*x+112*(x^4-1)^(1/4)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^3+44*Ro
otOf(256*_Z^4+16*_Z^2-1)*(x^4-1)^(1/2)*x^2+42*x^4*RootOf(256*_Z^4+16*_Z^2-1)+7*(x^4-1)^(3/4)*x+11*x^3*(x^4-1)^
(1/4)-128*RootOf(256*_Z^4+16*_Z^2-1)^3-14*RootOf(256*_Z^4+16*_Z^2-1))/(16*RootOf(256*_Z^4+16*_Z^2-1)^2*x^4+x^4
+1))*RootOf(256*_Z^4+16*_Z^2-1)^3+ln((2048*RootOf(256*_Z^4+16*_Z^2-1)^5*x^4+448*RootOf(256*_Z^4+16*_Z^2-1)^3*(
x^4-1)^(1/2)*x^2+608*x^4*RootOf(256*_Z^4+16*_Z^2-1)^3+64*RootOf(256*_Z^4+16*_Z^2-1)^2*(x^4-1)^(3/4)*x+112*(x^4
-1)^(1/4)*RootOf(256*_Z^4+16*_Z^2-1)^2*x^3+44*RootOf(256*_Z^4+16*_Z^2-1)*(x^4-1)^(1/2)*x^2+42*x^4*RootOf(256*_
Z^4+16*_Z^2-1)+7*(x^4-1)^(3/4)*x+11*x^3*(x^4-1)^(1/4)-128*RootOf(256*_Z^4+16*_Z^2-1)^3-14*RootOf(256*_Z^4+16*_
Z^2-1))/(16*RootOf(256*_Z^4+16*_Z^2-1)^2*x^4+x^4+1))*RootOf(256*_Z^4+16*_Z^2-1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="maxima")

[Out]

integrate((2*x^4 - 1)/((x^8 - x^4 - 1)*(x^4 - 1)^(1/4)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 990 vs. \(2 (125) = 250\).
time = 15.73, size = 990, normalized size = 5.13 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \arctan \left (-\frac {{\left (\sqrt {5} \sqrt {2} {\left (x^{8} + x^{4} - 1\right )} - \sqrt {2} {\left (5 \, x^{8} - 3 \, x^{4} - 1\right )} - 2 \, {\left (\sqrt {5} \sqrt {2} x^{6} - \sqrt {2} {\left (x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1}\right )} \sqrt {2 \, \sqrt {5} + 2} \sqrt {\sqrt {5} + 1} - 4 \, {\left ({\left (\sqrt {5} \sqrt {2} x^{5} - \sqrt {2} {\left (x^{5} + 2 \, x\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - {\left (\sqrt {5} \sqrt {2} x^{3} - \sqrt {2} {\left (2 \, x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {\sqrt {5} + 1}}{8 \, {\left (x^{8} - x^{4} - 1\right )}}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \arctan \left (\frac {{\left (\sqrt {5} \sqrt {2} {\left (x^{8} + x^{4} - 1\right )} + \sqrt {2} {\left (5 \, x^{8} - 3 \, x^{4} - 1\right )} + 2 \, {\left (\sqrt {5} \sqrt {2} x^{6} + \sqrt {2} {\left (x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1}\right )} \sqrt {2 \, \sqrt {5} - 2} \sqrt {\sqrt {5} - 1} - 4 \, {\left ({\left (\sqrt {5} \sqrt {2} x^{5} + \sqrt {2} {\left (x^{5} + 2 \, x\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + {\left (\sqrt {5} \sqrt {2} x^{3} + \sqrt {2} {\left (2 \, x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {\sqrt {5} - 1}}{8 \, {\left (x^{8} - x^{4} - 1\right )}}\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (\frac {2 \, {\left (2 \, x^{5} + \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + {\left (\sqrt {5} \sqrt {2} {\left (x^{8} - x^{4}\right )} + \sqrt {2} {\left (2 \, x^{4} - 1\right )} + \sqrt {x^{4} - 1} {\left (\sqrt {5} \sqrt {2} x^{2} + \sqrt {2} {\left (2 \, x^{6} - x^{2}\right )}\right )}\right )} \sqrt {\sqrt {5} - 1} - 2 \, {\left (x^{7} - 3 \, x^{3} - \sqrt {5} {\left (x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - x^{4} - 1}\right ) + \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (\frac {2 \, {\left (2 \, x^{5} + \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - {\left (\sqrt {5} \sqrt {2} {\left (x^{8} - x^{4}\right )} + \sqrt {2} {\left (2 \, x^{4} - 1\right )} + \sqrt {x^{4} - 1} {\left (\sqrt {5} \sqrt {2} x^{2} + \sqrt {2} {\left (2 \, x^{6} - x^{2}\right )}\right )}\right )} \sqrt {\sqrt {5} - 1} - 2 \, {\left (x^{7} - 3 \, x^{3} - \sqrt {5} {\left (x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - x^{4} - 1}\right ) + \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (\frac {2 \, {\left (2 \, x^{5} - \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + {\left (\sqrt {5} \sqrt {2} {\left (x^{8} - x^{4}\right )} - \sqrt {2} {\left (2 \, x^{4} - 1\right )} - \sqrt {x^{4} - 1} {\left (\sqrt {5} \sqrt {2} x^{2} - \sqrt {2} {\left (2 \, x^{6} - x^{2}\right )}\right )}\right )} \sqrt {\sqrt {5} + 1} + 2 \, {\left (x^{7} - 3 \, x^{3} + \sqrt {5} {\left (x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - x^{4} - 1}\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (\frac {2 \, {\left (2 \, x^{5} - \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - {\left (\sqrt {5} \sqrt {2} {\left (x^{8} - x^{4}\right )} - \sqrt {2} {\left (2 \, x^{4} - 1\right )} - \sqrt {x^{4} - 1} {\left (\sqrt {5} \sqrt {2} x^{2} - \sqrt {2} {\left (2 \, x^{6} - x^{2}\right )}\right )}\right )} \sqrt {\sqrt {5} + 1} + 2 \, {\left (x^{7} - 3 \, x^{3} + \sqrt {5} {\left (x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - x^{4} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*sqrt(sqrt(5) + 1)*arctan(-1/8*((sqrt(5)*sqrt(2)*(x^8 + x^4 - 1) - sqrt(2)*(5*x^8 - 3*x^4 - 1) - 2
*(sqrt(5)*sqrt(2)*x^6 - sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1))*sqrt(2*sqrt(5) + 2)*sqrt(sqrt(5) + 1) - 4*((sqrt
(5)*sqrt(2)*x^5 - sqrt(2)*(x^5 + 2*x))*(x^4 - 1)^(3/4) - (sqrt(5)*sqrt(2)*x^3 - sqrt(2)*(2*x^7 - x^3))*(x^4 -
1)^(1/4))*sqrt(sqrt(5) + 1))/(x^8 - x^4 - 1)) + 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*arctan(1/8*((sqrt(5)*sqrt(2)*(x^
8 + x^4 - 1) + sqrt(2)*(5*x^8 - 3*x^4 - 1) + 2*(sqrt(5)*sqrt(2)*x^6 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1))*sq
rt(2*sqrt(5) - 2)*sqrt(sqrt(5) - 1) - 4*((sqrt(5)*sqrt(2)*x^5 + sqrt(2)*(x^5 + 2*x))*(x^4 - 1)^(3/4) + (sqrt(5
)*sqrt(2)*x^3 + sqrt(2)*(2*x^7 - x^3))*(x^4 - 1)^(1/4))*sqrt(sqrt(5) - 1))/(x^8 - x^4 - 1)) - 1/16*sqrt(2)*sqr
t(sqrt(5) - 1)*log((2*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/4) + (sqrt(5)*sqrt(2)*(x^8 - x^4) + sqrt(2)*(2*x^4
- 1) + sqrt(x^4 - 1)*(sqrt(5)*sqrt(2)*x^2 + sqrt(2)*(2*x^6 - x^2)))*sqrt(sqrt(5) - 1) - 2*(x^7 - 3*x^3 - sqrt(
5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + 1/16*sqrt(2)*sqrt(sqrt(5) - 1)*log((2*(2*x^5 + sqrt(5)*x -
x)*(x^4 - 1)^(3/4) - (sqrt(5)*sqrt(2)*(x^8 - x^4) + sqrt(2)*(2*x^4 - 1) + sqrt(x^4 - 1)*(sqrt(5)*sqrt(2)*x^2
+ sqrt(2)*(2*x^6 - x^2)))*sqrt(sqrt(5) - 1) - 2*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^
4 - 1)) + 1/16*sqrt(2)*sqrt(sqrt(5) + 1)*log((2*(2*x^5 - sqrt(5)*x - x)*(x^4 - 1)^(3/4) + (sqrt(5)*sqrt(2)*(x^
8 - x^4) - sqrt(2)*(2*x^4 - 1) - sqrt(x^4 - 1)*(sqrt(5)*sqrt(2)*x^2 - sqrt(2)*(2*x^6 - x^2)))*sqrt(sqrt(5) + 1
) + 2*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) - 1/16*sqrt(2)*sqrt(sqrt(5) + 1)*l
og((2*(2*x^5 - sqrt(5)*x - x)*(x^4 - 1)^(3/4) - (sqrt(5)*sqrt(2)*(x^8 - x^4) - sqrt(2)*(2*x^4 - 1) - sqrt(x^4
- 1)*(sqrt(5)*sqrt(2)*x^2 - sqrt(2)*(2*x^6 - x^2)))*sqrt(sqrt(5) + 1) + 2*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*
(x^4 - 1)^(1/4))/(x^8 - x^4 - 1))

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Sympy [F(-1)] Timed out
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((2*x**4-1)/(x**4-1)**(1/4)/(x**8-x**4-1),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="giac")

[Out]

integrate((2*x^4 - 1)/((x^8 - x^4 - 1)*(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 {2\,x^4-1}{{\left (x^4-1\right )}^{1/4}\,\left (-x^8+x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^4 - 1)/((x^4 - 1)^(1/4)*(x^4 - x^8 + 1)),x)

[Out]

-int((2*x^4 - 1)/((x^4 - 1)^(1/4)*(x^4 - x^8 + 1)), x)

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