3.25.1 \(\int \frac {1}{\sqrt [4]{-1+x^4} (-1-x^4+x^8)} \, dx\) [2401]

Optimal. Leaf size=193 \[ -\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right ) \]

[Out]

-1/20*(-10+10*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))-1/20*(10+10*5^(1/2))^(1/2)*arcta
n(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))-1/20*(-10+10*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4
-1)^(1/4))-1/20*(10+10*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 = 0.14, antiderivative size = 209, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1442, 385, 218, 212, 209} \begin {gather*} -\frac {\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 )}{2 \sqrt {5}}-\frac {\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 )}{2 \sqrt {5}}-\frac {\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 )}{2 \sqrt {5}}-\frac {\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 )}{2 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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 1442

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx &=\frac {2 \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}\\ &=\frac {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 )}{\sqrt {5}}-\frac {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 )}{\sqrt {5}}\\ &=-\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 {10}}-\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 {10}}-\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 {10}}-\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 {10}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{-1+x^4}}\right )}{2^{3/4} \sqrt {5} \sqrt [4]{3+\sqrt {5}}}-\frac {\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 )}{2 \sqrt {5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{-1+x^4}}\right )}{2^{3/4} \sqrt {5} \sqrt [4]{3+\sqrt {5}}}-\frac {\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 )}{2 \sqrt {5}}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 166, normalized size = 0.86 \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 {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(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)])/Sqrt[10]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 10.96, size = 1704, normalized size = 8.83

method	result	size
trager	\(\text {Expression too large to display}\)	\(1704\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(6400*_Z^4+80*_Z^2-1)*ln(-(384000*RootOf(6400*_Z^4+80*_Z^2-1)^5*x^4+17600*RootOf(6400*_Z^4+80*_Z^2-1)^3*
(x^4-1)^(1/2)*x^2-13600*x^4*RootOf(6400*_Z^4+80*_Z^2-1)^3+1840*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^4-1)^(3/4)*x-3
120*(x^4-1)^(1/4)*RootOf(6400*_Z^4+80*_Z^2-1)^2*x^3-120*RootOf(6400*_Z^4+80*_Z^2-1)*(x^4-1)^(1/2)*x^2+100*Root
Of(6400*_Z^4+80*_Z^2-1)*x^4-16*(x^4-1)^(3/4)*x+23*x^3*(x^4-1)^(1/4)+4800*RootOf(6400*_Z^4+80*_Z^2-1)^3-50*Root
Of(6400*_Z^4+80*_Z^2-1))/(640*RootOf(6400*_Z^4+80*_Z^2-1)^3*x^2+4*RootOf(6400*_Z^4+80*_Z^2-1)*x^2-1)/(640*Root
Of(6400*_Z^4+80*_Z^2-1)^3*x^2+4*RootOf(6400*_Z^4+80*_Z^2-1)*x^2+1))-1/20*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_
Z^2-1)^2+5)*ln((6400*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1)^4*x^4-160*(x
^4-1)^(1/2)*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1)^2*x^2+320*RootOf(_Z^2
+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1)^2*x^4+480*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^4
-1)^(3/4)*x-640*(x^4-1)^(1/4)*RootOf(6400*_Z^4+80*_Z^2-1)^2*x^3-4*(x^4-1)^(1/2)*RootOf(_Z^2+400*RootOf(6400*_Z
^4+80*_Z^2-1)^2+5)*x^2+3*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*x^4+8*(x^4-1)^(3/4)*x-14*x^3*(x^4-1)
^(1/4)-80*RootOf(6400*_Z^4+80*_Z^2-1)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)-RootOf(_Z^2+400*RootO
f(6400*_Z^4+80*_Z^2-1)^2+5))/(80*x^4*RootOf(6400*_Z^4+80*_Z^2-1)^2+x^4+1))-4*RootOf(6400*_Z^4+80*_Z^2-1)^2*Roo
tOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*ln(-(6400*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootO
f(6400*_Z^4+80*_Z^2-1)^4*x^4-320*(x^4-1)^(1/2)*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z
^4+80*_Z^2-1)^2*x^2-240*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1)^2*x^4+480
*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^4-1)^(3/4)*x+640*(x^4-1)^(1/4)*RootOf(6400*_Z^4+80*_Z^2-1)^2*x^3+2*(x^4-1)^(
1/2)*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*x^2+2*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*x
^4-2*(x^4-1)^(3/4)*x-6*x^3*(x^4-1)^(1/4)+80*RootOf(6400*_Z^4+80*_Z^2-1)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*
_Z^2-1)^2+5)-RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5))/(640*RootOf(6400*_Z^4+80*_Z^2-1)^3*x^2+4*RootOf
(6400*_Z^4+80*_Z^2-1)*x^2-1)/(640*RootOf(6400*_Z^4+80*_Z^2-1)^3*x^2+4*RootOf(6400*_Z^4+80*_Z^2-1)*x^2+1))+80*l
n(-(64000*RootOf(6400*_Z^4+80*_Z^2-1)^5*x^4+3200*RootOf(6400*_Z^4+80*_Z^2-1)^3*(x^4-1)^(1/2)*x^2+4000*x^4*Root
Of(6400*_Z^4+80*_Z^2-1)^3-240*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^4-1)^(3/4)*x-320*(x^4-1)^(1/4)*RootOf(6400*_Z^4
+80*_Z^2-1)^2*x^3+60*RootOf(6400*_Z^4+80*_Z^2-1)*(x^4-1)^(1/2)*x^2+60*RootOf(6400*_Z^4+80*_Z^2-1)*x^4-4*(x^4-1
)^(3/4)*x-7*x^3*(x^4-1)^(1/4)-800*RootOf(6400*_Z^4+80*_Z^2-1)^3-20*RootOf(6400*_Z^4+80*_Z^2-1))/(80*x^4*RootOf
(6400*_Z^4+80*_Z^2-1)^2+x^4+1))*RootOf(6400*_Z^4+80*_Z^2-1)^3+ln(-(64000*RootOf(6400*_Z^4+80*_Z^2-1)^5*x^4+320
0*RootOf(6400*_Z^4+80*_Z^2-1)^3*(x^4-1)^(1/2)*x^2+4000*x^4*RootOf(6400*_Z^4+80*_Z^2-1)^3-240*RootOf(6400*_Z^4+
80*_Z^2-1)^2*(x^4-1)^(3/4)*x-320*(x^4-1)^(1/4)*RootOf(6400*_Z^4+80*_Z^2-1)^2*x^3+60*RootOf(6400*_Z^4+80*_Z^2-1
)*(x^4-1)^(1/2)*x^2+60*RootOf(6400*_Z^4+80*_Z^2-1)*x^4-4*(x^4-1)^(3/4)*x-7*x^3*(x^4-1)^(1/4)-800*RootOf(6400*_
Z^4+80*_Z^2-1)^3-20*RootOf(6400*_Z^4+80*_Z^2-1))/(80*x^4*RootOf(6400*_Z^4+80*_Z^2-1)^2+x^4+1))*RootOf(6400*_Z^
4+80*_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(1/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="maxima")

[Out]

integrate(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. 953 vs. \(2 (125) = 250\).
time = 15.61, size = 953, normalized size = 4.94 \begin {gather*} -\frac {1}{20} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \arctan \left (-\frac {\sqrt {2} {\left (2 \, \sqrt {10} {\left (5 \, x^{6} - \sqrt {5} {\left (x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} - \sqrt {10} {\left (5 \, x^{8} + 5 \, x^{4} - \sqrt {5} {\left (5 \, x^{8} - 3 \, x^{4} - 1\right )} - 5\right )}\right )} {\left (\sqrt {5} + 1\right )} + 4 \, {\left (\sqrt {10} {\left (5 \, x^{5} - \sqrt {5} {\left (x^{5} + 2 \, x\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - \sqrt {10} {\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (5 \, x^{3} - \sqrt {5} {\left (2 \, x^{7} - x^{3}\right )}\right )}\right )} \sqrt {\sqrt {5} + 1}}{40 \, {\left (x^{8} - x^{4} - 1\right )}}\right ) + \frac {1}{20} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {10} {\left (5 \, x^{6} + \sqrt {5} {\left (x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} + \sqrt {10} {\left (5 \, x^{8} + 5 \, x^{4} + \sqrt {5} {\left (5 \, x^{8} - 3 \, x^{4} - 1\right )} - 5\right )}\right )} {\left (\sqrt {5} - 1\right )} - 4 \, {\left (\sqrt {10} {\left (5 \, x^{5} + \sqrt {5} {\left (x^{5} + 2 \, x\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \sqrt {10} {\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (5 \, x^{3} + \sqrt {5} {\left (2 \, x^{7} - x^{3}\right )}\right )}\right )} \sqrt {\sqrt {5} - 1}}{40 \, {\left (x^{8} - x^{4} - 1\right )}}\right ) - \frac {1}{80} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\frac {10 \, {\left (2 \, x^{5} + \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + {\left (\sqrt {10} \sqrt {x^{4} - 1} {\left (5 \, x^{2} + \sqrt {5} {\left (2 \, x^{6} - x^{2}\right )}\right )} + \sqrt {10} {\left (5 \, x^{8} - 5 \, x^{4} + \sqrt {5} {\left (2 \, x^{4} - 1\right )}\right )}\right )} \sqrt {\sqrt {5} - 1} - 10 \, {\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}{80} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\frac {10 \, {\left (2 \, x^{5} + \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - {\left (\sqrt {10} \sqrt {x^{4} - 1} {\left (5 \, x^{2} + \sqrt {5} {\left (2 \, x^{6} - x^{2}\right )}\right )} + \sqrt {10} {\left (5 \, x^{8} - 5 \, x^{4} + \sqrt {5} {\left (2 \, x^{4} - 1\right )}\right )}\right )} \sqrt {\sqrt {5} - 1} - 10 \, {\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}{80} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\frac {10 \, {\left (2 \, x^{5} - \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + {\left (\sqrt {10} \sqrt {x^{4} - 1} {\left (5 \, x^{2} - \sqrt {5} {\left (2 \, x^{6} - x^{2}\right )}\right )} - \sqrt {10} {\left (5 \, x^{8} - 5 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 1\right )}\right )}\right )} \sqrt {\sqrt {5} + 1} + 10 \, {\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}{80} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\frac {10 \, {\left (2 \, x^{5} - \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - {\left (\sqrt {10} \sqrt {x^{4} - 1} {\left (5 \, x^{2} - \sqrt {5} {\left (2 \, x^{6} - x^{2}\right )}\right )} - \sqrt {10} {\left (5 \, x^{8} - 5 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 1\right )}\right )}\right )} \sqrt {\sqrt {5} + 1} + 10 \, {\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(1/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="fricas")

[Out]

-1/20*sqrt(10)*sqrt(sqrt(5) + 1)*arctan(-1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 - sqrt(5)*(x^6 + 2*x^2))*sqrt(x^4 -
1) - sqrt(10)*(5*x^8 + 5*x^4 - sqrt(5)*(5*x^8 - 3*x^4 - 1) - 5))*(sqrt(5) + 1) + 4*(sqrt(10)*(5*x^5 - sqrt(5)*
(x^5 + 2*x))*(x^4 - 1)^(3/4) - sqrt(10)*(x^4 - 1)^(1/4)*(5*x^3 - sqrt(5)*(2*x^7 - x^3)))*sqrt(sqrt(5) + 1))/(x
^8 - x^4 - 1)) + 1/20*sqrt(10)*sqrt(sqrt(5) - 1)*arctan(1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 + sqrt(5)*(x^6 + 2*x^
2))*sqrt(x^4 - 1) + sqrt(10)*(5*x^8 + 5*x^4 + sqrt(5)*(5*x^8 - 3*x^4 - 1) - 5))*(sqrt(5) - 1) - 4*(sqrt(10)*(5
*x^5 + sqrt(5)*(x^5 + 2*x))*(x^4 - 1)^(3/4) + sqrt(10)*(x^4 - 1)^(1/4)*(5*x^3 + sqrt(5)*(2*x^7 - x^3)))*sqrt(s
qrt(5) - 1))/(x^8 - x^4 - 1)) - 1/80*sqrt(10)*sqrt(sqrt(5) - 1)*log((10*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/4
) + (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2*x^6 - x^2)) + sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(2*x^4 - 1)))*
sqrt(sqrt(5) - 1) - 10*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + 1/80*sqrt(10)*s
qrt(sqrt(5) - 1)*log((10*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/4) - (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2
*x^6 - x^2)) + sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) - 1) - 10*(x^7 - 3*x^3 - sqrt(5)*(
x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + 1/80*sqrt(10)*sqrt(sqrt(5) + 1)*log((10*(2*x^5 - sqrt(5)*x - x
)*(x^4 - 1)^(3/4) + (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 - sqrt(5)*(2*x^6 - x^2)) - sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5
)*(2*x^4 - 1)))*sqrt(sqrt(5) + 1) + 10*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) -
1/80*sqrt(10)*sqrt(sqrt(5) + 1)*log((10*(2*x^5 - sqrt(5)*x - x)*(x^4 - 1)^(3/4) - (sqrt(10)*sqrt(x^4 - 1)*(5*
x^2 - sqrt(5)*(2*x^6 - x^2)) - sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) + 1) + 10*(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} - x^{4} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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