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

Optimal. Leaf size=85 \[ \frac {1}{4} \text {RootSum}\left [2 \text {$\#$1}^8-5 \text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)-\log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )+\log (x)}{4 \text {$\#$1}^5-5 \text {$\#$1}}\& \right ] \]

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Rubi [B]  time = 0.45, antiderivative size = 199, normalized size of antiderivative = 2.34, number of steps used = 10, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6728, 377, 212, 206, 203} \begin {gather*} -\frac {\sqrt [4]{487-79 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}+\frac {\sqrt [4]{487+79 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{487-79 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}+\frac {\sqrt [4]{487+79 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*((487 - 79*Sqrt[17])^(1/4)*ArcTan[((2/(5 + Sqrt[17]))^(1/4)*x)/(-1 + x^4)^(1/4)])/Sqrt[17] + ((487 + 79*S
qrt[17])^(1/4)*ArcTan[((5 + Sqrt[17])^(1/4)*x)/(Sqrt[2]*(-1 + x^4)^(1/4))])/(4*Sqrt[17]) - ((487 - 79*Sqrt[17]
)^(1/4)*ArcTanh[((2/(5 + Sqrt[17]))^(1/4)*x)/(-1 + x^4)^(1/4)])/(4*Sqrt[17]) + ((487 + 79*Sqrt[17])^(1/4)*ArcT
anh[((5 + Sqrt[17])^(1/4)*x)/(Sqrt[2]*(-1 + x^4)^(1/4))])/(4*Sqrt[17])

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 6728

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 (-2-x^4+2 x^8\right )} \, dx &=\int \left (\frac {2-\frac {2}{\sqrt {17}}}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {17}+4 x^4\right )}+\frac {2+\frac {2}{\sqrt {17}}}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {17}+4 x^4\right )}\right ) \, dx\\ &=\frac {1}{17} \left (2 \left (17-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {17}+4 x^4\right )} \, dx+\frac {1}{17} \left (2 \left (17+\sqrt {17}\right )\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {17}+4 x^4\right )} \, dx\\ &=\frac {1}{17} \left (2 \left (17-\sqrt {17}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {17}-\left (3-\sqrt {17}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{17} \left (2 \left (17+\sqrt {17}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {17}-\left (3+\sqrt {17}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{2} \sqrt {\frac {1}{34} \left (23-\sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \sqrt {\frac {1}{34} \left (23-\sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (23+\sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (23+\sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\frac {\sqrt [4]{487-79 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}+\frac {\sqrt [4]{487+79 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{487-79 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}+\frac {\sqrt [4]{487+79 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}\\ \end {align*}

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Mathematica [B]  time = 0.34, size = 192, normalized size = 2.26 \begin {gather*} \frac {-\frac {8 \sqrt [4]{23+\sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{1+\sqrt {17}}+\sqrt [4]{487+79 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )-\frac {8 \sqrt [4]{23+\sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{1+\sqrt {17}}+\sqrt [4]{487+79 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-8*(23 + Sqrt[17])^(1/4)*ArcTan[((2/(5 + Sqrt[17]))^(1/4)*x)/(-1 + x^4)^(1/4)])/(1 + Sqrt[17]) + (487 + 79*S
qrt[17])^(1/4)*ArcTan[((5 + Sqrt[17])^(1/4)*x)/(Sqrt[2]*(-1 + x^4)^(1/4))] - (8*(23 + Sqrt[17])^(1/4)*ArcTanh[
((2/(5 + Sqrt[17]))^(1/4)*x)/(-1 + x^4)^(1/4)])/(1 + Sqrt[17]) + (487 + 79*Sqrt[17])^(1/4)*ArcTanh[((5 + Sqrt[
17])^(1/4)*x)/(Sqrt[2]*(-1 + x^4)^(1/4))])/(4*Sqrt[17])

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IntegrateAlgebraic [A]  time = 0.27, size = 85, normalized size = 1.00 \begin {gather*} \frac {1}{4} \text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[1 - 5*#1^4 + 2*#1^8 & , (Log[x] - Log[(-1 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-1 + x^4)^(1/4) -
x*#1]*#1^4)/(-5*#1 + 4*#1^5) & ]/4

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fricas [B]  time = 19.62, size = 1200, normalized size = 14.12

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/68*sqrt(17)*(-79*sqrt(17) + 487)^(1/4)*arctan(-1/262144*(64*(107*x^5 + sqrt(17)*(51*x^5 + 14*x) + 190*x)*(x^
4 - 1)^(3/4)*sqrt(-79*sqrt(17) + 487) - sqrt(2)*(32*(14*x^6 + 5*x^2 + sqrt(17)*(2*x^6 + 3*x^2))*sqrt(x^4 - 1)
+ (40*x^8 - 7*x^4 + sqrt(17)*(8*x^8 + x^4 - 6) - 22)*sqrt(-79*sqrt(17) + 487))*sqrt(-(3071*sqrt(17) - 15081)*s
qrt(-79*sqrt(17) + 487)) - 16384*(5*x^7 - 14*x^3 - sqrt(17)*(3*x^7 - 2*x^3))*(x^4 - 1)^(1/4))*(-79*sqrt(17) +
487)^(1/4)/(2*x^8 - x^4 - 2)) + 1/68*sqrt(17)*(79*sqrt(17) + 487)^(1/4)*arctan(-1/262144*(sqrt(2)*(32*(14*x^6
+ 5*x^2 - sqrt(17)*(2*x^6 + 3*x^2))*sqrt(x^4 - 1) + (40*x^8 - 7*x^4 - sqrt(17)*(8*x^8 + x^4 - 6) - 22)*sqrt(79
*sqrt(17) + 487))*sqrt((3071*sqrt(17) + 15081)*sqrt(79*sqrt(17) + 487))*(79*sqrt(17) + 487)^(1/4) - 64*((107*x
^5 - sqrt(17)*(51*x^5 + 14*x) + 190*x)*(x^4 - 1)^(3/4)*sqrt(79*sqrt(17) + 487) - 256*(5*x^7 - 14*x^3 + sqrt(17
)*(3*x^7 - 2*x^3))*(x^4 - 1)^(1/4))*(79*sqrt(17) + 487)^(1/4))/(2*x^8 - x^4 - 2)) + 1/272*sqrt(17)*(79*sqrt(17
) + 487)^(1/4)*log((4096*(89*x^5 + sqrt(17)*(15*x^5 - 26*x) - 86*x)*(x^4 - 1)^(3/4) + 16*(1151*x^7 - 1210*x^3
+ sqrt(17)*(217*x^7 - 342*x^3))*(x^4 - 1)^(1/4)*sqrt(79*sqrt(17) + 487) + (91136*x^8 - 123008*x^4 + (3335*x^6
- 2538*x^2 + sqrt(17)*(401*x^6 - 934*x^2))*sqrt(x^4 - 1)*sqrt(79*sqrt(17) + 487) + 128*sqrt(17)*(120*x^8 - 231
*x^4 + 74) + 21248)*(79*sqrt(17) + 487)^(1/4))/(2*x^8 - x^4 - 2)) - 1/272*sqrt(17)*(79*sqrt(17) + 487)^(1/4)*l
og((4096*(89*x^5 + sqrt(17)*(15*x^5 - 26*x) - 86*x)*(x^4 - 1)^(3/4) + 16*(1151*x^7 - 1210*x^3 + sqrt(17)*(217*
x^7 - 342*x^3))*(x^4 - 1)^(1/4)*sqrt(79*sqrt(17) + 487) - (91136*x^8 - 123008*x^4 + (3335*x^6 - 2538*x^2 + sqr
t(17)*(401*x^6 - 934*x^2))*sqrt(x^4 - 1)*sqrt(79*sqrt(17) + 487) + 128*sqrt(17)*(120*x^8 - 231*x^4 + 74) + 212
48)*(79*sqrt(17) + 487)^(1/4))/(2*x^8 - x^4 - 2)) - 1/272*sqrt(17)*(-79*sqrt(17) + 487)^(1/4)*log((4096*(89*x^
5 - sqrt(17)*(15*x^5 - 26*x) - 86*x)*(x^4 - 1)^(3/4) + 16*(1151*x^7 - 1210*x^3 - sqrt(17)*(217*x^7 - 342*x^3))
*(x^4 - 1)^(1/4)*sqrt(-79*sqrt(17) + 487) + (91136*x^8 - 123008*x^4 + (3335*x^6 - 2538*x^2 - sqrt(17)*(401*x^6
- 934*x^2))*sqrt(x^4 - 1)*sqrt(-79*sqrt(17) + 487) - 128*sqrt(17)*(120*x^8 - 231*x^4 + 74) + 21248)*(-79*sqrt
(17) + 487)^(1/4))/(2*x^8 - x^4 - 2)) + 1/272*sqrt(17)*(-79*sqrt(17) + 487)^(1/4)*log((4096*(89*x^5 - sqrt(17)
*(15*x^5 - 26*x) - 86*x)*(x^4 - 1)^(3/4) + 16*(1151*x^7 - 1210*x^3 - sqrt(17)*(217*x^7 - 342*x^3))*(x^4 - 1)^(
1/4)*sqrt(-79*sqrt(17) + 487) - (91136*x^8 - 123008*x^4 + (3335*x^6 - 2538*x^2 - sqrt(17)*(401*x^6 - 934*x^2))
*sqrt(x^4 - 1)*sqrt(-79*sqrt(17) + 487) - 128*sqrt(17)*(120*x^8 - 231*x^4 + 74) + 21248)*(-79*sqrt(17) + 487)^
(1/4))/(2*x^8 - x^4 - 2))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {2 x^{4}-1}{\left (x^{4}-1\right )^{\frac {1}{4}} \left (2 x^{8}-x^{4}-2\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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sympy [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((2*x**4-1)/(x**4-1)**(1/4)/(2*x**8-x**4-2),x)

[Out]

Timed out

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