3.21.5 \(\int \frac {-b^8+2 a^4 x^4}{\sqrt [4]{-b^8+a^4 x^4} (-b^8-c x^4+a^8 x^8)} \, dx\)

Optimal. Leaf size=141 \[ \frac {1}{4} \text {RootSum}\left [-\text {$\#$1}^8+2 \text {$\#$1}^4 a^4+\text {$\#$1}^4 c+a^8 b^8-a^8-a^4 c\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a^4 x^4-b^8}-\text {$\#$1} x\right )-\text {$\#$1}^4 \log (x)+a^4 \log \left (\sqrt [4]{a^4 x^4-b^8}-\text {$\#$1} x\right )+a^4 (-\log (x))}{-2 \text {$\#$1}^5+2 \text {$\#$1} a^4+\text {$\#$1} c}\& \right ] \]

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Rubi [B]  time = 2.00, antiderivative size = 657, normalized size of antiderivative = 4.66, number of steps used = 10, number of rules used = 5, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {6728, 377, 212, 208, 205} \begin {gather*} \frac {a^3 \left (\frac {a^4 b^8-c}{\sqrt {4 a^8 b^8+c^2}}+1\right ) \tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^8-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^8}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^8-c}}-\frac {a^3 \left (1-\frac {a^4 b^8-c}{\sqrt {4 a^8 b^8+c^2}}\right ) \tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^8+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^8}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^8+c}}+\frac {a^3 \left (\frac {a^4 b^8-c}{\sqrt {4 a^8 b^8+c^2}}+1\right ) \tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^8-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^8}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^8-c}}-\frac {a^3 \left (1-\frac {a^4 b^8-c}{\sqrt {4 a^8 b^8+c^2}}\right ) \tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^8+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^8}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^8+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(1 + (a^4*b^8 - c)/Sqrt[4*a^8*b^8 + c^2])*ArcTan[(a*(2*a^4*b^8 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c
+ Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^8 + a^4*x^4)^(1/4))])/((-c + Sqrt[4*a^8*b^8 + c^2])^(3/4)*(2*a^4*b^8 - c +
Sqrt[4*a^8*b^8 + c^2])^(1/4)) - (a^3*(1 - (a^4*b^8 - c)/Sqrt[4*a^8*b^8 + c^2])*ArcTan[(a*(-2*a^4*b^8 + c + Sq
rt[4*a^8*b^8 + c^2])^(1/4)*x)/((c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^8 + a^4*x^4)^(1/4))])/((c + Sqrt[4*a^8*b^
8 + c^2])^(3/4)*(-2*a^4*b^8 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) + (a^3*(1 + (a^4*b^8 - c)/Sqrt[4*a^8*b^8 + c^2
])*ArcTanh[(a*(2*a^4*b^8 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^8 + a^4
*x^4)^(1/4))])/((-c + Sqrt[4*a^8*b^8 + c^2])^(3/4)*(2*a^4*b^8 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) - (a^3*(1 -
(a^4*b^8 - c)/Sqrt[4*a^8*b^8 + c^2])*ArcTanh[(a*(-2*a^4*b^8 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((c + Sqrt[4
*a^8*b^8 + c^2])^(1/4)*(-b^8 + a^4*x^4)^(1/4))])/((c + Sqrt[4*a^8*b^8 + c^2])^(3/4)*(-2*a^4*b^8 + c + Sqrt[4*a
^8*b^8 + c^2])^(1/4))

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

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Mathematica [B]  time = 1.51, size = 671, normalized size = 4.76 \begin {gather*} a^3 \left (\frac {\left (\sqrt {4 a^8 b^8+c^2}+a^4 b^8-c\right ) \tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^8-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^8}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^8-c}}-\frac {\left (\frac {c-a^4 b^8}{\sqrt {4 a^8 b^8+c^2}}+1\right ) \tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^8+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^8}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^8+c}}+\frac {\left (\sqrt {4 a^8 b^8+c^2}+a^4 b^8-c\right ) \tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^8-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^8}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^8-c}}-\frac {\left (\frac {c-a^4 b^8}{\sqrt {4 a^8 b^8+c^2}}+1\right ) \tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^8+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^8}}\right )}{\left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^8+c}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*(((a^4*b^8 - c + Sqrt[4*a^8*b^8 + c^2])*ArcTan[(a*(2*a^4*b^8 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c +
Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^8 + a^4*x^4)^(1/4))])/(Sqrt[4*a^8*b^8 + c^2]*(-c + Sqrt[4*a^8*b^8 + c^2])^(3/
4)*(2*a^4*b^8 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) - ((1 + (-(a^4*b^8) + c)/Sqrt[4*a^8*b^8 + c^2])*ArcTan[(a*(-
2*a^4*b^8 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^8 + a^4*x^4)^(1/4))])/(
(c + Sqrt[4*a^8*b^8 + c^2])^(3/4)*(-2*a^4*b^8 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) + ((a^4*b^8 - c + Sqrt[4*a^8
*b^8 + c^2])*ArcTanh[(a*(2*a^4*b^8 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(
-b^8 + a^4*x^4)^(1/4))])/(Sqrt[4*a^8*b^8 + c^2]*(-c + Sqrt[4*a^8*b^8 + c^2])^(3/4)*(2*a^4*b^8 - c + Sqrt[4*a^8
*b^8 + c^2])^(1/4)) - ((1 + (-(a^4*b^8) + c)/Sqrt[4*a^8*b^8 + c^2])*ArcTanh[(a*(-2*a^4*b^8 + c + Sqrt[4*a^8*b^
8 + c^2])^(1/4)*x)/((c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^8 + a^4*x^4)^(1/4))])/((c + Sqrt[4*a^8*b^8 + c^2])^(
3/4)*(-2*a^4*b^8 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)))

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IntegrateAlgebraic [A]  time = 1.56, size = 142, normalized size = 1.01 \begin {gather*} \frac {1}{4} \text {RootSum}\left [-a^8+a^8 b^8-a^4 c+2 a^4 \text {$\#$1}^4+c \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {a^4 \log (x)-a^4 \log \left (\sqrt [4]{-b^8+a^4 x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b^8+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 a^4 \text {$\#$1}-c \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[-a^8 + a^8*b^8 - a^4*c + 2*a^4*#1^4 + c*#1^4 - #1^8 & , (a^4*Log[x] - a^4*Log[(-b^8 + a^4*x^4)^(1/4) -
x*#1] + Log[x]*#1^4 - Log[(-b^8 + a^4*x^4)^(1/4) - x*#1]*#1^4)/(-2*a^4*#1 - c*#1 + 2*#1^5) & ]/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((2*a^4*x^4-b^8)/(a^4*x^4-b^8)^(1/4)/(a^8*x^8-b^8-c*x^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 {b^{8} - 2 \, a^{4} x^{4}}{{\left (a^{8} x^{8} - b^{8} - c x^{4}\right )} {\left (-b^{8} + a^{4} x^{4}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(b^8 - 2*a^4*x^4)/((a^8*x^8 - b^8 - c*x^4)*(-b^8 + a^4*x^4)^(1/4)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {2 a^{4} x^{4}-b^{8}}{\left (a^{4} x^{4}-b^{8}\right )^{\frac {1}{4}} \left (a^{8} x^{8}-b^{8}-c \,x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((b^8 - 2*a^4*x^4)/((a^8*x^8 - b^8 - c*x^4)*(-b^8 + a^4*x^4)^(1/4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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