3.25.42 \(\int \frac {b-2 x^4+a x^8}{\sqrt [4]{-b+a x^4} (b-2 x^4+2 a x^8)} \, dx\)
Optimal. Leaf size=198 \[ \frac {1}{16} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+2 \text {$\#$1}^4+a^2+2 a b-2 a\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+a \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-2 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-a \log (x)+2 \log (x)}{\text {$\#$1}^5-\text {$\#$1} a+\text {$\#$1}}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [4]{a}} \]
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Rubi [B] time = 1.53, antiderivative size = 555, normalized size of antiderivative = 2.80,
number of steps used = 16, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used
= {6728, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [4]{a}}+\frac {\left (1-\frac {1-a b}{\sqrt {1-2 a b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {1-2 a b}-2 b+1}}{\sqrt [4]{1-\sqrt {1-2 a b}} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt [4]{a} \left (1-\sqrt {1-2 a b}\right )^{3/4} \sqrt [4]{-\sqrt {1-2 a b}-2 b+1}}+\frac {\left (\frac {1-a b}{\sqrt {1-2 a b}}+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {1-2 a b}-2 b+1}}{\sqrt [4]{\sqrt {1-2 a b}+1} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt [4]{a} \left (\sqrt {1-2 a b}+1\right )^{3/4} \sqrt [4]{\sqrt {1-2 a b}-2 b+1}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [4]{a}}+\frac {\left (1-\frac {1-a b}{\sqrt {1-2 a b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {1-2 a b}-2 b+1}}{\sqrt [4]{1-\sqrt {1-2 a b}} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt [4]{a} \left (1-\sqrt {1-2 a b}\right )^{3/4} \sqrt [4]{-\sqrt {1-2 a b}-2 b+1}}+\frac {\left (\frac {1-a b}{\sqrt {1-2 a b}}+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {1-2 a b}-2 b+1}}{\sqrt [4]{\sqrt {1-2 a b}+1} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt [4]{a} \left (\sqrt {1-2 a b}+1\right )^{3/4} \sqrt [4]{\sqrt {1-2 a b}-2 b+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(b - 2*x^4 + a*x^8)/((-b + a*x^4)^(1/4)*(b - 2*x^4 + 2*a*x^8)),x]
[Out]
ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(4*a^(1/4)) + ((1 - (1 - a*b)/Sqrt[1 - 2*a*b])*ArcTan[(a^(1/4)*(1 - 2*b
- Sqrt[1 - 2*a*b])^(1/4)*x)/((1 - Sqrt[1 - 2*a*b])^(1/4)*(-b + a*x^4)^(1/4))])/(4*a^(1/4)*(1 - Sqrt[1 - 2*a*b
])^(3/4)*(1 - 2*b - Sqrt[1 - 2*a*b])^(1/4)) + ((1 + (1 - a*b)/Sqrt[1 - 2*a*b])*ArcTan[(a^(1/4)*(1 - 2*b + Sqrt
[1 - 2*a*b])^(1/4)*x)/((1 + Sqrt[1 - 2*a*b])^(1/4)*(-b + a*x^4)^(1/4))])/(4*a^(1/4)*(1 + Sqrt[1 - 2*a*b])^(3/4
)*(1 - 2*b + Sqrt[1 - 2*a*b])^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(4*a^(1/4)) + ((1 - (1 - a*b)/S
qrt[1 - 2*a*b])*ArcTanh[(a^(1/4)*(1 - 2*b - Sqrt[1 - 2*a*b])^(1/4)*x)/((1 - Sqrt[1 - 2*a*b])^(1/4)*(-b + a*x^4
)^(1/4))])/(4*a^(1/4)*(1 - Sqrt[1 - 2*a*b])^(3/4)*(1 - 2*b - Sqrt[1 - 2*a*b])^(1/4)) + ((1 + (1 - a*b)/Sqrt[1
- 2*a*b])*ArcTanh[(a^(1/4)*(1 - 2*b + Sqrt[1 - 2*a*b])^(1/4)*x)/((1 + Sqrt[1 - 2*a*b])^(1/4)*(-b + a*x^4)^(1/4
))])/(4*a^(1/4)*(1 + Sqrt[1 - 2*a*b])^(3/4)*(1 - 2*b + Sqrt[1 - 2*a*b])^(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 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 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 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 240
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]
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-2 x^4+a x^8}{\sqrt [4]{-b+a x^4} \left (b-2 x^4+2 a x^8\right )} \, dx &=\int \left (\frac {1}{2 \sqrt [4]{-b+a x^4}}+\frac {b-2 x^4}{2 \sqrt [4]{-b+a x^4} \left (b-2 x^4+2 a x^8\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx+\frac {1}{2} \int \frac {b-2 x^4}{\sqrt [4]{-b+a x^4} \left (b-2 x^4+2 a x^8\right )} \, dx\\ &=\frac {1}{2} \int \left (\frac {-2+\frac {2 (-1+a b)}{\sqrt {1-2 a b}}}{\sqrt [4]{-b+a x^4} \left (-2-2 \sqrt {1-2 a b}+4 a x^4\right )}+\frac {-2-\frac {2 (-1+a b)}{\sqrt {1-2 a b}}}{\sqrt [4]{-b+a x^4} \left (-2+2 \sqrt {1-2 a b}+4 a x^4\right )}\right ) \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (-1-\frac {1-a b}{\sqrt {1-2 a b}}\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-2-2 \sqrt {1-2 a b}+4 a x^4\right )} \, dx+\left (-1+\frac {1-a b}{\sqrt {1-2 a b}}\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-2+2 \sqrt {1-2 a b}+4 a x^4\right )} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a}}+\left (-1-\frac {1-a b}{\sqrt {1-2 a b}}\right ) \operatorname {Subst}\left (\int \frac {1}{-2-2 \sqrt {1-2 a b}-\left (4 a b+a \left (-2-2 \sqrt {1-2 a b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (-1+\frac {1-a b}{\sqrt {1-2 a b}}\right ) \operatorname {Subst}\left (\int \frac {1}{-2+2 \sqrt {1-2 a b}-\left (4 a b+a \left (-2+2 \sqrt {1-2 a b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a}}+\frac {\left (1-\frac {1-a b}{\sqrt {1-2 a b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\sqrt {1-2 a b}}-\sqrt {a} \sqrt {1-2 b-\sqrt {1-2 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {1-\sqrt {1-2 a b}}}+\frac {\left (1-\frac {1-a b}{\sqrt {1-2 a b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\sqrt {1-2 a b}}+\sqrt {a} \sqrt {1-2 b-\sqrt {1-2 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {1-\sqrt {1-2 a b}}}+\frac {\left (1+\frac {1-a b}{\sqrt {1-2 a b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {1-2 a b}}-\sqrt {a} \sqrt {1-2 b+\sqrt {1-2 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {1+\sqrt {1-2 a b}}}+\frac {\left (1+\frac {1-a b}{\sqrt {1-2 a b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {1-2 a b}}+\sqrt {a} \sqrt {1-2 b+\sqrt {1-2 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {1+\sqrt {1-2 a b}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a}}+\frac {\left (1-\frac {1-a b}{\sqrt {1-2 a b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b-\sqrt {1-2 a b}} x}{\sqrt [4]{1-\sqrt {1-2 a b}} \sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a} \left (1-\sqrt {1-2 a b}\right )^{3/4} \sqrt [4]{1-2 b-\sqrt {1-2 a b}}}+\frac {\left (1+\frac {1-a b}{\sqrt {1-2 a b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b+\sqrt {1-2 a b}} x}{\sqrt [4]{1+\sqrt {1-2 a b}} \sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a} \left (1+\sqrt {1-2 a b}\right )^{3/4} \sqrt [4]{1-2 b+\sqrt {1-2 a b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a}}+\frac {\left (1-\frac {1-a b}{\sqrt {1-2 a b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b-\sqrt {1-2 a b}} x}{\sqrt [4]{1-\sqrt {1-2 a b}} \sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a} \left (1-\sqrt {1-2 a b}\right )^{3/4} \sqrt [4]{1-2 b-\sqrt {1-2 a b}}}+\frac {\left (1+\frac {1-a b}{\sqrt {1-2 a b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b+\sqrt {1-2 a b}} x}{\sqrt [4]{1+\sqrt {1-2 a b}} \sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a} \left (1+\sqrt {1-2 a b}\right )^{3/4} \sqrt [4]{1-2 b+\sqrt {1-2 a b}}}\\ \end {align*}
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Mathematica [B] time = 0.52, size = 524, normalized size = 2.65 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+\frac {\left (\frac {a b-1}{\sqrt {1-2 a b}}+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {1-2 a b}-2 b+1}}{\sqrt [4]{1-\sqrt {1-2 a b}} \sqrt [4]{a x^4-b}}\right )}{\left (1-\sqrt {1-2 a b}\right )^{3/4} \sqrt [4]{-\sqrt {1-2 a b}-2 b+1}}+\frac {\left (-a b+\sqrt {1-2 a b}+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {1-2 a b}-2 b+1}}{\sqrt [4]{\sqrt {1-2 a b}+1} \sqrt [4]{a x^4-b}}\right )}{\sqrt {1-2 a b} \left (\sqrt {1-2 a b}+1\right )^{3/4} \sqrt [4]{\sqrt {1-2 a b}-2 b+1}}+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+\frac {\left (\frac {a b-1}{\sqrt {1-2 a b}}+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {1-2 a b}-2 b+1}}{\sqrt [4]{1-\sqrt {1-2 a b}} \sqrt [4]{a x^4-b}}\right )}{\left (1-\sqrt {1-2 a b}\right )^{3/4} \sqrt [4]{-\sqrt {1-2 a b}-2 b+1}}+\frac {\left (-a b+\sqrt {1-2 a b}+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {1-2 a b}-2 b+1}}{\sqrt [4]{\sqrt {1-2 a b}+1} \sqrt [4]{a x^4-b}}\right )}{\sqrt {1-2 a b} \left (\sqrt {1-2 a b}+1\right )^{3/4} \sqrt [4]{\sqrt {1-2 a b}-2 b+1}}}{4 \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(b - 2*x^4 + a*x^8)/((-b + a*x^4)^(1/4)*(b - 2*x^4 + 2*a*x^8)),x]
[Out]
(ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] + ((1 + (-1 + a*b)/Sqrt[1 - 2*a*b])*ArcTan[(a^(1/4)*(1 - 2*b - Sqrt[1
- 2*a*b])^(1/4)*x)/((1 - Sqrt[1 - 2*a*b])^(1/4)*(-b + a*x^4)^(1/4))])/((1 - Sqrt[1 - 2*a*b])^(3/4)*(1 - 2*b -
Sqrt[1 - 2*a*b])^(1/4)) + ((1 - a*b + Sqrt[1 - 2*a*b])*ArcTan[(a^(1/4)*(1 - 2*b + Sqrt[1 - 2*a*b])^(1/4)*x)/((
1 + Sqrt[1 - 2*a*b])^(1/4)*(-b + a*x^4)^(1/4))])/(Sqrt[1 - 2*a*b]*(1 + Sqrt[1 - 2*a*b])^(3/4)*(1 - 2*b + Sqrt[
1 - 2*a*b])^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] + ((1 + (-1 + a*b)/Sqrt[1 - 2*a*b])*ArcTanh[(a^(1
/4)*(1 - 2*b - Sqrt[1 - 2*a*b])^(1/4)*x)/((1 - Sqrt[1 - 2*a*b])^(1/4)*(-b + a*x^4)^(1/4))])/((1 - Sqrt[1 - 2*a
*b])^(3/4)*(1 - 2*b - Sqrt[1 - 2*a*b])^(1/4)) + ((1 - a*b + Sqrt[1 - 2*a*b])*ArcTanh[(a^(1/4)*(1 - 2*b + Sqrt[
1 - 2*a*b])^(1/4)*x)/((1 + Sqrt[1 - 2*a*b])^(1/4)*(-b + a*x^4)^(1/4))])/(Sqrt[1 - 2*a*b]*(1 + Sqrt[1 - 2*a*b])
^(3/4)*(1 - 2*b + Sqrt[1 - 2*a*b])^(1/4)))/(4*a^(1/4))
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IntegrateAlgebraic [A] time = 0.00, size = 198, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{a}}+\frac {1}{16} \text {RootSum}\left [-2 a+a^2+2 a b+2 \text {$\#$1}^4-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 \log (x)-a \log (x)-2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{\text {$\#$1}-a \text {$\#$1}+\text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(b - 2*x^4 + a*x^8)/((-b + a*x^4)^(1/4)*(b - 2*x^4 + 2*a*x^8)),x]
[Out]
ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(4*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(4*a^(1/4)) + Roo
tSum[-2*a + a^2 + 2*a*b + 2*#1^4 - 2*a*#1^4 + #1^8 & , (2*Log[x] - a*Log[x] - 2*Log[(-b + a*x^4)^(1/4) - x*#1]
+ a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(#1 - a*#1 + #1^5) &
]/16
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fricas [B] time = 2.21, size = 10215, normalized size = 51.59
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x^8-2*x^4+b)/(a*x^4-b)^(1/4)/(2*a*x^8-2*x^4+b),x, algorithm="fricas")
[Out]
-1/2*sqrt(1/2)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b + (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2
*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^
3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 +
12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a
^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(1
/4)*arctan(sqrt(1/2)*(((8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*x*sqrt(
-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/
(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*
(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) - (2*a^3*b^3 + (
4*a^3 - 13*a^2)*b^2 - 2*(3*a^2 - 7*a)*b + 2*a - 4)*x)*sqrt(((a^10*b^10 + 4*(a^10 - 3*a^9)*b^9 + 4*(a^10 - 7*a^
9 + 11*a^8)*b^8 - 8*(a^9 - 5*a^8 + 6*a^7)*b^7 + 4*(a^8 - 4*a^7 + 4*a^6)*b^6)*sqrt(a*x^4 - b) - ((16*a^11*b^11
+ 8*(a^12 + 4*a^11 - 19*a^10)*b^10 + 4*(6*a^12 - 23*a^11 - 28*a^10 + 115*a^9)*b^9 + 2*(8*a^12 - 82*a^11 + 207*
a^10 - 4*a^9 - 305*a^8)*b^8 - (40*a^11 - 306*a^10 + 685*a^9 - 268*a^8 - 400*a^7)*b^7 + (36*a^10 - 243*a^9 + 51
4*a^8 - 280*a^7 - 128*a^6)*b^6 - 2*(7*a^9 - 44*a^8 + 90*a^7 - 56*a^6 - 8*a^5)*b^5 + 2*(a^8 - 6*a^7 + 12*a^6 -
8*a^5)*b^4)*x^2*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 +
6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*
a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)
*b)) - (2*(a^11 - 3*a^10)*b^10 + (8*a^11 - 51*a^10 + 79*a^9)*b^9 + (8*a^11 - 92*a^10 + 321*a^9 - 350*a^8)*b^8
- 4*(7*a^10 - 61*a^9 + 171*a^8 - 155*a^7)*b^7 + 4*(9*a^9 - 67*a^8 + 163*a^7 - 130*a^6)*b^6 - 4*(5*a^8 - 33*a^7
+ 72*a^6 - 52*a^5)*b^5 + 4*(a^7 - 6*a^6 + 12*a^5 - 8*a^4)*b^4)*x^2)*sqrt(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)
*b + (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 -
3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*
a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*
a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a
^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)))/x^2) + (2*a^8*b^8 + (8*a^8 - 25*a^7)*b^7 + 4*(2*a^8 -
15*a^7 + 25*a^6)*b^6 - 4*(5*a^7 - 27*a^6 + 35*a^5)*b^5 + 8*(2*a^6 - 9*a^5 + 10*a^4)*b^4 - 4*(a^5 - 4*a^4 + 4*
a^3)*b^3 - (8*a^8*b^8 + 4*(a^9 + 2*a^8 - 14*a^7)*b^7 + 2*(4*a^9 - 22*a^8 + 12*a^7 + 41*a^6)*b^6 - (16*a^8 - 73
*a^7 + 62*a^6 + 44*a^5)*b^5 + 2*(5*a^7 - 21*a^6 + 20*a^5 + 4*a^4)*b^4 - 2*(a^6 - 4*a^5 + 4*a^4)*b^3)*sqrt(-(a^
4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*
a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a
^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)))*(a*x^4 - b)^(1/4))*
(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b + (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 -
a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a
)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)
*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b))
+ 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(1/4)/((a^8*b^8 +
4*(a^8 - 3*a^7)*b^7 + 4*(a^8 - 7*a^7 + 11*a^6)*b^6 - 8*(a^7 - 5*a^6 + 6*a^5)*b^5 + 4*(a^6 - 4*a^5 + 4*a^4)*b^
4)*x)) + 1/2*sqrt(1/2)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b - (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^
2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2
- 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 -
2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^
4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b -
2*a))^(1/4)*arctan((sqrt(1/2)*((8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)
*x*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*
a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*
a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + (2*a^3
*b^3 + (4*a^3 - 13*a^2)*b^2 - 2*(3*a^2 - 7*a)*b + 2*a - 4)*x)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b - (8*a^3
*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3
+ 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5
- 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^
3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)
*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(1/4)*sqrt(((a^10*b^10 + 4*(a^10 - 3*a^9)*b^9 + 4*(a^10 - 7*a^9 +
11*a^8)*b^8 - 8*(a^9 - 5*a^8 + 6*a^7)*b^7 + 4*(a^8 - 4*a^7 + 4*a^6)*b^6)*sqrt(a*x^4 - b) + ((16*a^11*b^11 + 8
*(a^12 + 4*a^11 - 19*a^10)*b^10 + 4*(6*a^12 - 23*a^11 - 28*a^10 + 115*a^9)*b^9 + 2*(8*a^12 - 82*a^11 + 207*a^1
0 - 4*a^9 - 305*a^8)*b^8 - (40*a^11 - 306*a^10 + 685*a^9 - 268*a^8 - 400*a^7)*b^7 + (36*a^10 - 243*a^9 + 514*a
^8 - 280*a^7 - 128*a^6)*b^6 - 2*(7*a^9 - 44*a^8 + 90*a^7 - 56*a^6 - 8*a^5)*b^5 + 2*(a^8 - 6*a^7 + 12*a^6 - 8*a
^5)*b^4)*x^2*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*
a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3
)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)
) + (2*(a^11 - 3*a^10)*b^10 + (8*a^11 - 51*a^10 + 79*a^9)*b^9 + (8*a^11 - 92*a^10 + 321*a^9 - 350*a^8)*b^8 - 4
*(7*a^10 - 61*a^9 + 171*a^8 - 155*a^7)*b^7 + 4*(9*a^9 - 67*a^8 + 163*a^7 - 130*a^6)*b^6 - 4*(5*a^8 - 33*a^7 +
72*a^6 - 52*a^5)*b^5 + 4*(a^7 - 6*a^6 + 12*a^5 - 8*a^4)*b^4)*x^2)*sqrt(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b
- (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*
a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6
- 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4
+ 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3
- 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)))/x^2) - sqrt(1/2)*(2*a^8*b^8 + (8*a^8 - 25*a^7)*b^7 + 4*(
2*a^8 - 15*a^7 + 25*a^6)*b^6 - 4*(5*a^7 - 27*a^6 + 35*a^5)*b^5 + 8*(2*a^6 - 9*a^5 + 10*a^4)*b^4 - 4*(a^5 - 4*a
^4 + 4*a^3)*b^3 + (8*a^8*b^8 + 4*(a^9 + 2*a^8 - 14*a^7)*b^7 + 2*(4*a^9 - 22*a^8 + 12*a^7 + 41*a^6)*b^6 - (16*a
^8 - 73*a^7 + 62*a^6 + 44*a^5)*b^5 + 2*(5*a^7 - 21*a^6 + 20*a^5 + 4*a^4)*b^4 - 2*(a^6 - 4*a^5 + 4*a^4)*b^3)*sq
rt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 1
6)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 -
4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)))*(a*x^4 - b)^
(1/4)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b - (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*
a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2
+ 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 +
3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^
2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(1/4))/((a^
8*b^8 + 4*(a^8 - 3*a^7)*b^7 + 4*(a^8 - 7*a^7 + 11*a^6)*b^6 - 8*(a^7 - 5*a^6 + 6*a^5)*b^5 + 4*(a^6 - 4*a^5 + 4*
a^4)*b^4)*x)) - 1/8*sqrt(1/2)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b + (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b
^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2
+ 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4
*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5
- 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 -
a)*b - 2*a))^(1/4)*log((sqrt(1/2)*((32*a^6*b^6 + 16*(a^7 + a^6 - 11*a^5)*b^5 + 8*(3*a^7 - 17*a^6 + 9*a^5 + 35*
a^4)*b^4 - 4*(13*a^6 - 61*a^5 + 49*a^4 + 49*a^3)*b^3 + 2*a^3 + 2*(21*a^5 - 91*a^4 + 83*a^3 + 32*a^2)*b^2 - 8*a
^2 - (15*a^4 - 62*a^3 + 60*a^2 + 8*a)*b + 8*a)*x*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^
2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(
a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*
(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) - (4*(a^6 - 4*a^5)*b^5 + 4*(2*a^6 - 17*a^5 + 32*a^4)*b^4 - (32*a^5 - 193
*a^4 + 276*a^3)*b^3 + 2*(21*a^4 - 104*a^3 + 126*a^2)*b^2 + 4*a^2 - 2*(11*a^3 - 48*a^2 + 52*a)*b - 16*a + 16)*x
)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b + (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2
- a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6
*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^
3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b
)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(3/4) + (a^5*b^
5 + 2*(a^5 - 3*a^4)*b^4 - 2*(a^4 - 2*a^3)*b^3)*(a*x^4 - b)^(1/4))/x) + 1/8*sqrt(1/2)*(((a^3 - 5*a^2)*b^2 - 2*(
2*a^2 - 5*a)*b + (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^
4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*
b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 -
12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 +
4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(1/4)*log(-(sqrt(1/2)*((32*a^6*b^6 + 16*(a
^7 + a^6 - 11*a^5)*b^5 + 8*(3*a^7 - 17*a^6 + 9*a^5 + 35*a^4)*b^4 - 4*(13*a^6 - 61*a^5 + 49*a^4 + 49*a^3)*b^3 +
2*a^3 + 2*(21*a^5 - 91*a^4 + 83*a^3 + 32*a^2)*b^2 - 8*a^2 - (15*a^4 - 62*a^3 + 60*a^2 + 8*a)*b + 8*a)*x*sqrt(
-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/
(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*
(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) - (4*(a^6 - 4*a^
5)*b^5 + 4*(2*a^6 - 17*a^5 + 32*a^4)*b^4 - (32*a^5 - 193*a^4 + 276*a^3)*b^3 + 2*(21*a^4 - 104*a^3 + 126*a^2)*b
^2 + 4*a^2 - 2*(11*a^3 - 48*a^2 + 52*a)*b - 16*a + 16)*x)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b + (8*a^3*b^3
+ 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4
*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3
*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 +
a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2
+ a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(3/4) - (a^5*b^5 + 2*(a^5 - 3*a^4)*b^4 - 2*(a^4 - 2*a^3)*b^3)*(a*x^4
- b)^(1/4))/x) + 1/8*sqrt(1/2)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b - (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*
b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2
+ 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 -
4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5
- 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 -
a)*b - 2*a))^(1/4)*log((sqrt(1/2)*((32*a^6*b^6 + 16*(a^7 + a^6 - 11*a^5)*b^5 + 8*(3*a^7 - 17*a^6 + 9*a^5 + 35
*a^4)*b^4 - 4*(13*a^6 - 61*a^5 + 49*a^4 + 49*a^3)*b^3 + 2*a^3 + 2*(21*a^5 - 91*a^4 + 83*a^3 + 32*a^2)*b^2 - 8*
a^2 - (15*a^4 - 62*a^3 + 60*a^2 + 8*a)*b + 8*a)*x*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a
^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*
(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2
*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + (4*(a^6 - 4*a^5)*b^5 + 4*(2*a^6 - 17*a^5 + 32*a^4)*b^4 - (32*a^5 - 19
3*a^4 + 276*a^3)*b^3 + 2*(21*a^4 - 104*a^3 + 126*a^2)*b^2 + 4*a^2 - 2*(11*a^3 - 48*a^2 + 52*a)*b - 16*a + 16)*
x)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b - (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2
- a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 +
6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a
^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*
b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(3/4) + (a^5*b
^5 + 2*(a^5 - 3*a^4)*b^4 - 2*(a^4 - 2*a^3)*b^3)*(a*x^4 - b)^(1/4))/x) - 1/8*sqrt(1/2)*(((a^3 - 5*a^2)*b^2 - 2*
(2*a^2 - 5*a)*b - (8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b
^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5
*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6
- 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 +
4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(1/4)*log(-(sqrt(1/2)*((32*a^6*b^6 + 16*(
a^7 + a^6 - 11*a^5)*b^5 + 8*(3*a^7 - 17*a^6 + 9*a^5 + 35*a^4)*b^4 - 4*(13*a^6 - 61*a^5 + 49*a^4 + 49*a^3)*b^3
+ 2*a^3 + 2*(21*a^5 - 91*a^4 + 83*a^3 + 32*a^2)*b^2 - 8*a^2 - (15*a^4 - 62*a^3 + 60*a^2 + 8*a)*b + 8*a)*x*sqrt
(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 + 4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)
/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 - 3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4
*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 + a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + (4*(a^6 - 4*a
^5)*b^5 + 4*(2*a^6 - 17*a^5 + 32*a^4)*b^4 - (32*a^5 - 193*a^4 + 276*a^3)*b^3 + 2*(21*a^4 - 104*a^3 + 126*a^2)*
b^2 + 4*a^2 - 2*(11*a^3 - 48*a^2 + 52*a)*b - 16*a + 16)*x)*(((a^3 - 5*a^2)*b^2 - 2*(2*a^2 - 5*a)*b - (8*a^3*b^
3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a)*sqrt(-(a^4*b^4 + 4*(a^4 - 3*a^3)*b^3 +
4*(a^4 - 7*a^3 + 11*a^2)*b^2 + 4*a^2 - 8*(a^3 - 5*a^2 + 6*a)*b - 16*a + 16)/(32*a^5*b^5 + 16*(2*a^6 - 4*a^5 -
3*a^4)*b^4 - a^4 + 8*(a^7 - 4*a^6 - 2*a^5 + 12*a^4 + 3*a^3)*b^3 + 4*a^3 - 4*(3*a^6 - 12*a^5 + 6*a^4 + 12*a^3 +
a^2)*b^2 - 4*a^2 + 2*(3*a^5 - 12*a^4 + 10*a^3 + 4*a^2)*b)) + 2*a - 4)/(8*a^3*b^3 + 4*(a^4 - 2*a^3 - 2*a^2)*b^
2 + a^2 - 2*(2*a^3 - 4*a^2 - a)*b - 2*a))^(3/4) - (a^5*b^5 + 2*(a^5 - 3*a^4)*b^4 - 2*(a^4 - 2*a^3)*b^3)*(a*x^4
- b)^(1/4))/x) + 1/2*arctan((x*sqrt((sqrt(a)*x^2 + sqrt(a*x^4 - b))/x^2)/a^(1/4) - (a*x^4 - b)^(1/4)/a^(1/4))
/x)/a^(1/4) + 1/8*log((a^(1/4)*x + (a*x^4 - b)^(1/4))/x)/a^(1/4) - 1/8*log(-(a^(1/4)*x - (a*x^4 - b)^(1/4))/x)
/a^(1/4)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{8} - 2 \, x^{4} + b}{{\left (2 \, a x^{8} - 2 \, x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x^8-2*x^4+b)/(a*x^4-b)^(1/4)/(2*a*x^8-2*x^4+b),x, algorithm="giac")
[Out]
integrate((a*x^8 - 2*x^4 + b)/((2*a*x^8 - 2*x^4 + b)*(a*x^4 - b)^(1/4)), x)
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{8}-2 x^{4}+b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (2 a \,x^{8}-2 x^{4}+b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x^8-2*x^4+b)/(a*x^4-b)^(1/4)/(2*a*x^8-2*x^4+b),x)
[Out]
int((a*x^8-2*x^4+b)/(a*x^4-b)^(1/4)/(2*a*x^8-2*x^4+b),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{8} - 2 \, x^{4} + b}{{\left (2 \, a x^{8} - 2 \, x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x^8-2*x^4+b)/(a*x^4-b)^(1/4)/(2*a*x^8-2*x^4+b),x, algorithm="maxima")
[Out]
integrate((a*x^8 - 2*x^4 + b)/((2*a*x^8 - 2*x^4 + b)*(a*x^4 - b)^(1/4)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x^8-2\,x^4+b}{{\left (a\,x^4-b\right )}^{1/4}\,\left (2\,a\,x^8-2\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b + a*x^8 - 2*x^4)/((a*x^4 - b)^(1/4)*(b + 2*a*x^8 - 2*x^4)),x)
[Out]
int((b + a*x^8 - 2*x^4)/((a*x^4 - b)^(1/4)*(b + 2*a*x^8 - 2*x^4)), 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((a*x**8-2*x**4+b)/(a*x**4-b)**(1/4)/(2*a*x**8-2*x**4+b),x)
[Out]
Timed out
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