\(\int \frac {1}{\sqrt [4]{-b+a x^4} (-b-x^4+a x^8)} \, dx\) [1724]

Optimal result

Integrand size = 30, antiderivative size = 116 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a+a^2-a b-\text {$\#$1}^4-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-2 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(447\) vs. \(2(116)=232\).

Time = 0.25 (sec) , antiderivative size = 447, normalized size of antiderivative = 3.85, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1442, 385, 218, 214, 211} \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\frac {a^{3/4} \arctan \left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}-\frac {a^{3/4} \arctan \left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}+\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}-\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{\sqrt {4 a b+1} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}} \]

[In]

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

[Out]

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

Rule 211

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

Rule 214

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

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*} \text {integral}& = \frac {(2 a) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-1-\sqrt {1+4 a b}+2 a x^4\right )} \, dx}{\sqrt {1+4 a b}}-\frac {(2 a) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-1+\sqrt {1+4 a b}+2 a x^4\right )} \, dx}{\sqrt {1+4 a b}} \\ & = \frac {(2 a) \text {Subst}\left (\int \frac {1}{-1-\sqrt {1+4 a b}-\left (2 a b+a \left (-1-\sqrt {1+4 a b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b}}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{-1+\sqrt {1+4 a b}-\left (2 a b+a \left (-1+\sqrt {1+4 a b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b}} \\ & = \frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1-\sqrt {1+4 a b}}-\sqrt {a} \sqrt {1-2 b-\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1-\sqrt {1+4 a b}}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1-\sqrt {1+4 a b}}+\sqrt {a} \sqrt {1-2 b-\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1-\sqrt {1+4 a b}}}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {1+4 a b}}-\sqrt {a} \sqrt {1-2 b+\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1+\sqrt {1+4 a b}}}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {1+4 a b}}+\sqrt {a} \sqrt {1-2 b+\sqrt {1+4 a b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \sqrt {1+\sqrt {1+4 a b}}} \\ & = \frac {a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b-\sqrt {1+4 a b}} x}{\sqrt [4]{1-\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1-\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b-\sqrt {1+4 a b}}}-\frac {a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b+\sqrt {1+4 a b}} x}{\sqrt [4]{1+\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1+\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b+\sqrt {1+4 a b}}}+\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b-\sqrt {1+4 a b}} x}{\sqrt [4]{1-\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1-\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b-\sqrt {1+4 a b}}}-\frac {a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{1-2 b+\sqrt {1+4 a b}} x}{\sqrt [4]{1+\sqrt {1+4 a b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt {1+4 a b} \left (1+\sqrt {1+4 a b}\right )^{3/4} \sqrt [4]{1-2 b+\sqrt {1+4 a b}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a+a^2-a b-\text {$\#$1}^4-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-2 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-2 a -1\right ) \textit {\_Z}^{4}+a^{2}-a b +a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-2 a -1\right )}}{4 b}\)	\(75\)

[In]

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

[Out]

1/4*sum(1/_R*(_R^4-a)*ln((-_R*x+(a*x^4-b)^(1/4))/x)/(2*_R^4-2*a-1),_R=RootOf(_Z^8+(-2*a-1)*_Z^4+a^2-a*b+a))/b

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 7.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} - b} \left (a x^{8} - b - x^{4}\right )}\, dx \]

[In]

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

[Out]

Integral(1/((a*x**4 - b)**(1/4)*(a*x**8 - b - x**4)), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} - x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} - x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\int \frac {1}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-a\,x^8+x^4+b\right )} \,d x \]

[In]

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

[Out]

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