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

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

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Rubi [B]  time = 0.65, antiderivative size = 447, normalized size of antiderivative = 3.85, number of steps used = 9, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1428, 377, 212, 208, 205} \begin {gather*} \frac {a^{3/4} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 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 1428

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]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx &=\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) \operatorname {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) \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [B]  time = 0.15, size = 404, normalized size = 3.48 \begin {gather*} -\frac {a^{3/4} \left (-\frac {\tan ^{-1}\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 )}{\left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}+\frac {\tan ^{-1}\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 )}{\left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}-\frac {\tanh ^{-1}\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 )}{\left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}+\frac {\tanh ^{-1}\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 )}{\left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}\right )}{\sqrt {4 a b+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[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))]/((1 - Sqrt[1 + 4*a*b])^(3/4)*(1 - 2*b - Sqrt[1 + 4*a*b])^(1/4))) + 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))]/((1 + Sqrt[1 + 4*a*b])^(3/4)*(1 - 2*b + Sqr
t[1 + 4*a*b])^(1/4)) - 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))]/((1 - Sqrt[1 + 4*a*b])^(3/4)*(1 - 2*b - Sqrt[1 + 4*a*b])^(1/4)) + 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))]/((1 + Sqrt[1 + 4*a*b])^(3/4)*(1 -
2*b + Sqrt[1 + 4*a*b])^(1/4))))/Sqrt[1 + 4*a*b])

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IntegrateAlgebraic [A]  time = 0.00, size = 116, normalized size = 1.00 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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

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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(1/(a*x^4-b)^(1/4)/(a*x^8-x^4-b),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 {1}{{\left (a x^{8} - 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(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{a x^{4} - b} \left (a x^{8} - b - x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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