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

Optimal. Leaf size=143 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8 b^4-2 \text {$\#$1}^4 a^4 b^4-\text {$\#$1}^4 c+a^4 c\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a^4 x^4-b^4}-\text {$\#$1} x\right )-\text {$\#$1}^4 \log (x)-a^4 \log \left (\sqrt [4]{a^4 x^4-b^4}-\text {$\#$1} x\right )+a^4 \log (x)}{-2 \text {$\#$1}^5 b^4+2 \text {$\#$1} a^4 b^4+\text {$\#$1} c}\& \right ]}{4 b^4} \]

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Rubi [B]  time = 1.06, antiderivative size = 601, normalized size of antiderivative = 4.20, number of steps used = 9, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1428, 377, 212, 208, 205} \begin {gather*} -\frac {a^7 \tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\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^4-c}}-\frac {a^7 \tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\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^4+c}}-\frac {a^7 \tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\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^4-c}}-\frac {a^7 \tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\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^4+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^7*ArcTan[(a*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 +
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^4 - c + Sqrt[4*a^8*b^8 +
c^2])^(1/4))) - (a^7*ArcTan[(a*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((c + Sqrt[4*a^8*b^8 + c^2])^
(1/4)*(-b^4 + 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^4 + c + Sqr
t[4*a^8*b^8 + c^2])^(1/4)) - (a^7*ArcTanh[(a*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c + Sqrt[4*a^
8*b^8 + c^2])^(1/4)*(-b^4 + 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^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) - (a^7*ArcTanh[(a*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/(
(c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 + 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^4 + 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 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^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx &=\frac {\left (2 a^8\right ) \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-c-\sqrt {4 a^8 b^8+c^2}+2 a^8 x^4\right )} \, dx}{\sqrt {4 a^8 b^8+c^2}}-\frac {\left (2 a^8\right ) \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-c+\sqrt {4 a^8 b^8+c^2}+2 a^8 x^4\right )} \, dx}{\sqrt {4 a^8 b^8+c^2}}\\ &=\frac {\left (2 a^8\right ) \operatorname {Subst}\left (\int \frac {1}{-c-\sqrt {4 a^8 b^8+c^2}-\left (2 a^8 b^4+a^4 \left (-c-\sqrt {4 a^8 b^8+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2}}-\frac {\left (2 a^8\right ) \operatorname {Subst}\left (\int \frac {1}{-c+\sqrt {4 a^8 b^8+c^2}-\left (2 a^8 b^4+a^4 \left (-c+\sqrt {4 a^8 b^8+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2}}\\ &=-\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {4 a^8 b^8+c^2}}-a^2 \sqrt {2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \sqrt {-c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {4 a^8 b^8+c^2}}+a^2 \sqrt {2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \sqrt {-c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {4 a^8 b^8+c^2}}-a^2 \sqrt {-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \sqrt {c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {4 a^8 b^8+c^2}}+a^2 \sqrt {-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \sqrt {c+\sqrt {4 a^8 b^8+c^2}}}\\ &=-\frac {a^7 \tan ^{-1}\left (\frac {a \sqrt [4]{2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x}{\sqrt [4]{-c+\sqrt {4 a^8 b^8+c^2}} \sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (-c+\sqrt {4 a^8 b^8+c^2}\right )^{3/4} \sqrt [4]{2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^7 \tan ^{-1}\left (\frac {a \sqrt [4]{-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x}{\sqrt [4]{c+\sqrt {4 a^8 b^8+c^2}} \sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (c+\sqrt {4 a^8 b^8+c^2}\right )^{3/4} \sqrt [4]{-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^7 \tanh ^{-1}\left (\frac {a \sqrt [4]{2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x}{\sqrt [4]{-c+\sqrt {4 a^8 b^8+c^2}} \sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (-c+\sqrt {4 a^8 b^8+c^2}\right )^{3/4} \sqrt [4]{2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}}}-\frac {a^7 \tanh ^{-1}\left (\frac {a \sqrt [4]{-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x}{\sqrt [4]{c+\sqrt {4 a^8 b^8+c^2}} \sqrt [4]{-b^4+a^4 x^4}}\right )}{\sqrt {4 a^8 b^8+c^2} \left (c+\sqrt {4 a^8 b^8+c^2}\right )^{3/4} \sqrt [4]{-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}}}\\ \end {align*}

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Mathematica [B]  time = 1.08, size = 542, normalized size = 3.79 \begin {gather*} -\frac {a^7 \left (\frac {\tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\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^4-c}}+\frac {\tan ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\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^4+c}}+\frac {\tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\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^4-c}}+\frac {\tanh ^{-1}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\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^4+c}}\right )}{\sqrt {4 a^8 b^8+c^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^7*(ArcTan[(a*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 +
a^4*x^4)^(1/4))]/((-c + Sqrt[4*a^8*b^8 + c^2])^(3/4)*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) + ArcTan[
(a*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 + a^4*x^4)^(1/4)
)]/((c + Sqrt[4*a^8*b^8 + c^2])^(3/4)*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) + ArcTanh[(a*(2*a^4*b^4
- c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 + a^4*x^4)^(1/4))]/((-c + Sqrt
[4*a^8*b^8 + c^2])^(3/4)*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) + ArcTanh[(a*(-2*a^4*b^4 + c + Sqrt[4*
a^8*b^8 + c^2])^(1/4)*x)/((c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 + a^4*x^4)^(1/4))]/((c + Sqrt[4*a^8*b^8 + c^
2])^(3/4)*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4))))/Sqrt[4*a^8*b^8 + c^2])

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IntegrateAlgebraic [A]  time = 0.78, size = 144, normalized size = 1.01 \begin {gather*} -\frac {\text {RootSum}\left [a^4 c-2 a^4 b^4 \text {$\#$1}^4-c \text {$\#$1}^4+b^4 \text {$\#$1}^8\&,\frac {-a^4 \log (x)+a^4 \log \left (\sqrt [4]{-b^4+a^4 x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 a^4 b^4 \text {$\#$1}-c \text {$\#$1}+2 b^4 \text {$\#$1}^5}\&\right ]}{4 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{4} x^{4}-b^{4}\right )^{\frac {1}{4}} \left (a^{8} x^{8}-b^{8}-c \,x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a^4*x^4-b^4)^(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 {1}{{\left (a^{8} x^{8} - b^{8} - c x^{4}\right )} {\left (a^{4} x^{4} - b^{4}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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