\(\int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} (-b^8+a^8 x^8)} \, dx\) [2859]

Optimal result

Integrand size = 47, antiderivative size = 299 \[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\frac {\left (1+2 a^4 b^4\right ) x \sqrt {-b^4+a^4 x^4}}{4 a^4 b^4 \left (b^4-a^4 x^4\right )}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (-1+2 a^4 b^4\right ) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a^5 b^5}+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \left (-1+2 a^4 b^4\right ) \text {arctanh}\left (\frac {(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt {-b^4+a^4 x^4}}{2 \sqrt {3-2 \sqrt {2}} a b x}\right )}{a^5 b^5}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \left (-1+2 a^4 b^4\right ) \text {arctanh}\left (\frac {(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt {-b^4+a^4 x^4}}{2 \sqrt {3+2 \sqrt {2}} a b x}\right )}{a^5 b^5} \]

[Out]

1/4*(2*a^4*b^4+1)*x*(a^4*x^4-b^4)^(1/2)/a^4/b^4/(-a^4*x^4+b^4)+(-1/8+1/8*I)*(2*a^4*b^4-1)*arctan((1+I)*a*b*x/(
I*b^2+a^2*x^2+(a^4*x^4-b^4)^(1/2)))/a^5/b^5+(1/16-1/16*I)*(2*a^4*b^4-1)*arctanh(1/2*((1+I)*b^2+(1-I)*a^2*x^2+(
1-I)*(a^4*x^4-b^4)^(1/2))/(2^(1/2)-1)/a/b/x)/a^5/b^5+(-1/16+1/16*I)*(2*a^4*b^4-1)*arctanh(1/2*((1+I)*b^2+(1-I)
*a^2*x^2+(1-I)*(a^4*x^4-b^4)^(1/2))/(1+2^(1/2))/a/b/x)/a^5/b^5

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.51 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.13, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.234, Rules used = {6857, 230, 227, 1469, 541, 537, 418, 1225, 1713, 209, 212} \[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {a^4 x^4-b^4}}-\frac {\left (1-2 a^4 b^4\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \left (-a^4\right )^{5/4} b^5}-\frac {\left (1-2 a^4 b^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \left (-a^4\right )^{5/4} b^5}-\frac {x \left (\frac {1}{a^4 b^4}+2\right )}{4 \sqrt {a^4 x^4-b^4}}+\frac {\left (1-2 a^4 b^4\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a^5 b^3 \sqrt {a^4 x^4-b^4}}-\frac {\left (2 a^4 b^4+1\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a^5 b^3 \sqrt {a^4 x^4-b^4}} \]

[In]

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

[Out]

-1/4*((2 + 1/(a^4*b^4))*x)/Sqrt[-b^4 + a^4*x^4] - ((1 - 2*a^4*b^4)*ArcTan[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4
+ a^4*x^4]])/(8*Sqrt[2]*(-a^4)^(5/4)*b^5) - ((1 - 2*a^4*b^4)*ArcTanh[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a
^4*x^4]])/(8*Sqrt[2]*(-a^4)^(5/4)*b^5) + (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(a*Sqrt[-b
^4 + a^4*x^4]) + ((1 - 2*a^4*b^4)*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(4*a^5*b^3*Sqrt[-b^4
+ a^4*x^4]) - ((1 + 2*a^4*b^4)*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(4*a^5*b^3*Sqrt[-b^4 +
a^4*x^4])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 1225

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x],
x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d
^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]

Rule 1469

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :
> Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, f, g, n, q, r}, x] && Eq
Q[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rule 1713

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {-b^4+a^4 x^4}}+\frac {2 b^8+x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )}\right ) \, dx \\ & = \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx+\int \frac {2 b^8+x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx \\ & = \frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{\sqrt {-b^4+a^4 x^4}}+\int \frac {2 b^8+x^4}{\left (-b^4+a^4 x^4\right )^{3/2} \left (b^4+a^4 x^4\right )} \, dx \\ & = -\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {\int \frac {b^8 \left (1-6 a^4 b^4\right )-a^4 b^4 \left (1+2 a^4 b^4\right ) x^4}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{4 a^4 b^8} \\ & = -\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} \left (2+\frac {1}{a^4 b^4}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{2} \left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx \\ & = -\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{4 b^4}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{4 b^4}-\frac {\left (\left (2+\frac {1}{a^4 b^4}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {\left (2+\frac {1}{a^4 b^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}+2 \frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{8 b^4}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1-\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 b^4}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1+\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 b^4} \\ & = -\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {\left (2+\frac {1}{a^4 b^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \text {Subst}\left (\int \frac {1}{1-2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 b^4}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \text {Subst}\left (\int \frac {1}{1+2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 b^4}+2 \frac {\left (\left (\frac {1}{a^4}-2 b^4\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 b^4 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b^5}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b^5}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {\left (2+\frac {1}{a^4 b^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a b^3 \sqrt {-b^4+a^4 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.51 \[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) \left (-\frac {(2-2 i) a b \left (1+2 a^4 b^4\right ) x}{\sqrt {-b^4+a^4 x^4}}+\left (-1+2 a^4 b^4\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b^4+a^4 x^4}}{a b x}\right )+2 i \left (-1+2 a^4 b^4\right ) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )\right )}{a^5 b^5} \]

[In]

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

[Out]

((1/16 + I/16)*(((-2 + 2*I)*a*b*(1 + 2*a^4*b^4)*x)/Sqrt[-b^4 + a^4*x^4] + (-1 + 2*a^4*b^4)*ArcTan[((1/2 + I/2)
*Sqrt[-b^4 + a^4*x^4])/(a*b*x)] + (2*I)*(-1 + 2*a^4*b^4)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 +
a^4*x^4])]))/(a^5*b^5)

Maple [A] (verified)

Time = 5.02 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.86

method	result	size
elliptic	\(\frac {\left (-\frac {\left (2 a^{4} b^{4}+1\right ) \sqrt {2}\, x}{4 a^{4} b^{4} \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\left (2 a^{4} b^{4}-1\right ) \sqrt {2}\, \left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )\right )}{32 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\)	\(258\)
default	\(-\frac {i \left (\left (4 x \sqrt {a^{4} x^{4}-b^{4}}\, \left (2 a^{4} b^{4}+1\right ) \sqrt {i a^{2} b^{2}}+\left (\ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )+2 \ln \left (2\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right ) \left (a^{4} b^{4}-\frac {1}{2}\right )\right ) \sqrt {-i a^{2} b^{2}}+2 \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right ) \sqrt {i a^{2} b^{2}}\, \left (\ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+\ln \left (2\right )\right ) \left (a^{4} b^{4}-\frac {1}{2}\right )\right )}{8 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) a^{2} \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right ) \left (a x -b \right ) \left (a x +b \right ) b^{2} \left (a x +i b \right )}\)	\(536\)
pseudoelliptic	\(-\frac {i \left (\left (4 x \sqrt {a^{4} x^{4}-b^{4}}\, \left (2 a^{4} b^{4}+1\right ) \sqrt {i a^{2} b^{2}}+\left (\ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )+\ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )+2 \ln \left (2\right )\right ) \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right ) \left (a^{4} b^{4}-\frac {1}{2}\right )\right ) \sqrt {-i a^{2} b^{2}}+2 \left (a^{2} x^{2}+b^{2}\right ) \left (a x -b \right ) \sqrt {2}\, \left (a x +b \right ) \sqrt {i a^{2} b^{2}}\, \left (\ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+\ln \left (2\right )\right ) \left (a^{4} b^{4}-\frac {1}{2}\right )\right )}{8 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) a^{2} \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (-a x +i b \right ) \left (a x -b \right ) \left (a x +b \right ) b^{2} \left (a x +i b \right )}\)	\(536\)

[In]

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

[Out]

1/2*(-1/4/a^4*(2*a^4*b^4+1)/b^4/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x+1/32*(2*a^4*b^4-1)/a^4/b^4/(a^4*b^4)^(1/4)*2^(1/
2)*(ln((1/2*(a^4*x^4-b^4)/x^2-(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+(a
^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2)))+2*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+1)+2*ar
ctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x-1)))*2^(1/2)

Fricas [A] (verification not implemented)

none

Time = 1.10 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.72 \[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=-\frac {4 \, {\left (2 \, a^{5} b^{5} + a b\right )} \sqrt {a^{4} x^{4} - b^{4}} x + 2 \, {\left (2 \, a^{4} b^{8} - {\left (2 \, a^{8} b^{4} - a^{4}\right )} x^{4} - b^{4}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) + {\left (2 \, a^{4} b^{8} - {\left (2 \, a^{8} b^{4} - a^{4}\right )} x^{4} - b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} - b^{4} - 2 \, \sqrt {a^{4} x^{4} - b^{4}} a b x}{a^{4} x^{4} + b^{4}}\right )}{16 \, {\left (a^{9} b^{5} x^{4} - a^{5} b^{9}\right )}} \]

[In]

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

[Out]

-1/16*(4*(2*a^5*b^5 + a*b)*sqrt(a^4*x^4 - b^4)*x + 2*(2*a^4*b^8 - (2*a^8*b^4 - a^4)*x^4 - b^4)*arctan(sqrt(a^4
*x^4 - b^4)*a*x/(a^2*b*x^2 + b^3)) + (2*a^4*b^8 - (2*a^8*b^4 - a^4)*x^4 - b^4)*log((a^4*x^4 + 2*a^2*b^2*x^2 -
b^4 - 2*sqrt(a^4*x^4 - b^4)*a*b*x)/(a^4*x^4 + b^4)))/(a^9*b^5*x^4 - a^5*b^9)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} + b^{8} + x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx=\int { \frac {a^{8} x^{8} + b^{8} + x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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