\(\int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} (b^8+a^8 x^8)} \, dx\) [1095]

Optimal result

Integrand size = 42, antiderivative size = 81 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \]

[Out]

-1/4*arctan(2^(1/4)*a*b*x/(a^4*x^4+b^4)^(1/2))*2^(3/4)/a/b-1/4*arctanh(2^(1/4)*a*b*x/(a^4*x^4+b^4)^(1/2))*2^(3
/4)/a/b

Rubi [C] (warning: unable to verify)

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

Time = 3.09 (sec) , antiderivative size = 1639, normalized size of antiderivative = 20.23, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1600, 6857, 415, 226, 418, 1231, 1721} \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {\left (\sqrt {-a^8}-a^4\right )^{3/2} \arctan \left (\frac {\sqrt [8]{-a^8} \sqrt {\sqrt {-a^8}-a^4} b x}{a^2 \sqrt {b^4+a^4 x^4}}\right ) a^6}{8 \left (-a^8\right )^{13/8} b}-\frac {\left (\sqrt {-a^8}-a^4\right )^{3/2} \arctan \left (\frac {\sqrt {\sqrt {-a^8}-a^4} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{8 \left (-\sqrt {-a^8}\right )^{3/4} b a^4}-\frac {\left (a^4+\sqrt {-a^8}\right )^{3/2} \arctan \left (\frac {\sqrt {a^4+\sqrt {-a^8}} b x}{\sqrt [8]{-a^8} \sqrt {b^4+a^4 x^4}}\right )}{8 \left (-a^8\right )^{3/8} b a^4}+\frac {\left (a^4-\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 b \sqrt {b^4+a^4 x^4} a^5}+\frac {\left (a^4+\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 b \sqrt {b^4+a^4 x^4} a^5}-\frac {\left (a^2-\sqrt [4]{-a^8}\right )^3 \left (a^2+\sqrt [4]{-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (a^2+\sqrt [4]{-a^8}\right )^2}{4 a^2 \sqrt [4]{-a^8}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 \sqrt {-a^8} b \sqrt {b^4+a^4 x^4} a^5}-\frac {\left (a^2-\sqrt [4]{-a^8}\right ) \left (a^4+\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{8 b \sqrt {b^4+a^4 x^4} a^7}-\frac {\sqrt {-a^8} \left (a^2+\sqrt [4]{-a^8}\right ) \left (a^4-\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{8 b \sqrt {b^4+a^4 x^4} a^{11}}+\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2-\sqrt {-\sqrt {-a^8}}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{8 b \sqrt {b^4+a^4 x^4} a^{11}}+\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2+\sqrt {-\sqrt {-a^8}}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{8 b \sqrt {b^4+a^4 x^4} a^{11}}+\frac {\left (-\sqrt {-a^8}\right )^{5/4} \left (a^4-\sqrt {-a^8}\right )^{3/2} \arctan \left (\frac {\sqrt {a^4-\sqrt {-a^8}} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{8 b a^{12}}+\frac {\sqrt {-a^8} \left (a^2+\sqrt [4]{-a^8}\right )^2 \left (a^4-\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {a^6 \left (a^2-\sqrt [4]{-a^8}\right )^2}{4 \left (-a^8\right )^{5/4}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 b \sqrt {b^4+a^4 x^4} a^{13}}-\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2+\sqrt {-\sqrt {-a^8}}\right )^2 \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (a^2-\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 b \sqrt {b^4+a^4 x^4} a^{13}}-\frac {\sqrt {-a^8} \left (a^4+\sqrt {-a^8}\right ) \left (a^2-\sqrt {-\sqrt {-a^8}}\right )^2 \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (a^2+\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}},2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{16 b \sqrt {b^4+a^4 x^4} a^{13}} \]

[In]

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

[Out]

((-Sqrt[-a^8])^(5/4)*(a^4 - Sqrt[-a^8])^(3/2)*ArcTan[(Sqrt[a^4 - Sqrt[-a^8]]*b*x)/((-Sqrt[-a^8])^(1/4)*Sqrt[b^
4 + a^4*x^4])])/(8*a^12*b) - (a^6*(-a^4 + Sqrt[-a^8])^(3/2)*ArcTan[((-a^8)^(1/8)*Sqrt[-a^4 + Sqrt[-a^8]]*b*x)/
(a^2*Sqrt[b^4 + a^4*x^4])])/(8*(-a^8)^(13/8)*b) - ((-a^4 + Sqrt[-a^8])^(3/2)*ArcTan[(Sqrt[-a^4 + Sqrt[-a^8]]*b
*x)/((-Sqrt[-a^8])^(1/4)*Sqrt[b^4 + a^4*x^4])])/(8*a^4*(-Sqrt[-a^8])^(3/4)*b) - ((a^4 + Sqrt[-a^8])^(3/2)*ArcT
an[(Sqrt[a^4 + Sqrt[-a^8]]*b*x)/((-a^8)^(1/8)*Sqrt[b^4 + a^4*x^4])])/(8*a^4*(-a^8)^(3/8)*b) + ((a^4 - Sqrt[-a^
8])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(4*a^5*b*Sqrt[b
^4 + a^4*x^4]) - (Sqrt[-a^8]*(a^2 + (-a^8)^(1/4))*(a^4 - Sqrt[-a^8])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2
+ a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(8*a^11*b*Sqrt[b^4 + a^4*x^4]) + ((a^4 + Sqrt[-a^8])*(b^2 +
a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(4*a^5*b*Sqrt[b^4 + a^4*x^
4]) - ((a^2 - (-a^8)^(1/4))*(a^4 + Sqrt[-a^8])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*Ellipti
cF[2*ArcTan[(a*x)/b], 1/2])/(8*a^7*b*Sqrt[b^4 + a^4*x^4]) + (Sqrt[-a^8]*(a^4 + Sqrt[-a^8])*(a^2 - Sqrt[-Sqrt[-
a^8]])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(8*a^11*b*Sq
rt[b^4 + a^4*x^4]) + (Sqrt[-a^8]*(a^4 + Sqrt[-a^8])*(a^2 + Sqrt[-Sqrt[-a^8]])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*
x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(8*a^11*b*Sqrt[b^4 + a^4*x^4]) + (Sqrt[-a^8]*(a^2 +
(-a^8)^(1/4))^2*(a^4 - Sqrt[-a^8])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticPi[(a^6*(a
^2 - (-a^8)^(1/4))^2)/(4*(-a^8)^(5/4)), 2*ArcTan[(a*x)/b], 1/2])/(16*a^13*b*Sqrt[b^4 + a^4*x^4]) - ((a^2 - (-a
^8)^(1/4))^3*(a^2 + (-a^8)^(1/4))*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticPi[(a^2 + (-
a^8)^(1/4))^2/(4*a^2*(-a^8)^(1/4)), 2*ArcTan[(a*x)/b], 1/2])/(16*a^5*Sqrt[-a^8]*b*Sqrt[b^4 + a^4*x^4]) - (Sqrt
[-a^8]*(a^4 + Sqrt[-a^8])*(a^2 + Sqrt[-Sqrt[-a^8]])^2*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*
EllipticPi[-1/4*(a^2 - Sqrt[-Sqrt[-a^8]])^2/(a^2*Sqrt[-Sqrt[-a^8]]), 2*ArcTan[(a*x)/b], 1/2])/(16*a^13*b*Sqrt[
b^4 + a^4*x^4]) - (Sqrt[-a^8]*(a^4 + Sqrt[-a^8])*(a^2 - Sqrt[-Sqrt[-a^8]])^2*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x
^4)/(b^2 + a^2*x^2)^2]*EllipticPi[(a^2 + Sqrt[-Sqrt[-a^8]])^2/(4*a^2*Sqrt[-Sqrt[-a^8]]), 2*ArcTan[(a*x)/b], 1/
2])/(16*a^13*b*Sqrt[b^4 + a^4*x^4])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 415

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[1/Sqrt[a + b*x^4], x], x] - Di
st[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 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 1231

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

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 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 \frac {\left (-b^4+a^4 x^4\right ) \sqrt {b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx \\ & = \int \left (-\frac {\sqrt {-a^8} \left (a^4 b^4-\sqrt {-a^8} b^4\right ) \sqrt {b^4+a^4 x^4}}{2 a^8 b^4 \left (b^4-\sqrt {-a^8} x^4\right )}+\frac {\sqrt {-a^8} \left (a^4 b^4+\sqrt {-a^8} b^4\right ) \sqrt {b^4+a^4 x^4}}{2 a^8 b^4 \left (b^4+\sqrt {-a^8} x^4\right )}\right ) \, dx \\ & = -\frac {\left (a^4+\sqrt {-a^8}\right ) \int \frac {\sqrt {b^4+a^4 x^4}}{b^4-\sqrt {-a^8} x^4} \, dx}{2 a^4}+\frac {\left (\sqrt {-a^8} \left (a^4 b^4+\sqrt {-a^8} b^4\right )\right ) \int \frac {\sqrt {b^4+a^4 x^4}}{b^4+\sqrt {-a^8} x^4} \, dx}{2 a^8 b^4} \\ & = \frac {1}{2} \left (1+\frac {a^4}{\sqrt {-a^8}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx+\frac {\left (a^4+\sqrt {-a^8}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 a^4}-b^4 \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^4-\sqrt {-a^8} x^4\right )} \, dx-b^4 \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^4+\sqrt {-a^8} x^4\right )} \, dx \\ & = \frac {\left (1+\frac {a^4}{\sqrt {-a^8}}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 a b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^4+\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 a^5 b \sqrt {b^4+a^4 x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx \\ & = \frac {\left (1+\frac {a^4}{\sqrt {-a^8}}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 a b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^4+\sqrt {-a^8}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{4 a^5 b \sqrt {b^4+a^4 x^4}}-\frac {a^2 \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^2-\sqrt [4]{-a^8}\right )}+\frac {\sqrt [4]{-a^8} \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^2-\sqrt [4]{-a^8}\right )}-\frac {a^2 \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^2+\sqrt [4]{-a^8}\right )}-\frac {\sqrt [4]{-a^8} \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^2+\sqrt [4]{-a^8}\right )}-\frac {\left (a^2 \left (a^2-\sqrt {-\sqrt {-a^8}}\right )\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^4+\sqrt {-a^8}\right )}-\frac {\left (a^2 \left (a^2+\sqrt {-\sqrt {-a^8}}\right )\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^4+\sqrt {-a^8}\right )}+\frac {\left (\sqrt {-\sqrt {-a^8}} \left (a^2+\sqrt {-\sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^4+\sqrt {-a^8}\right )}-\frac {\left (\sqrt {-a^8}+a^2 \sqrt {-\sqrt {-a^8}}\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{2 \left (a^4+\sqrt {-a^8}\right )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 9.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.41

method	result	size
default	\(-\frac {\left (-2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 2^{\frac {3}{4}}}{2 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}{2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}\right )\right ) 2^{\frac {3}{4}}}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\)	\(114\)
pseudoelliptic	\(-\frac {\left (-2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, 2^{\frac {3}{4}}}{2 x \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}{2^{\frac {1}{4}} \left (a^{4} b^{4}\right )^{\frac {1}{4}} x -\sqrt {a^{4} x^{4}+b^{4}}}\right )\right ) 2^{\frac {3}{4}}}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\)	\(114\)
elliptic	\(\frac {2 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right )-\ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\)	\(144\)

[In]

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

[Out]

-1/8*(-2*arctan(1/2*(a^4*x^4+b^4)^(1/2)/x*2^(3/4)/(a^4*b^4)^(1/4))+ln((-2^(1/4)*(a^4*b^4)^(1/4)*x-(a^4*x^4+b^4
)^(1/2))/(2^(1/4)*(a^4*b^4)^(1/4)*x-(a^4*x^4+b^4)^(1/2))))*2^(3/4)/(a^4*b^4)^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.14 (sec) , antiderivative size = 623, normalized size of antiderivative = 7.69 \[ \int \frac {-b^8+a^8 x^8}{\sqrt {b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx=-\frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} + a^{4} x^{5} + b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 4 \, a^{4} b^{4} x^{4} + b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) + \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} + a^{4} x^{5} + b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 4 \, a^{4} b^{4} x^{4} + b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) + \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (i \, a^{8} b^{4} x^{6} + i \, a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} - a^{4} x^{5} - b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-i \, a^{8} x^{8} - 4 i \, a^{4} b^{4} x^{4} - i \, b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) - \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (-i \, a^{8} b^{4} x^{6} - i \, a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} - 2 \, {\left (2 \, \sqrt {\frac {1}{2}} a^{4} b^{4} x^{3} \sqrt {\frac {1}{a^{4} b^{4}}} - a^{4} x^{5} - b^{4} x\right )} \sqrt {a^{4} x^{4} + b^{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (i \, a^{8} x^{8} + 4 i \, a^{4} b^{4} x^{4} + i \, b^{8}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a^{8} x^{8} + b^{8}\right )}}\right ) \]

[In]

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

[Out]

-1/8*(1/2)^(1/4)*(1/(a^4*b^4))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*(a^8*b^4*x^6 + a^4*b^8*x^2)*(1/(a^4*b^4))^(3/4) +
2*(2*sqrt(1/2)*a^4*b^4*x^3*sqrt(1/(a^4*b^4)) + a^4*x^5 + b^4*x)*sqrt(a^4*x^4 + b^4) + (1/2)^(1/4)*(a^8*x^8 +
4*a^4*b^4*x^4 + b^8)*(1/(a^4*b^4))^(1/4))/(a^8*x^8 + b^8)) + 1/8*(1/2)^(1/4)*(1/(a^4*b^4))^(1/4)*log(1/2*(4*(1
/2)^(3/4)*(a^8*b^4*x^6 + a^4*b^8*x^2)*(1/(a^4*b^4))^(3/4) - 2*(2*sqrt(1/2)*a^4*b^4*x^3*sqrt(1/(a^4*b^4)) + a^4
*x^5 + b^4*x)*sqrt(a^4*x^4 + b^4) + (1/2)^(1/4)*(a^8*x^8 + 4*a^4*b^4*x^4 + b^8)*(1/(a^4*b^4))^(1/4))/(a^8*x^8
+ b^8)) + 1/8*I*(1/2)^(1/4)*(1/(a^4*b^4))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*(I*a^8*b^4*x^6 + I*a^4*b^8*x^2)*(1/(a^
4*b^4))^(3/4) - 2*(2*sqrt(1/2)*a^4*b^4*x^3*sqrt(1/(a^4*b^4)) - a^4*x^5 - b^4*x)*sqrt(a^4*x^4 + b^4) + (1/2)^(1
/4)*(-I*a^8*x^8 - 4*I*a^4*b^4*x^4 - I*b^8)*(1/(a^4*b^4))^(1/4))/(a^8*x^8 + b^8)) - 1/8*I*(1/2)^(1/4)*(1/(a^4*b
^4))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*(-I*a^8*b^4*x^6 - I*a^4*b^8*x^2)*(1/(a^4*b^4))^(3/4) - 2*(2*sqrt(1/2)*a^4*b
^4*x^3*sqrt(1/(a^4*b^4)) - a^4*x^5 - b^4*x)*sqrt(a^4*x^4 + b^4) + (1/2)^(1/4)*(I*a^8*x^8 + 4*I*a^4*b^4*x^4 + I
*b^8)*(1/(a^4*b^4))^(1/4))/(a^8*x^8 + b^8))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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