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

Optimal. Leaf size=81 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{2 \sqrt [4]{2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{2 \sqrt [4]{2} a b} \]

________________________________________________________________________________________

Rubi [C]  time = 2.74, antiderivative size = 1639, normalized size of antiderivative = 20.23, number of steps used = 21, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1586, 6725, 406, 220, 409, 1217, 1707}

result too large to display

Warning: Unable to verify antiderivative.

[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 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 406

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 409

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 1217

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 1586

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 1707

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)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), 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 6725

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*} \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 (-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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 )}\\ &=-\frac {\sqrt [4]{-\sqrt {-a^8}} \tan ^{-1}\left (\frac {\sqrt {a^4-\sqrt {-a^8}} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {a^4-\sqrt {-a^8}} b}-\frac {a^2 \tan ^{-1}\left (\frac {\sqrt [8]{-a^8} \sqrt {-a^4+\sqrt {-a^8}} b x}{a^2 \sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt [8]{-a^8} \sqrt {-a^4+\sqrt {-a^8}} b}-\frac {\sqrt [4]{-\sqrt {-a^8}} \tan ^{-1}\left (\frac {\sqrt {-a^4+\sqrt {-a^8}} b x}{\sqrt [4]{-\sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {-a^4+\sqrt {-a^8}} b}-\frac {\sqrt [8]{-a^8} \tan ^{-1}\left (\frac {\sqrt {a^4+\sqrt {-a^8}} b x}{\sqrt [8]{-a^8} \sqrt {b^4+a^4 x^4}}\right )}{4 \sqrt {a^4+\sqrt {-a^8}} b}+\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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 a b \sqrt {b^4+a^4 x^4}}-\frac {a \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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \left (a^2-\sqrt [4]{-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {a \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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \left (a^2+\sqrt [4]{-a^8}\right ) 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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 a^5 b \sqrt {b^4+a^4 x^4}}-\frac {a \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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \left (a^4+\sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {a \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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \left (a^4+\sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt [4]{-a^8} \left (a^4+a^2 \sqrt [4]{-a^8}+\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}} \Pi \left (\frac {a^6 \left (a^2-\sqrt [4]{-a^8}\right )^2}{4 \left (-a^8\right )^{5/4}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{8 a^7 b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt [4]{-a^8} \left (a^4+\sqrt {-a^8}+\frac {\left (-a^8\right )^{5/4}}{a^6}\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}} \Pi \left (\frac {\left (a^2+\sqrt [4]{-a^8}\right )^2}{4 a^2 \sqrt [4]{-a^8}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{8 a^7 b \sqrt {b^4+a^4 x^4}}+\frac {\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}} \Pi \left (-\frac {\left (a^2-\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{8 a \left (a^4+\sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\left (a^4-\sqrt {-a^8}-2 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}} \Pi \left (\frac {\left (a^2+\sqrt {-\sqrt {-a^8}}\right )^2}{4 a^2 \sqrt {-\sqrt {-a^8}}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{8 a \left (a^4+\sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.56, size = 201, normalized size = 2.48 \begin {gather*} -\frac {i \sqrt {\frac {a^4 x^4}{b^4}+1} \left (2 F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-(-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left ((-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )\right )}{2 \sqrt {\frac {i a^2}{b^2}} \sqrt {a^4 x^4+b^4}} \end {gather*}

Antiderivative was successfully verified.

[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*I)*Sqrt[1 + (a^4*x^4)/b^4]*(2*EllipticF[I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] - EllipticPi[-(-1)^(1/4), I
*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] - EllipticPi[(-1)^(1/4), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] - EllipticPi[-
(-1)^(3/4), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] - EllipticPi[(-1)^(3/4), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1]))
/(Sqrt[(I*a^2)/b^2]*Sqrt[b^4 + a^4*x^4])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.53, size = 81, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{2 \sqrt [4]{2} a b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

fricas [B]  time = 4.86, size = 500, normalized size = 6.17 \begin {gather*} -\frac {1}{2} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (2 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} a^{4} b^{4} x^{3} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{8} b^{4} x^{5} + a^{4} b^{8} x\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{4} x^{4} + b^{4}} + {\left (\left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a^{12} b^{4} x^{8} + 4 \, a^{8} b^{8} x^{4} + a^{4} b^{12}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a^{8} b^{4} x^{6} + a^{4} b^{8} x^{2}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}}\right )} \sqrt {\sqrt {\frac {1}{2}} \sqrt {\frac {1}{a^{4} b^{4}}}}\right )}}{a^{8} x^{8} + b^{8}}\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} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

-1/2*(1/2)^(1/4)*(1/(a^4*b^4))^(1/4)*arctan(2*(2*((1/2)^(1/4)*a^4*b^4*x^3*(1/(a^4*b^4))^(1/4) + (1/2)^(3/4)*(a
^8*b^4*x^5 + a^4*b^8*x)*(1/(a^4*b^4))^(3/4))*sqrt(a^4*x^4 + b^4) + ((1/2)^(3/4)*(a^12*b^4*x^8 + 4*a^8*b^8*x^4
+ a^4*b^12)*(1/(a^4*b^4))^(3/4) + 2*(1/2)^(1/4)*(a^8*b^4*x^6 + a^4*b^8*x^2)*(1/(a^4*b^4))^(1/4))*sqrt(sqrt(1/2
)*sqrt(1/(a^4*b^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*(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))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [B]  time = 0.16, size = 168, normalized size = 2.07

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right )}{2 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, \ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\)	\(168\)
default	\(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b^{8} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{8} \textit {\_Z}^{8}+b^{8}\right )}{\sum }\frac {-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-a^{4} \underline {\hspace {1.25 ex}}\alpha ^{6}+b^{4} x^{2}\right ) a^{4}}{b^{4} \sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+b^{4}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{7} a^{8} \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}}{b^{6}}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}+b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{7}}\right )}{8 a^{8}}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________