3.30.20 \(\int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} (b^{12}+a^{12} x^{12})} \, dx\) [2920]

Optimal. Leaf size=333 \[ -\frac {x}{3 \sqrt {b^4+a^4 x^4}}+\frac {1}{6} \text {RootSum}\left [16 a^8 b^8-32 a^6 b^6 \text {$\#$1}^2-24 a^4 b^4 \text {$\#$1}^4-8 a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right )-4 a^4 b^4 \log (x) \text {$\#$1}^2+4 a^4 b^4 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a^2 b^2 \log (x) \text {$\#$1}^4+2 a^2 b^2 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+\log (x) \text {$\#$1}^6-\log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{8 a^6 b^6 \text {$\#$1}+12 a^4 b^4 \text {$\#$1}^3+6 a^2 b^2 \text {$\#$1}^5-\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(2431\) vs. \(2(333)=666\).
time = 11.48, antiderivative size = 2431, normalized size of antiderivative = 7.30, number of steps used = 25, number of rules used = 8, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6857, 226, 2098, 425, 537, 418, 1231, 1721} \begin {gather*} -\frac {2 \sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt {3 a^4-\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4-\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right ) a^{12}}{\sqrt {3} \sqrt {-a^8} \left (a^4-\sqrt {3} \sqrt {-a^8}\right )^{3/4} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right )^{3/2} b}+\frac {2 \sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt {\sqrt {3} \sqrt {-a^8}-3 a^4} b x}{\sqrt [4]{2} \sqrt [4]{a^4-\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right ) a^{12}}{\sqrt {3} \sqrt {-a^8} \left (a^4-\sqrt {3} \sqrt {-a^8}\right )^{3/4} \left (\sqrt {3} \sqrt {-a^8}-3 a^4\right )^{3/2} b}-\frac {2 \sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt {-3 a^4-\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4+\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right ) a^{12}}{\sqrt {3} \sqrt {-a^8} \left (-3 a^4-\sqrt {3} \sqrt {-a^8}\right )^{3/2} \left (a^4+\sqrt {3} \sqrt {-a^8}\right )^{3/4} b}+\frac {2 \sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt {3 a^4+\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4+\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right ) a^{12}}{\sqrt {3} \sqrt {-a^8} \left (a^4+\sqrt {3} \sqrt {-a^8}\right )^{3/4} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right )^{3/2} b}+\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \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 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right ) a^7}{\sqrt {3} \sqrt {-a^8} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\left (\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}}+1\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 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right ) a^7}{\sqrt {3} \sqrt {-a^8} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \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 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right ) a^7}{\sqrt {3} \sqrt {-a^8} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}}+1\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 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right ) a^7}{\sqrt {3} \sqrt {-a^8} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {2 x a^4}{\sqrt {3} \left (\sqrt {3} a^4-3 \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 x a^4}{\sqrt {3} \left (\sqrt {3} a^4+3 \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {\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 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right ) a^3}{\sqrt {3} \left (\sqrt {3} a^4-3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\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 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right ) a^3}{\sqrt {3} \left (\sqrt {3} a^4+3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}+\frac {\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 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 b \sqrt {b^4+a^4 x^4} a}-\frac {\left (\sqrt {2} a^2+\sqrt {a^4-\sqrt {3} \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 (\sqrt {2} a^2-\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}};2 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \sqrt {3} \left (\sqrt {3} a^4+3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4} a}-\frac {\left (\sqrt {2} a^2-\sqrt {a^4-\sqrt {3} \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 (\sqrt {2} a^2+\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}};2 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \sqrt {3} \left (\sqrt {3} a^4+3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4} a}-\frac {\left (\sqrt {2} a^2+\sqrt {a^4+\sqrt {3} \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 (\sqrt {2} a^2-\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}};2 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \sqrt {3} \left (\sqrt {3} a^4-3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4} a}-\frac {\left (\sqrt {2} a^2-\sqrt {a^4+\sqrt {3} \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 (\sqrt {2} a^2+\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}};2 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{4 \sqrt {3} \left (\sqrt {3} a^4-3 \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4} a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-b^12 + a^12*x^12)/(Sqrt[b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]

[Out]

(-2*a^4*x)/(Sqrt[3]*(Sqrt[3]*a^4 - 3*Sqrt[-a^8])*Sqrt[b^4 + a^4*x^4]) - (2*a^4*x)/(Sqrt[3]*(Sqrt[3]*a^4 + 3*Sq
rt[-a^8])*Sqrt[b^4 + a^4*x^4]) - (2*2^(1/4)*a^12*ArcTan[(Sqrt[3*a^4 - Sqrt[3]*Sqrt[-a^8]]*b*x)/(2^(1/4)*(a^4 -
Sqrt[3]*Sqrt[-a^8])^(1/4)*Sqrt[b^4 + a^4*x^4])])/(Sqrt[3]*Sqrt[-a^8]*(a^4 - Sqrt[3]*Sqrt[-a^8])^(3/4)*(3*a^4
- Sqrt[3]*Sqrt[-a^8])^(3/2)*b) + (2*2^(1/4)*a^12*ArcTan[(Sqrt[-3*a^4 + Sqrt[3]*Sqrt[-a^8]]*b*x)/(2^(1/4)*(a^4
- Sqrt[3]*Sqrt[-a^8])^(1/4)*Sqrt[b^4 + a^4*x^4])])/(Sqrt[3]*Sqrt[-a^8]*(a^4 - Sqrt[3]*Sqrt[-a^8])^(3/4)*(-3*a^
4 + Sqrt[3]*Sqrt[-a^8])^(3/2)*b) - (2*2^(1/4)*a^12*ArcTan[(Sqrt[-3*a^4 - Sqrt[3]*Sqrt[-a^8]]*b*x)/(2^(1/4)*(a^
4 + Sqrt[3]*Sqrt[-a^8])^(1/4)*Sqrt[b^4 + a^4*x^4])])/(Sqrt[3]*Sqrt[-a^8]*(-3*a^4 - Sqrt[3]*Sqrt[-a^8])^(3/2)*(
a^4 + Sqrt[3]*Sqrt[-a^8])^(3/4)*b) + (2*2^(1/4)*a^12*ArcTan[(Sqrt[3*a^4 + Sqrt[3]*Sqrt[-a^8]]*b*x)/(2^(1/4)*(a
^4 + Sqrt[3]*Sqrt[-a^8])^(1/4)*Sqrt[b^4 + a^4*x^4])])/(Sqrt[3]*Sqrt[-a^8]*(a^4 + Sqrt[3]*Sqrt[-a^8])^(3/4)*(3*
a^4 + Sqrt[3]*Sqrt[-a^8])^(3/2)*b) + ((b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcT
an[(a*x)/b], 1/2])/(2*a*b*Sqrt[b^4 + a^4*x^4]) - (a^3*(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])/(Sqrt[3]*(Sqrt[3]*a^4 - 3*Sqrt[-a^8])*b*Sqrt[b^4 + a^4*x^4]) - (a^3*(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])/(Sqrt[3]*(Sqrt[3]*a^4 + 3*
Sqrt[-a^8])*b*Sqrt[b^4 + a^4*x^4]) + (a^7*(1 - (Sqrt[2]*a^2)/Sqrt[a^4 - Sqrt[3]*Sqrt[-a^8]])*(b^2 + a^2*x^2)*S
qrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(Sqrt[3]*Sqrt[-a^8]*(3*a^4 + Sqrt[3]
*Sqrt[-a^8])*b*Sqrt[b^4 + a^4*x^4]) + (a^7*(1 + (Sqrt[2]*a^2)/Sqrt[a^4 - Sqrt[3]*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])/(Sqrt[3]*Sqrt[-a^8]*(3*a^4 + Sqrt[3
]*Sqrt[-a^8])*b*Sqrt[b^4 + a^4*x^4]) - (a^7*(1 - (Sqrt[2]*a^2)/Sqrt[a^4 + Sqrt[3]*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])/(Sqrt[3]*Sqrt[-a^8]*(3*a^4 - Sqrt[
3]*Sqrt[-a^8])*b*Sqrt[b^4 + a^4*x^4]) - (a^7*(1 + (Sqrt[2]*a^2)/Sqrt[a^4 + Sqrt[3]*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])/(Sqrt[3]*Sqrt[-a^8]*(3*a^4 - Sqrt
[3]*Sqrt[-a^8])*b*Sqrt[b^4 + a^4*x^4]) - ((Sqrt[2]*a^2 + Sqrt[a^4 - Sqrt[3]*Sqrt[-a^8]])^2*(b^2 + a^2*x^2)*Sqr
t[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticPi[-1/4*(Sqrt[2]*a^2 - Sqrt[a^4 - Sqrt[3]*Sqrt[-a^8]])^2/(Sqrt[2]
*a^2*Sqrt[a^4 - Sqrt[3]*Sqrt[-a^8]]), 2*ArcTan[(a*x)/b], 1/2])/(4*Sqrt[3]*a*(Sqrt[3]*a^4 + 3*Sqrt[-a^8])*b*Sqr
t[b^4 + a^4*x^4]) - ((Sqrt[2]*a^2 - Sqrt[a^4 - Sqrt[3]*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[(Sqrt[2]*a^2 + Sqrt[a^4 - Sqrt[3]*Sqrt[-a^8]])^2/(4*Sqrt[2]*a^2*Sqrt[a^4 - Sqrt[3]*
Sqrt[-a^8]]), 2*ArcTan[(a*x)/b], 1/2])/(4*Sqrt[3]*a*(Sqrt[3]*a^4 + 3*Sqrt[-a^8])*b*Sqrt[b^4 + a^4*x^4]) - ((Sq
rt[2]*a^2 + Sqrt[a^4 + Sqrt[3]*Sqrt[-a^8]])^2*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*Elliptic
Pi[-1/4*(Sqrt[2]*a^2 - Sqrt[a^4 + Sqrt[3]*Sqrt[-a^8]])^2/(Sqrt[2]*a^2*Sqrt[a^4 + Sqrt[3]*Sqrt[-a^8]]), 2*ArcTa
n[(a*x)/b], 1/2])/(4*Sqrt[3]*a*(Sqrt[3]*a^4 - 3*Sqrt[-a^8])*b*Sqrt[b^4 + a^4*x^4]) - ((Sqrt[2]*a^2 - Sqrt[a^4
+ Sqrt[3]*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[(Sqrt[2]*a^2 + Sqr
t[a^4 + Sqrt[3]*Sqrt[-a^8]])^2/(4*Sqrt[2]*a^2*Sqrt[a^4 + Sqrt[3]*Sqrt[-a^8]]), 2*ArcTan[(a*x)/b], 1/2])/(4*Sqr
t[3]*a*(Sqrt[3]*a^4 - 3*Sqrt[-a^8])*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 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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

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 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 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 2098

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
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*} \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx &=\int \left (\frac {1}{\sqrt {b^4+a^4 x^4}}-\frac {2 b^{12}}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )}\right ) \, dx\\ &=-\left (\left (2 b^{12}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx\right )+\int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}-\left (2 b^{12}\right ) \int \left (-\frac {2 a^8}{\sqrt {3} \sqrt {-a^8} b^4 \left (b^4+a^4 x^4\right )^{3/2} \left (a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4-2 a^8 x^4\right )}-\frac {2 a^8}{\sqrt {3} \sqrt {-a^8} b^4 \left (b^4+a^4 x^4\right )^{3/2} \left (-a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4+2 a^8 x^4\right )}\right ) \, dx\\ &=\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (4 \sqrt {-a^8} b^8\right ) \int \frac {1}{\left (b^4+a^4 x^4\right )^{3/2} \left (a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4-2 a^8 x^4\right )} \, dx}{\sqrt {3}}-\frac {\left (4 \sqrt {-a^8} b^8\right ) \int \frac {1}{\left (b^4+a^4 x^4\right )^{3/2} \left (-a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4+2 a^8 x^4\right )} \, dx}{\sqrt {3}}\\ &=\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}+\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (2 \sqrt {-a^8}\right ) \int \frac {a^4 \left (5 a^4-\sqrt {3} \sqrt {-a^8}\right ) b^4-2 a^{12} x^4}{\sqrt {b^4+a^4 x^4} \left (-a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4+2 a^8 x^4\right )} \, dx}{\sqrt {3} a^4 \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right )}+\frac {\left (2 \sqrt {-a^8}\right ) \int \frac {-a^4 \left (5 a^4+\sqrt {3} \sqrt {-a^8}\right ) b^4+2 a^{12} x^4}{\sqrt {b^4+a^4 x^4} \left (a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4-2 a^8 x^4\right )} \, dx}{\sqrt {3} a^4 \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right )}\\ &=\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}+\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {\left (2 \sqrt {-a^8}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (2 \sqrt {-a^8}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (8 a^4 \sqrt {-a^8} b^4\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (-a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4+2 a^8 x^4\right )} \, dx}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (8 a^4 \sqrt {-a^8} b^4\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (a^4 b^4+\sqrt {3} \sqrt {-a^8} b^4-2 a^8 x^4\right )} \, dx}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right )}\\ &=\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}+\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt {-a^8} \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 )}{\sqrt {3} a \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt {-a^8} \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 )}{\sqrt {3} a \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} \int \frac {1}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}+\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt {-a^8} \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 )}{\sqrt {3} a \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt {-a^8} \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 )}{\sqrt {3} a \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (1-\frac {2 a^4}{a^4-\sqrt {3} \sqrt {-a^8}}\right )}-\frac {\left (1+\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (1-\frac {2 a^4}{a^4-\sqrt {3} \sqrt {-a^8}}\right )}-\frac {\left (a^2 \left (2 a^2-\sqrt {2} \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (a^4+\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (a^2 \left (2 a^2+\sqrt {2} \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4-\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (a^4+\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (1-\frac {2 a^4}{a^4+\sqrt {3} \sqrt {-a^8}}\right )}-\frac {\left (1+\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 \left (1-\frac {2 a^4}{a^4+\sqrt {3} \sqrt {-a^8}}\right )}-\frac {\left (a^2 \left (2 a^2-\sqrt {2} \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1-\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (a^4-\sqrt {3} \sqrt {-a^8}\right )}-\frac {\left (a^2 \left (2 a^2+\sqrt {2} \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (1+\frac {\sqrt {2} a^4 x^2}{\sqrt {a^4+\sqrt {3} \sqrt {-a^8}} b^2}\right ) \sqrt {b^4+a^4 x^4}} \, dx}{3 \left (a^4-\sqrt {3} \sqrt {-a^8}\right )}\\ &=\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {2 \sqrt {-a^8} x}{\sqrt {3} \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) \sqrt {b^4+a^4 x^4}}-\frac {\sqrt [4]{a^4-\sqrt {3} \sqrt {-a^8}} \tan ^{-1}\left (\frac {\sqrt {3 a^4-\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4-\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{3\ 2^{3/4} \sqrt {3 a^4-\sqrt {3} \sqrt {-a^8}} b}-\frac {\sqrt [4]{a^4-\sqrt {3} \sqrt {-a^8}} \tan ^{-1}\left (\frac {\sqrt {-3 a^4+\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4-\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{3\ 2^{3/4} \sqrt {-3 a^4+\sqrt {3} \sqrt {-a^8}} b}-\frac {\sqrt [4]{a^4+\sqrt {3} \sqrt {-a^8}} \tan ^{-1}\left (\frac {\sqrt {-3 a^4-\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4+\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{3\ 2^{3/4} \sqrt {-3 a^4-\sqrt {3} \sqrt {-a^8}} b}-\frac {\sqrt [4]{a^4+\sqrt {3} \sqrt {-a^8}} \tan ^{-1}\left (\frac {\sqrt {3 a^4+\sqrt {3} \sqrt {-a^8}} b x}{\sqrt [4]{2} \sqrt [4]{a^4+\sqrt {3} \sqrt {-a^8}} \sqrt {b^4+a^4 x^4}}\right )}{3\ 2^{3/4} \sqrt {3 a^4+\sqrt {3} \sqrt {-a^8}} b}+\frac {\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 )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {\sqrt {-a^8} \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 )}{\sqrt {3} a \left (3 a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\sqrt {-a^8} \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 )}{\sqrt {3} a \left (3 a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \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 )}{6 a \left (1-\frac {2 a^4}{a^4-\sqrt {3} \sqrt {-a^8}}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (1+\frac {\sqrt {2} a^2}{\sqrt {a^4-\sqrt {3} \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 )}{6 a \left (1-\frac {2 a^4}{a^4-\sqrt {3} \sqrt {-a^8}}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (1-\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \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 )}{6 a \left (1-\frac {2 a^4}{a^4+\sqrt {3} \sqrt {-a^8}}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (1+\frac {\sqrt {2} a^2}{\sqrt {a^4+\sqrt {3} \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 )}{6 a \left (1-\frac {2 a^4}{a^4+\sqrt {3} \sqrt {-a^8}}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (2 a^2+\sqrt {2} \sqrt {a^4-\sqrt {3} \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 (\sqrt {2} a^2-\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{24 a \left (a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (2 a^2-\sqrt {2} \sqrt {a^4-\sqrt {3} \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 (\sqrt {2} a^2+\sqrt {a^4-\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4-\sqrt {3} \sqrt {-a^8}}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{24 a \left (a^4+\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (2 a^2+\sqrt {2} \sqrt {a^4+\sqrt {3} \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 (\sqrt {2} a^2-\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{24 a \left (a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}-\frac {\left (2 a^2-\sqrt {2} \sqrt {a^4+\sqrt {3} \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 (\sqrt {2} a^2+\sqrt {a^4+\sqrt {3} \sqrt {-a^8}}\right )^2}{4 \sqrt {2} a^2 \sqrt {a^4+\sqrt {3} \sqrt {-a^8}}};2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{24 a \left (a^4-\sqrt {3} \sqrt {-a^8}\right ) b \sqrt {b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [A]
time = 1.30, size = 93, normalized size = 0.28 \begin {gather*} -\frac {\frac {3 a b x}{\sqrt {b^4+a^4 x^4}}+3^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{3} a b x}{\sqrt {b^4+a^4 x^4}}\right )+3^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{9 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b^12 + a^12*x^12)/(Sqrt[b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]

[Out]

-1/9*((3*a*b*x)/Sqrt[b^4 + a^4*x^4] + 3^(3/4)*ArcTan[(3^(1/4)*a*b*x)/Sqrt[b^4 + a^4*x^4]] + 3^(3/4)*ArcTanh[(3
^(1/4)*a*b*x)/Sqrt[b^4 + a^4*x^4]])/(a*b)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.20, size = 467, normalized size = 1.40

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{x \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}\right )}{3 \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, \ln \left (\frac {\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}{2}}{\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}-\frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}{2}}\right )}{6 \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, x}{3 \sqrt {a^{4} x^{4}+b^{4}}}\right ) \sqrt {2}}{2}\)	\(187\)
default	\(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b^{4} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8} a^{8}-\textit {\_Z}^{4} a^{4} b^{4}+b^{8}\right )}{\sum }\frac {\left (-a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+2 b^{4}\right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{6} a^{4}+\underline {\hspace {1.25 ex}}\alpha ^{2} b^{4}+b^{4} x^{2}\right ) a^{4}}{b^{4} \sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+b^{4}}}+\frac {2 a^{4} \underline {\hspace {1.25 ex}}\alpha ^{3} \left (a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}\right ) \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \left (a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}\right ) a^{2}}{b^{6}}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \left (2 a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}\right )}\right )}{12 a^{4}}-\frac {2 b^{4} \left (\frac {x}{2 b^{4} \sqrt {\left (x^{4}+\frac {b^{4}}{a^{4}}\right ) a^{4}}}+\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{2 b^{4} \sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{3}\)	\(467\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x,method=_RETURNVERBOSE)

[Out]

1/(I*a^2/b^2)^(1/2)*(1-I*a^2/b^2*x^2)^(1/2)*(1+I*a^2/b^2*x^2)^(1/2)/(a^4*x^4+b^4)^(1/2)*EllipticF(x*(I*a^2/b^2
)^(1/2),I)-1/12*b^4/a^4*sum((-_alpha^4*a^4+2*b^4)/_alpha^3/(2*_alpha^4*a^4-b^4)*(-1/(_alpha^4*a^4+b^4)^(1/2)*a
rctanh(_alpha^2/b^4*(-_alpha^6*a^4+_alpha^2*b^4+b^4*x^2)*a^4/(_alpha^4*a^4+b^4)^(1/2)/(a^4*x^4+b^4)^(1/2))+2/(
I*a^2/b^2)^(1/2)*a^4*_alpha^3*(_alpha^4*a^4-b^4)/b^8*(1-I*a^2/b^2*x^2)^(1/2)*(1+I*a^2/b^2*x^2)^(1/2)/(a^4*x^4+
b^4)^(1/2)*EllipticPi(x*(I*a^2/b^2)^(1/2),I*_alpha^2*(_alpha^4*a^4-b^4)*a^2/b^6,(-I*a^2/b^2)^(1/2)/(I*a^2/b^2)
^(1/2))),_alpha=RootOf(_Z^8*a^8-_Z^4*a^4*b^4+b^8))-2/3*b^4*(1/2/b^4*x/((x^4+b^4/a^4)*a^4)^(1/2)+1/2/b^4/(I*a^2
/b^2)^(1/2)*(1-I*a^2/b^2*x^2)^(1/2)*(1+I*a^2/b^2*x^2)^(1/2)/(a^4*x^4+b^4)^(1/2)*EllipticF(x*(I*a^2/b^2)^(1/2),
I))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x, algorithm="maxima")

[Out]

integrate((a^12*x^12 - b^12)/((a^12*x^12 + b^12)*sqrt(a^4*x^4 + b^4)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 8.28, size = 596, normalized size = 1.79 \begin {gather*} -\frac {4 \, \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{4} x^{4} + b^{4}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {3 \, {\left (2 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{4} b^{4} x^{3} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\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}{3}\right )^{\frac {3}{4}} {\left (a^{12} b^{4} x^{8} + 5 \, 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}{3}\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}{3}} \sqrt {\frac {1}{a^{4} b^{4}}}}\right )}}{a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{4} x^{4} + b^{4}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {6 \, \left (\frac {1}{3}\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 (3 \, \sqrt {\frac {1}{3}} 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}{3}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 5 \, 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} - a^{4} b^{4} x^{4} + b^{8}\right )}}\right ) - \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{4} x^{4} + b^{4}\right )} \left (\frac {1}{a^{4} b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {6 \, \left (\frac {1}{3}\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 (3 \, \sqrt {\frac {1}{3}} 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}{3}\right )^{\frac {1}{4}} {\left (a^{8} x^{8} + 5 \, 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} - a^{4} b^{4} x^{4} + b^{8}\right )}}\right ) + 4 \, \sqrt {a^{4} x^{4} + b^{4}} x}{12 \, {\left (a^{4} x^{4} + b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
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((a**12*x**12-b**12)/(a**4*x**4+b**4)**(1/2)/(a**12*x**12+b**12),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^12*x^12-b^12)/(a^4*x^4+b^4)^(1/2)/(a^12*x^12+b^12),x, algorithm="giac")

[Out]

integrate((a^12*x^12 - b^12)/((a^12*x^12 + b^12)*sqrt(a^4*x^4 + b^4)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {b^{12}-a^{12}\,x^{12}}{\sqrt {a^4\,x^4+b^4}\,\left (a^{12}\,x^{12}+b^{12}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^12 - a^12*x^12)/((b^4 + a^4*x^4)^(1/2)*(b^12 + a^12*x^12)),x)

[Out]

int(-(b^12 - a^12*x^12)/((b^4 + a^4*x^4)^(1/2)*(b^12 + a^12*x^12)), x)

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