\(\int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} (b^6+a^6 x^6)} \, dx\) [2233]

Optimal result

Integrand size = 42, antiderivative size = 166 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}+\frac {2 \text {arctanh}\left (\frac {\sqrt {4-2 \sqrt {3}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {2 \text {arctanh}\left (\frac {\sqrt {4+2 \sqrt {3}} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]

[Out]

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

Rubi [C] (verified)

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

Time = 1.39 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.44, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6857, 226, 2098, 1225, 1713, 211, 6860, 1231, 1721} \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {2 \arctan \left (\frac {\sqrt {-a^2} b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {-a^2} b}+\frac {\left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}-\frac {\left (a^2-\sqrt {3} \sqrt {-a^4}\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 a \left (3 a^2-\sqrt {3} \sqrt {-a^4}\right ) b \sqrt {a^4 x^4+b^4}}-\frac {\left (\sqrt {3} \sqrt {-a^4}+a^2\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 a \left (\sqrt {3} \sqrt {-a^4}+3 a^2\right ) b \sqrt {a^4 x^4+b^4}} \]

[In]

Int[(-b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(b^6 + a^6*x^6)),x]

[Out]

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

Rule 211

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

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 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 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 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 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]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.46 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )+2 \text {arctanh}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]

[In]

Integrate[(-b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(b^6 + a^6*x^6)),x]

[Out]

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

Maple [A] (verified)

Time = 2.95 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.47

method	result	size
elliptic	\(\frac {\left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{3 a b}\right ) \sqrt {2}}{2}\)	\(78\)
default	\(-\frac {\left (\sqrt {a^{2} b^{2}}\, \ln \left (2\right )+2 \sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (2\right )+\sqrt {a^{2} b^{2}}\, \ln \left (\frac {\left (-2 x \,a^{2} b^{2}+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (-\frac {2 \left (-\frac {\sqrt {a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}{2}+\frac {\sqrt {3}\, \left (a^{2} x^{2}+b^{2}\right ) \sqrt {a^{2} b^{2}}}{2}+x \,a^{2} b^{2}\right ) a^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +b^{2}}\right )+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}{2}-\frac {\sqrt {3}\, \left (a^{2} x^{2}+b^{2}\right ) \sqrt {a^{2} b^{2}}}{2}+x \,a^{2} b^{2}\right )}{a^{2} x^{2}-\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +b^{2}}\right )\right ) \sqrt {2}}{6 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}}\)	\(331\)
pseudoelliptic	\(-\frac {\left (\sqrt {a^{2} b^{2}}\, \ln \left (2\right )+2 \sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (2\right )+\sqrt {a^{2} b^{2}}\, \ln \left (\frac {\left (-2 x \,a^{2} b^{2}+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}\right ) a^{2}}{a^{2} x^{2}+b^{2}}\right )+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (-\frac {2 \left (-\frac {\sqrt {a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}{2}+\frac {\sqrt {3}\, \left (a^{2} x^{2}+b^{2}\right ) \sqrt {a^{2} b^{2}}}{2}+x \,a^{2} b^{2}\right ) a^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +b^{2}}\right )+\sqrt {2}\, \sqrt {-a^{2} b^{2}}\, \ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}{2}-\frac {\sqrt {3}\, \left (a^{2} x^{2}+b^{2}\right ) \sqrt {a^{2} b^{2}}}{2}+x \,a^{2} b^{2}\right )}{a^{2} x^{2}-\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +b^{2}}\right )\right ) \sqrt {2}}{6 \sqrt {-a^{2} b^{2}}\, \sqrt {a^{2} b^{2}}}\)	\(331\)

[In]

int((a^6*x^6-b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6+b^6),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.61 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} a b x}{\sqrt {a^{4} x^{4} + b^{4}}}\right ) - 2 \, \log \left (\frac {a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4} - 2 \, \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right )}{6 \, a b} \]

[In]

integrate((a^6*x^6-b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6+b^6),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\left (a^{2} x^{2} + b^{2}\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \]

[In]

integrate((a**6*x**6-b**6)/(a**4*x**4+b**4)**(1/2)/(a**6*x**6+b**6),x)

[Out]

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

Maxima [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]

[In]

integrate((a^6*x^6-b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6+b^6),x, algorithm="maxima")

[Out]

integrate((a^6*x^6 - b^6)/((a^6*x^6 + b^6)*sqrt(a^4*x^4 + b^4)), x)

Giac [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{4} x^{4} + b^{4}}} \,d x } \]

[In]

integrate((a^6*x^6-b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6+b^6),x, algorithm="giac")

[Out]

integrate((a^6*x^6 - b^6)/((a^6*x^6 + b^6)*sqrt(a^4*x^4 + b^4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx=\int -\frac {b^6-a^6\,x^6}{\sqrt {a^4\,x^4+b^4}\,\left (a^6\,x^6+b^6\right )} \,d x \]

[In]

int(-(b^6 - a^6*x^6)/((b^4 + a^4*x^4)^(1/2)*(b^6 + a^6*x^6)),x)

[Out]

int(-(b^6 - a^6*x^6)/((b^4 + a^4*x^4)^(1/2)*(b^6 + a^6*x^6)), x)