\(\int \frac {-b^3+a^3 x^3}{(b^3+a^3 x^3) \sqrt {b^4+a^4 x^4}} \, dx\) [1558]

Optimal result

Integrand size = 42, antiderivative size = 107 \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=-\frac {4 \arctan \left (\frac {a b x}{b^2-a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]

[Out]

-4/3*arctan(a*b*x/(b^2-a*b*x+a^2*x^2+(a^4*x^4+b^4)^(1/2)))/a/b-1/3*2^(1/2)*arctanh(2^(1/2)*a*b*x/(b^2+2*a*b*x+
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 = 1.58 (sec) , antiderivative size = 662, normalized size of antiderivative = 6.19, number of steps used = 29, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {6857, 226, 2099, 1739, 1225, 1713, 214, 1262, 739, 212, 6860, 1231, 1721} \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=-\frac {2 \arctan \left (\frac {a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 a b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{6 a^2 b \sqrt {a^4 x^4+b^4}}-\frac {\left (\sqrt {3} \sqrt {-a^2}+a\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{6 a^2 b \sqrt {a^4 x^4+b^4}}+\frac {\left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {a x}{b}\right ),\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}+\frac {\text {arctanh}\left (\frac {a^2 x^2+b^2}{\sqrt {2} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \text {arctanh}\left (\frac {\sqrt {a} \left (\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 x^2+4 b^2\right )}{2 \sqrt {2} \sqrt {\sqrt {3} \sqrt {-a^2}+a} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {\sqrt {3} \sqrt {-a^2}+a} b}-\frac {\left (\sqrt {3} \sqrt {-a^2}+a\right ) \text {arctanh}\left (\frac {\sqrt {a} \left (\left (\sqrt {3} \sqrt {-a^2}+a\right )^2 x^2+4 b^2\right )}{2 \sqrt {2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} \sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} b} \]

[In]

Int[(-b^3 + a^3*x^3)/((b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),x]

[Out]

(-2*ArcTan[(a*b*x)/Sqrt[b^4 + a^4*x^4]])/(3*a*b) - ArcTanh[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(3*Sqrt[2]*a*b
) + ArcTanh[(b^2 + a^2*x^2)/(Sqrt[2]*Sqrt[b^4 + a^4*x^4])]/(3*Sqrt[2]*a*b) - ((a - Sqrt[3]*Sqrt[-a^2])*ArcTanh
[(Sqrt[a]*(4*b^2 + (a - Sqrt[3]*Sqrt[-a^2])^2*x^2))/(2*Sqrt[2]*Sqrt[a + Sqrt[3]*Sqrt[-a^2]]*Sqrt[b^4 + a^4*x^4
])])/(3*Sqrt[2]*a^(3/2)*Sqrt[a + Sqrt[3]*Sqrt[-a^2]]*b) - ((a + Sqrt[3]*Sqrt[-a^2])*ArcTanh[(Sqrt[a]*(4*b^2 +
(a + Sqrt[3]*Sqrt[-a^2])^2*x^2))/(2*Sqrt[2]*Sqrt[a - Sqrt[3]*Sqrt[-a^2]]*Sqrt[b^4 + a^4*x^4])])/(3*Sqrt[2]*a^(
3/2)*Sqrt[a - Sqrt[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*Ar
cTan[(a*x)/b], 1/2])/(3*a*b*Sqrt[b^4 + a^4*x^4]) - ((a - Sqrt[3]*Sqrt[-a^2])*(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])/(6*a^2*b*Sqrt[b^4 + a^4*x^4]) - ((a + Sqrt[3]*Sqrt[-
a^2])*(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])/(6*a^2*b*Sqrt
[b^4 + a^4*x^4])

Rule 212

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[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 739

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

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 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

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 1739

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

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

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^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx \\ & = -\left (\left (2 b^3\right ) \int \frac {1}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, 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^3\right ) \int \left (\frac {1}{3 b^2 (b+a x) \sqrt {b^4+a^4 x^4}}+\frac {2 b-a x}{3 b^2 \left (b^2-a b x+a^2 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 )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} (2 b) \int \frac {1}{(b+a x) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} (2 b) \int \frac {2 b-a x}{\left (b^2-a b x+a^2 x^2\right ) \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}}-\frac {1}{3} (2 b) \int \left (\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}}+\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx+\frac {1}{3} (2 a b) \int \frac {x}{\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 {1}{\left (b^2-a^2 x^2\right ) \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}}-\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} (a b) \text {Subst}\left (\int \frac {1}{\left (b^2-a^2 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {b^4+a^4 x^4}} \, dx+\frac {1}{3} \left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \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 )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} (a b) \text {Subst}\left (\int \frac {1}{2 a^4 b^4-x^2} \, dx,x,\frac {-a^2 b^4-a^4 b^2 x^2}{\sqrt {b^4+a^4 x^4}}\right )-\frac {1}{3} \left (4 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (4 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\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-\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\text {arctanh}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \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 (a-\sqrt {3} \sqrt {-a^2}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 a}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{3 a}-\frac {1}{3} \left (2 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \text {Subst}\left (\int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (2 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \text {Subst}\left (\int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (4 a \left (a-\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (4 a \left (a+\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx \\ & = -\frac {2 \arctan \left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\text {arctanh}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \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 (a-\sqrt {3} \sqrt {-a^2}\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 )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\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 )}{6 a^2 b \sqrt {b^4+a^4 x^4}}+\frac {1}{3} \left (2 a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \text {Subst}\left (\int \frac {1}{16 a^8 b^4+a^4 \left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {-4 a^4 b^4-a^4 \left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {b^4+a^4 x^4}}\right )+\frac {1}{3} \left (2 a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \text {Subst}\left (\int \frac {1}{16 a^8 b^4+a^4 \left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {-4 a^4 b^4-a^4 \left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {b^4+a^4 x^4}}\right ) \\ & = -\frac {2 \arctan \left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\text {arctanh}\left (\frac {b^2+a^2 x^2}{\sqrt {2} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \text {arctanh}\left (\frac {\sqrt {a} \left (4 b^2+\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {a+\sqrt {3} \sqrt {-a^2}} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a+\sqrt {3} \sqrt {-a^2}} b}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \text {arctanh}\left (\frac {\sqrt {a} \left (4 b^2+\left (a+\sqrt {3} \sqrt {-a^2}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {a-\sqrt {3} \sqrt {-a^2}} \sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a^{3/2} \sqrt {a-\sqrt {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-\sqrt {3} \sqrt {-a^2}\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 )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\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 )}{6 a^2 b \sqrt {b^4+a^4 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.92 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=-\frac {4 \arctan \left (\frac {a b x}{b^2-a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]

[In]

Integrate[(-b^3 + a^3*x^3)/((b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),x]

[Out]

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

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {\left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 a^{3} b \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+\ln \left (2\right )\right ) \sqrt {2}\, \sqrt {-a^{2} b^{2}}+4 \sqrt {a^{2} b^{2}}\, \left (\ln \left (\frac {\left (\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+a b \left (a x -b \right )^{2}\right ) a^{2}}{a^{2} x^{2}-a b x +b^{2}}\right )+\ln \left (2\right )\right )}{6 \sqrt {a^{2} b^{2}}\, \sqrt {-a^{2} b^{2}}}\)	\(174\)
pseudoelliptic	\(-\frac {\left (\ln \left (\frac {\sqrt {2}\, \sqrt {a^{2} b^{2}}\, a^{2} \sqrt {a^{4} x^{4}+b^{4}}-2 a^{3} b \left (a^{2} x^{2}+a b x +b^{2}\right )}{\left (a x +b \right )^{2}}\right )+\ln \left (2\right )\right ) \sqrt {2}\, \sqrt {-a^{2} b^{2}}+4 \sqrt {a^{2} b^{2}}\, \left (\ln \left (\frac {\left (\sqrt {-a^{2} b^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}+a b \left (a x -b \right )^{2}\right ) a^{2}}{a^{2} x^{2}-a b x +b^{2}}\right )+\ln \left (2\right )\right )}{6 \sqrt {a^{2} b^{2}}\, \sqrt {-a^{2} b^{2}}}\)	\(174\)
elliptic	\(a^{3} b^{3} \left (-\frac {2 b^{2} \sqrt {2}\, \ln \left (\frac {4 b^{4}+2 a^{2} b^{2} \left (x^{2}-\frac {b^{2}}{a^{2}}\right )+2 \sqrt {2}\, \sqrt {b^{4}}\, \sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right )^{2} a^{4}+2 a^{2} b^{2} \left (x^{2}-\frac {b^{2}}{a^{2}}\right )+2 b^{4}}}{x^{2}-\frac {b^{2}}{a^{2}}}\right )}{\left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {b^{4}}}-\frac {\left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}+\frac {a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{a^{2} \left (3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {-3 a^{4} b^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}+\frac {\left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {2}\, \ln \left (\frac {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}+\left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}\, \sqrt {4 \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )^{2} a^{4}+4 \left (-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \left (x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}\right )+\frac {2 b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}{2}}{x^{2}-\frac {-a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}}{2 a^{4}}}\right )}{a^{2} \left (-3 a^{2} b^{2}+\sqrt {-3 a^{4} b^{4}}\right ) \sqrt {-3 a^{4} b^{4}}\, \sqrt {\frac {b^{2} \left (a^{2} b^{2}-\sqrt {-3 a^{4} b^{4}}\right )}{a^{2}}}}\right )+\frac {\left (-\frac {\ln \left (a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}+\frac {\ln \left (-a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}\right ) \sqrt {2}}{2}\)	\(951\)

[In]

int((a^3*x^3-b^3)/(a^3*x^3+b^3)/(a^4*x^4+b^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*((ln((2^(1/2)*(a^2*b^2)^(1/2)*a^2*(a^4*x^4+b^4)^(1/2)-2*a^3*b*(a^2*x^2+a*b*x+b^2))/(a*x+b)^2)+ln(2))*2^(1
/2)*(-a^2*b^2)^(1/2)+4*(a^2*b^2)^(1/2)*(ln(((-a^2*b^2)^(1/2)*(a^4*x^4+b^4)^(1/2)+a*b*(a*x-b)^2)*a^2/(a^2*x^2-a
*b*x+b^2))+ln(2)))/(a^2*b^2)^(1/2)/(-a^2*b^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.54 \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\frac {\sqrt {2} \log \left (-\frac {3 \, a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + 3 \, b^{4} + 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} {\left (a^{2} x^{2} + a b x + b^{2}\right )}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ) - 8 \, \arctan \left (\frac {\sqrt {a^{4} x^{4} + b^{4}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right )}{12 \, a b} \]

[In]

integrate((a^3*x^3-b^3)/(a^3*x^3+b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int \frac {\left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\left (a x + b\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{2} x^{2} - a b x + b^{2}\right )}\, dx \]

[In]

integrate((a**3*x**3-b**3)/(a**3*x**3+b**3)/(a**4*x**4+b**4)**(1/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int { \frac {a^{3} x^{3} - b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} + b^{3}\right )}} \,d x } \]

[In]

integrate((a^3*x^3-b^3)/(a^3*x^3+b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="maxima")

[Out]

integrate((a^3*x^3 - b^3)/(sqrt(a^4*x^4 + b^4)*(a^3*x^3 + b^3)), x)

Giac [F]

\[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int { \frac {a^{3} x^{3} - b^{3}}{\sqrt {a^{4} x^{4} + b^{4}} {\left (a^{3} x^{3} + b^{3}\right )}} \,d x } \]

[In]

integrate((a^3*x^3-b^3)/(a^3*x^3+b^3)/(a^4*x^4+b^4)^(1/2),x, algorithm="giac")

[Out]

integrate((a^3*x^3 - b^3)/(sqrt(a^4*x^4 + b^4)*(a^3*x^3 + b^3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-b^3+a^3 x^3}{\left (b^3+a^3 x^3\right ) \sqrt {b^4+a^4 x^4}} \, dx=\int -\frac {b^3-a^3\,x^3}{\left (a^3\,x^3+b^3\right )\,\sqrt {a^4\,x^4+b^4}} \,d x \]

[In]

int(-(b^3 - a^3*x^3)/((b^3 + a^3*x^3)*(b^4 + a^4*x^4)^(1/2)),x)

[Out]

int(-(b^3 - a^3*x^3)/((b^3 + a^3*x^3)*(b^4 + a^4*x^4)^(1/2)), x)