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

Optimal. Leaf size=107 \[ -\frac {4 \text {ArcTan}\left (\frac {a b x}{b^2-a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \]

[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

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.79, antiderivative size = 662, normalized size of antiderivative = 6.19, number of steps used = 29, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {6857, 226, 2099, 1739, 1225, 1713, 214, 1262, 739, 212, 6860, 1231, 1721} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\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}} F\left (2 \text {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}} F\left (2 \text {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}} F\left (2 \text {ArcTan}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}}+\frac {\tanh ^{-1}\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 \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}} 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^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}} 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 {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}} 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 {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}} 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 {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}} F\left (2 \tan ^{-1}\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 {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\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}} F\left (2 \tan ^{-1}\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 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\tanh ^{-1}\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [A]
time = 1.56, size = 100, normalized size = 0.93 \begin {gather*} -\frac {4 \text {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} \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.17, size = 457, normalized size = 4.27

method	result	size
default	\(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {2 b \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 a^{2} b^{2} x^{2}+2 b^{4}\right ) \sqrt {2}}{4 \sqrt {b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 \sqrt {b^{4}}}+\frac {a \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, -i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b \sqrt {a^{4} x^{4}+b^{4}}}\right )}{3 a}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a +2 b \right ) \left (-\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) a b \left (a \,x^{2}-b \underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {-b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {-b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}}+\frac {2 a \left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{2} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{3 a}\)	\(457\)
elliptic	\(\text {Expression too large to display}\)	\(1539\)

Verification of antiderivative is not currently implemented for this CAS.

[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/(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)-2/3*b/a*(-1/4*2^(1/2)/(b^4)^(1/2)*arctanh(1/4*(2*a^2*b^2*x^2+2*b^4)*2^(1/2)/(b^4)^(1/2)/(a^4*x^4+b^
4)^(1/2))+1/(I*a^2/b^2)^(1/2)/b*a*(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)*Elliptic
Pi(x*(I*a^2/b^2)^(1/2),-I,(-I*a^2/b^2)^(1/2)/(I*a^2/b^2)^(1/2)))-1/3*b/a*sum((-_alpha*a+2*b)/(2*_alpha*a-b)*(-
1/(-b^3*(_alpha*a-b))^(1/2)*arctanh((_alpha*a-b)*a*b*(a*x^2-_alpha*b)/(-b^3*(_alpha*a-b))^(1/2)/(a^4*x^4+b^4)^
(1/2))+2/(I*a^2/b^2)^(1/2)*a*(_alpha*a-b)/b^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)*EllipticPi(x*(I*a^2/b^2)^(1/2),I*_alpha*a/b,(-I*a^2/b^2)^(1/2)/(I*a^2/b^2)^(1/2))),_alpha=RootOf(_Z^2*a^2-
_Z*a*b+b^2))

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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^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)

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Fricas [A]
time = 0.52, size = 165, normalized size = 1.54 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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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^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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