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

Optimal. Leaf size=166 \[ -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{3 a b}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {4-2 \sqrt {3}} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{3 a b}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {4+2 \sqrt {3}} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{3 a b} \]

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Rubi [C]  time = 2.20, antiderivative size = 405, normalized size of antiderivative = 2.44, number of steps used = 17, number of rules used = 9, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6725, 220, 2073, 1211, 1699, 205, 6728, 1217, 1707} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 205

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

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1211

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 1217

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d +
e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
&& PosQ[c/a]

Rule 1699

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 1707

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, -Simp[((B*d - A*e)*ArcTan[(Rt[(c*d)/e + (a*e)/d, 2]*x)/Sqrt[a + c*x^4]])/(2*d*e*Rt[(c*d)/e + (a*e)/d, 2]),
x] + Simp[((B*d + A*e)*(A + B*x^2)*Sqrt[(A^2*(a + c*x^4))/(a*(A + B*x^2)^2)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 2073

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 6725

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

Rule 6728

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^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^6+a^6 x^6\right )} \, dx &=\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}} 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^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}} 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} \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}} 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} \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}} 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} b^2 \operatorname {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 {\tan ^{-1}\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}} 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 (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 {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}-\frac {2 \tan ^{-1}\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}} 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^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}} F\left (2 \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} \Pi \left (\frac {3}{4};2 \tan ^{-1}\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}} \Pi \left (\frac {3}{4};2 \tan ^{-1}\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*}

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Mathematica [C]  time = 0.71, size = 195, normalized size = 1.17 \begin {gather*} -\frac {(-1)^{5/6} \sqrt {\frac {a^4 x^4}{b^4}+1} \left (\left (3-3 i \sqrt {3}\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+4 \left ((-1)^{2/3} \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+\left (\sqrt [3]{-1}-1\right ) \Pi \left (\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+(-1)^{2/3} \Pi \left ((-1)^{5/6};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )\right )\right )}{6 \sqrt {\frac {i a^2}{b^2}} \sqrt {a^4 x^4+b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*((-1)^(5/6)*Sqrt[1 + (a^4*x^4)/b^4]*((3 - (3*I)*Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] +
4*((-1)^(2/3)*EllipticPi[-I, I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] + (-1 + (-1)^(1/3))*EllipticPi[(-1)^(1/6), I*
ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] + (-1)^(2/3)*EllipticPi[(-1)^(5/6), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1])))/(
Sqrt[(I*a^2)/b^2]*Sqrt[b^4 + a^4*x^4])

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IntegrateAlgebraic [A]  time = 1.21, size = 84, normalized size = 0.51 \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\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 \tanh ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 1.08, size = 101, normalized size = 0.61 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.21, size = 78, normalized size = 0.47

method	result	size
elliptic	\(\frac {\left (-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{3 a b}\right ) \sqrt {2}}{2}\)	\(78\)
default	\(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {2 \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{3 \sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{4} \textit {\_Z}^{4}-a^{2} b^{2} \textit {\_Z}^{2}+b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 b^{2}\right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a^{2} x^{2}+b^{2}\right ) a^{2}}{\sqrt {a^{2} b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {a^{2} b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 a^{2} \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right )}{b^{2}}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right )}\right )}{6 a^{2}}\)	\(441\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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