3.29.87 \(\int \frac {b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} (-b^8+a^8 x^8)} \, dx\)

Optimal. Leaf size=313 \[ -\frac {x}{2 \sqrt {a^4 x^4-b^4}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {a^4 x^4-b^4}}{\sqrt {3-2 \sqrt {2}} a b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3-2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3-2 \sqrt {2}} a}}{x}\right )}{a b}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {a^4 x^4-b^4}}{\sqrt {3+2 \sqrt {2}} a b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3+2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3+2 \sqrt {2}} a}}{x}\right )}{a b}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2}\right )}{a b} \]

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Rubi [A]  time = 0.71, antiderivative size = 135, normalized size of antiderivative = 0.43, number of steps used = 20, number of rules used = 11, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6725, 224, 221, 1404, 414, 523, 409, 1211, 1699, 203, 206} \begin {gather*} -\frac {x}{2 \sqrt {a^4 x^4-b^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{4 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{4 \sqrt {2} \sqrt [4]{-a^4} b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b^8 + a^8*x^8)/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]

[Out]

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

Rule 203

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

Rule 206

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 409

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 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 523

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 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 1404

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

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 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]

Rubi steps

\begin {align*} \int \frac {b^8+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx &=\int \left (\frac {1}{\sqrt {-b^4+a^4 x^4}}+\frac {2 b^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )}\right ) \, dx\\ &=\left (2 b^8\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx+\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx\\ &=\left (2 b^8\right ) \int \frac {1}{\left (-b^4+a^4 x^4\right )^{3/2} \left (b^4+a^4 x^4\right )} \, dx+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{\sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {\int \frac {-3 a^4 b^4-a^8 x^4}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{2 a^4}\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{2} \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx-b^4 \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}-2 \left (\frac {1}{4} \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx\right )-\frac {1}{4} \int \frac {1-\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{4} \int \frac {1+\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )-2 \frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{4 \sqrt {2} \sqrt [4]{-a^4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{4 \sqrt {2} \sqrt [4]{-a^4} b}\\ \end {align*}

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Mathematica [C]  time = 0.63, size = 201, normalized size = 0.64 \begin {gather*} \frac {x \left (-\frac {5 \left (b^8-a^4 b^4 x^4\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )}{\left (a^4 x^4+b^4\right ) \left (5 b^4 F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )-2 a^4 x^4 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )+F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )\right )\right )}-1\right )}{2 \sqrt {a^4 x^4-b^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(b^8 + a^8*x^8)/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]

[Out]

(x*(-1 - (5*(b^8 - a^4*b^4*x^4)*AppellF1[1/4, -1/2, 1, 5/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)])/((b^4 + a^4*x^4)
*(5*b^4*AppellF1[1/4, -1/2, 1, 5/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)] - 2*a^4*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4
, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)] + AppellF1[5/4, 1/2, 1, 9/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)])))))/(2*Sqrt[
-b^4 + a^4*x^4])

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IntegrateAlgebraic [A]  time = 1.09, size = 247, normalized size = 0.79 \begin {gather*} -\frac {x}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a b}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tanh ^{-1}\left (\frac {b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-a^4 x^4+\left (-i b^2-(1-i) a b x-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left (b^2+(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right )}{a b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b^8 + a^8*x^8)/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]

[Out]

-1/2*x/Sqrt[-b^4 + a^4*x^4] - ((1/4 - I/4)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(
a*b) + ((1/8 - I/8)*ArcTanh[(b^4 - (1 + I)*a*b^3*x - (1 - I)*a^3*b*x^3 - a^4*x^4 + ((-I)*b^2 - (1 - I)*a*b*x -
 a^2*x^2)*Sqrt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 + (1 + I)*a*b*
x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])])/(a*b)

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fricas [A]  time = 1.03, size = 162, normalized size = 0.52 \begin {gather*} -\frac {4 \, \sqrt {a^{4} x^{4} - b^{4}} a b x - 2 \, {\left (a^{4} x^{4} - b^{4}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) - {\left (a^{4} x^{4} - b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} - b^{4} - 2 \, \sqrt {a^{4} x^{4} - b^{4}} a b x}{a^{4} x^{4} + b^{4}}\right )}{8 \, {\left (a^{5} b x^{4} - a b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^8*x^8+b^8)/(a^4*x^4-b^4)^(1/2)/(a^8*x^8-b^8),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{8} x^{8} + b^{8}}{{\left (a^{8} x^{8} - b^{8}\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^8*x^8+b^8)/(a^4*x^4-b^4)^(1/2)/(a^8*x^8-b^8),x, algorithm="giac")

[Out]

integrate((a^8*x^8 + b^8)/((a^8*x^8 - b^8)*sqrt(a^4*x^4 - b^4)), x)

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maple [A]  time = 0.43, size = 249, normalized size = 0.80

method result size
elliptic \(\frac {\left (\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{16 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, x}{2 \sqrt {a^{4} x^{4}-b^{4}}}\right ) \sqrt {2}}{2}\) \(249\)
default \(\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {b \left (\frac {a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}\right )}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{4}+\frac {b \left (-\frac {a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}\right )}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{4}-\frac {b^{4} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}+b^{4}\right )}{\sum }\frac {-\frac {\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) a^{4}}{\sqrt {-2 b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {-b^{4}}}+\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{4} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}}{b^{2}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{16 a^{4}}-\frac {b^{2} \left (-\frac {\left (a^{4} x^{2}-a^{2} b^{2}\right ) x}{2 b^{4} a^{2} \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) \left (a^{4} x^{2}-a^{2} b^{2}\right )}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{2}\) \(981\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^8*x^8+b^8)/(a^4*x^4-b^4)^(1/2)/(a^8*x^8-b^8),x,method=_RETURNVERBOSE)

[Out]

1/2*(1/16/(a^4*b^4)^(1/4)*2^(1/2)*ln((1/2*(a^4*x^4-b^4)/x^2-(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1
/2))/(1/2*(a^4*x^4-b^4)/x^2+(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2)))+1/8/(a^4*b^4)^(1/4)*2^(1/2
)*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+1)+1/8/(a^4*b^4)^(1/4)*2^(1/2)*arctan(1/(a^4*b^4)^(1/4)*(a^4*
x^4-b^4)^(1/2)/x-1)-1/2/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x)*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{8} x^{8} + b^{8}}{{\left (a^{8} x^{8} - b^{8}\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^8*x^8+b^8)/(a^4*x^4-b^4)^(1/2)/(a^8*x^8-b^8),x, algorithm="maxima")

[Out]

integrate((a^8*x^8 + b^8)/((a^8*x^8 - b^8)*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 {a^8\,x^8+b^8}{\sqrt {a^4\,x^4-b^4}\,\left (b^8-a^8\,x^8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^8 + a^8*x^8)/((a^4*x^4 - b^4)^(1/2)*(b^8 - a^8*x^8)),x)

[Out]

int(-(b^8 + a^8*x^8)/((a^4*x^4 - b^4)^(1/2)*(b^8 - a^8*x^8)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{8} x^{8} + b^{8}}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**8*x**8+b**8)/(a**4*x**4-b**4)**(1/2)/(a**8*x**8-b**8),x)

[Out]

Integral((a**8*x**8 + b**8)/(sqrt((a*x - b)*(a*x + b)*(a**2*x**2 + b**2))*(a*x - b)*(a*x + b)*(a**2*x**2 + b**
2)*(a**4*x**4 + b**4)), x)

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