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

Optimal. Leaf size=150 \[ -\frac {\tan ^{-1}\left (\frac {-\frac {a^3 x^4}{2^{3/4} b}+\frac {b^3}{2^{3/4} a}+\frac {a b x^2}{\sqrt [4]{2}}}{x \sqrt {a^4 x^4-b^4}}\right )}{2\ 2^{3/4} a b}-\frac {\tanh ^{-1}\left (\frac {2^{3/4} a b x \sqrt {a^4 x^4-b^4}}{a^4 x^4+\sqrt {2} a^2 b^2 x^2-b^4}\right )}{2\ 2^{3/4} a b} \]

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Rubi [C]  time = 0.73, antiderivative size = 400, normalized size of antiderivative = 2.67, number of steps used = 18, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6725, 406, 224, 221, 409, 1219, 1218} \begin {gather*} -\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}}+\frac {\left (a^4-\sqrt {-a^8}\right ) 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^5 \sqrt {a^4 x^4-b^4}}+\frac {\left (\sqrt {-a^8}+a^4\right ) 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^5 \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {a^4 x^4-b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a^4 - Sqrt[-a^8])*b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(2*a^5*Sqrt[-b^4 + a^4*x^4]) + (
(a^4 + Sqrt[-a^8])*b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(2*a^5*Sqrt[-b^4 + a^4*x^4]) - (b
*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[a^6/(-a^8)^(3/4), ArcSin[(a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sq
rt[1 - (a^4*x^4)/b^4]*EllipticPi[(-a^8)^(1/4)/a^2, ArcSin[(a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[
1 - (a^4*x^4)/b^4]*EllipticPi[-(Sqrt[-Sqrt[-a^8]]/a^2), ArcSin[(a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4]) - (b*
Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[Sqrt[-Sqrt[-a^8]]/a^2, ArcSin[(a*x)/b], -1])/(2*a*Sqrt[-b^4 + a^4*x^4])

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 406

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[1/Sqrt[a + b*x^4], x], x] - Di
st[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 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 1218

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

Rule 1219

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + (c*x^4)/a]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 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 {\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}{b^8+a^8 x^8} \, dx &=\int \left (-\frac {\sqrt {-a^8} \left (a^4 b^4+\sqrt {-a^8} b^4\right ) \sqrt {-b^4+a^4 x^4}}{2 a^8 b^4 \left (b^4-\sqrt {-a^8} x^4\right )}+\frac {\sqrt {-a^8} \left (a^4 b^4-\sqrt {-a^8} b^4\right ) \sqrt {-b^4+a^4 x^4}}{2 a^8 b^4 \left (b^4+\sqrt {-a^8} x^4\right )}\right ) \, dx\\ &=\frac {\left (a^4+\sqrt {-a^8}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b^4+\sqrt {-a^8} x^4} \, dx}{2 a^4}-\frac {\left (\sqrt {-a^8} \left (a^4 b^4+\sqrt {-a^8} b^4\right )\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b^4-\sqrt {-a^8} x^4} \, dx}{2 a^8 b^4}\\ &=\frac {1}{2} \left (1+\frac {a^4}{\sqrt {-a^8}}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx+\frac {\left (a^4+\sqrt {-a^8}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{2 a^4}-b^4 \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4-\sqrt {-a^8} x^4\right )} \, dx-b^4 \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+\sqrt {-a^8} x^4\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx\right )-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {\left (\left (1+\frac {a^4}{\sqrt {-a^8}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (a^4+\sqrt {-a^8}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 a^4 \sqrt {-b^4+a^4 x^4}}\\ &=\frac {\left (1+\frac {a^4}{\sqrt {-a^8}}\right ) 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 {\left (a^4+\sqrt {-a^8}\right ) 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^5 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{2 \sqrt {-b^4+a^4 x^4}}\\ &=\frac {\left (1+\frac {a^4}{\sqrt {-a^8}}\right ) 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 {\left (a^4+\sqrt {-a^8}\right ) 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^5 \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{2 a \sqrt {-b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.87, size = 192, normalized size = 1.28 \begin {gather*} -\frac {i \sqrt {1-\frac {a^4 x^4}{b^4}} \left (2 F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-(-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left ((-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )\right )}{2 \sqrt {-\frac {a^2}{b^2}} \sqrt {a^4 x^4-b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.66, size = 156, normalized size = 1.04 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\frac {b^3}{2^{3/4} a}+\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{2\ 2^{3/4} a b}+\frac {\tanh ^{-1}\left (\frac {\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{2\ 2^{3/4} a b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.13, size = 284, normalized size = 1.89

method	result	size
default	\(\frac {\left (\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )}{2 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{2 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\)	\(284\)
elliptic	\(\frac {\left (\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )}{4 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )}{2 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{2 \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\)	\(284\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a^4\,x^4+b^4\right )\,\sqrt {a^4\,x^4-b^4}}{a^8\,x^8+b^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((b^4 + a^4*x^4)*(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 {\sqrt {\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 )}{a^{8} x^{8} + b^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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