3.6.30 \(\int \frac {(-1+x^4) \sqrt {1+x^4} (1+x^2+x^4)}{x^4 (1-x^2+x^4)} \, dx\)

Optimal. Leaf size=41 \[ \frac {\sqrt {x^4+1} \left (x^4+6 x^2+1\right )}{3 x^3}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right ) \]

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Rubi [C]  time = 1.19, antiderivative size = 334, normalized size of antiderivative = 8.15, number of steps used = 26, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6728, 195, 220, 277, 305, 1196, 1209, 1198, 1217, 1707} \begin {gather*} \frac {1}{3} \sqrt {x^4+1} x+\frac {2 \sqrt {x^4+1}}{x}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )+\frac {\sqrt {x^4+1}}{3 x^3}-\frac {\left (\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\left (\sqrt {3}+3 i\right ) \sqrt {x^4+1}}+\frac {\left (3+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {x^4+1}}+\frac {\left (3-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {x^4+1}}-\frac {\left (-\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\left (-\sqrt {3}+3 i\right ) \sqrt {x^4+1}}-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {x^4+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^4)*Sqrt[1 + x^4]*(1 + x^2 + x^4))/(x^4*(1 - x^2 + x^4)),x]

[Out]

Sqrt[1 + x^4]/(3*x^3) + (2*Sqrt[1 + x^4])/x + (x*Sqrt[1 + x^4])/3 - 2*ArcTanh[x/Sqrt[1 + x^4]] - (2*(1 + x^2)*
Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/Sqrt[1 + x^4] - ((I - Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^
4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/((3*I - Sqrt[3])*Sqrt[1 + x^4]) + ((3 - I*Sqrt[3])*(1 + x^2)*Sqrt
[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(2*Sqrt[1 + x^4]) + ((3 + I*Sqrt[3])*(1 + x^2)*Sqrt[(1 +
x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(2*Sqrt[1 + x^4]) - ((I + Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1
+ x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/((3*I + Sqrt[3])*Sqrt[1 + x^4])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 305

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

Rule 1196

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

Rule 1198

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

Rule 1209

Int[((a_) + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(e^2)^(-1), Int[(c*d - c*e*x^2)*(a +
c*x^4)^(p - 1), x], x] + Dist[(c*d^2 + a*e^2)/e^2, Int[(a + c*x^4)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, c,
d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p + 1/2, 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 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 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 {\left (-1+x^4\right ) \sqrt {1+x^4} \left (1+x^2+x^4\right )}{x^4 \left (1-x^2+x^4\right )} \, dx &=\int \left (\sqrt {1+x^4}-\frac {\sqrt {1+x^4}}{x^4}-\frac {2 \sqrt {1+x^4}}{x^2}+\frac {2 \left (-1+2 x^2\right ) \sqrt {1+x^4}}{1-x^2+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^4}}{x^2} \, dx\right )+2 \int \frac {\left (-1+2 x^2\right ) \sqrt {1+x^4}}{1-x^2+x^4} \, dx+\int \sqrt {1+x^4} \, dx-\int \frac {\sqrt {1+x^4}}{x^4} \, dx\\ &=\frac {\sqrt {1+x^4}}{3 x^3}+\frac {2 \sqrt {1+x^4}}{x}+\frac {1}{3} x \sqrt {1+x^4}+2 \int \left (\frac {2 \sqrt {1+x^4}}{-1-i \sqrt {3}+2 x^2}+\frac {2 \sqrt {1+x^4}}{-1+i \sqrt {3}+2 x^2}\right ) \, dx-4 \int \frac {x^2}{\sqrt {1+x^4}} \, dx\\ &=\frac {\sqrt {1+x^4}}{3 x^3}+\frac {2 \sqrt {1+x^4}}{x}+\frac {1}{3} x \sqrt {1+x^4}-4 \int \frac {1}{\sqrt {1+x^4}} \, dx+4 \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+4 \int \frac {\sqrt {1+x^4}}{-1-i \sqrt {3}+2 x^2} \, dx+4 \int \frac {\sqrt {1+x^4}}{-1+i \sqrt {3}+2 x^2} \, dx\\ &=\frac {\sqrt {1+x^4}}{3 x^3}+\frac {2 \sqrt {1+x^4}}{x}+\frac {1}{3} x \sqrt {1+x^4}-\frac {4 x \sqrt {1+x^4}}{1+x^2}+\frac {4 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {1+x^4}}-\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {1+x^4}}+\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}} \, dx+\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}} \, dx-\int \frac {-1-i \sqrt {3}-2 x^2}{\sqrt {1+x^4}} \, dx-\int \frac {-1+i \sqrt {3}-2 x^2}{\sqrt {1+x^4}} \, dx\\ &=\frac {\sqrt {1+x^4}}{3 x^3}+\frac {2 \sqrt {1+x^4}}{x}+\frac {1}{3} x \sqrt {1+x^4}-\frac {4 x \sqrt {1+x^4}}{1+x^2}+\frac {4 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {1+x^4}}-\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {1+x^4}}-2 \left (2 \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-\frac {\left (2 \left (i-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{3 i-\sqrt {3}}+\frac {\left (4 \left (i-\sqrt {3}\right )\right ) \int \frac {1+x^2}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{3 i-\sqrt {3}}-\left (-3+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\left (3+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {\left (2 \left (i+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{3 i+\sqrt {3}}+\frac {\left (4 \left (i+\sqrt {3}\right )\right ) \int \frac {1+x^2}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{3 i+\sqrt {3}}\\ &=\frac {\sqrt {1+x^4}}{3 x^3}+\frac {2 \sqrt {1+x^4}}{x}+\frac {1}{3} x \sqrt {1+x^4}-\frac {4 x \sqrt {1+x^4}}{1+x^2}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {4 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {1+x^4}}-2 \left (-\frac {2 x \sqrt {1+x^4}}{1+x^2}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {1+x^4}}\right )-\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {\left (3+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.48, size = 145, normalized size = 3.54 \begin {gather*} \frac {x^8+6 x^6+2 x^4+6 x^2-6 \sqrt [4]{-1} \sqrt {x^4+1} x^3 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+6 \sqrt [4]{-1} \sqrt {x^4+1} x^3 \Pi \left (\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+6 \sqrt [4]{-1} \sqrt {x^4+1} x^3 \Pi \left ((-1)^{5/6};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+1}{3 x^3 \sqrt {x^4+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^4)*Sqrt[1 + x^4]*(1 + x^2 + x^4))/(x^4*(1 - x^2 + x^4)),x]

[Out]

(1 + 6*x^2 + 2*x^4 + 6*x^6 + x^8 - 6*(-1)^(1/4)*x^3*Sqrt[1 + x^4]*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] + 6*(
-1)^(1/4)*x^3*Sqrt[1 + x^4]*EllipticPi[(-1)^(1/6), I*ArcSinh[(-1)^(1/4)*x], -1] + 6*(-1)^(1/4)*x^3*Sqrt[1 + x^
4]*EllipticPi[(-1)^(5/6), I*ArcSinh[(-1)^(1/4)*x], -1])/(3*x^3*Sqrt[1 + x^4])

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IntegrateAlgebraic [A]  time = 0.71, size = 41, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^4} \left (1+6 x^2+x^4\right )}{3 x^3}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^4]*(1 + x^2 + x^4))/(x^4*(1 - x^2 + x^4)),x]

[Out]

(Sqrt[1 + x^4]*(1 + 6*x^2 + x^4))/(3*x^3) - 2*ArcTanh[x/Sqrt[1 + x^4]]

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fricas [A]  time = 0.52, size = 62, normalized size = 1.51 \begin {gather*} \frac {3 \, x^{3} \log \left (-\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 1} x + 1}{x^{4} - x^{2} + 1}\right ) + {\left (x^{4} + 6 \, x^{2} + 1\right )} \sqrt {x^{4} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*x^3*log(-(x^4 + x^2 - 2*sqrt(x^4 + 1)*x + 1)/(x^4 - x^2 + 1)) + (x^4 + 6*x^2 + 1)*sqrt(x^4 + 1))/x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{2} + 1\right )} \sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} - x^{2} + 1\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + x^2 + 1)*sqrt(x^4 + 1)*(x^4 - 1)/((x^4 - x^2 + 1)*x^4), x)

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maple [A]  time = 0.32, size = 54, normalized size = 1.32

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \left (x^{4}+1\right )^{\frac {3}{2}}}{3 x^{3}}+\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}-2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+1}}{x}\right )\right ) \sqrt {2}}{2}\)	\(54\)
trager	\(\frac {\sqrt {x^{4}+1}\, \left (x^{4}+6 x^{2}+1\right )}{3 x^{3}}-\ln \left (\frac {x^{4}+2 \sqrt {x^{4}+1}\, x +x^{2}+1}{x^{4}-x^{2}+1}\right )\)	\(58\)
risch	\(\frac {x^{8}+6 x^{6}+2 x^{4}+6 x^{2}+1}{3 x^{3} \sqrt {x^{4}+1}}+\frac {2 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{2}-i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{2}\)	\(203\)
default	\(\frac {\sqrt {x^{4}+1}\, x}{3}+\frac {2 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\sqrt {x^{4}+1}}{3 x^{3}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{2}-i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{2}+\frac {2 \sqrt {x^{4}+1}}{x}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(1/3*2^(1/2)/x^3*(x^4+1)^(3/2)+2*2^(1/2)/x*(x^4+1)^(1/2)-2*2^(1/2)*arctanh((x^4+1)^(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 {{\left (x^{4} + x^{2} + 1\right )} \sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} - x^{2} + 1\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + x^2 + 1)*sqrt(x^4 + 1)*(x^4 - 1)/((x^4 - x^2 + 1)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^4-1\right )\,\sqrt {x^4+1}\,\left (x^4+x^2+1\right )}{x^4\,\left (x^4-x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} + 1} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{x^{4} \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x - 1)*(x + 1)*(x**2 + 1)*sqrt(x**4 + 1)*(x**2 - x + 1)*(x**2 + x + 1)/(x**4*(x**4 - x**2 + 1)), x)

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