3.14.11 \(\int \frac {1+3 x^4+x^8}{x^2 (1+x^4)^{3/4} (1+3 x^4+3 x^8)} \, dx\)

Optimal. Leaf size=94 \[ -\frac {\sqrt [4]{x^4+1}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )}{\sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right )}{\sqrt {3}} \]

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Rubi [C]  time = 0.79, antiderivative size = 347, normalized size of antiderivative = 3.69, number of steps used = 13, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6728, 264, 1528, 494, 298, 205, 208} \begin {gather*} -\frac {\sqrt [4]{x^4+1}}{x}-\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+1}}\right )-\frac {\left (3-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}}}+\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+1}}\right )+\frac {\left (3-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(1 + 3*x^4 + x^8)/(x^2*(1 + x^4)^(3/4)*(1 + 3*x^4 + 3*x^8)),x]

[Out]

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

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 208

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

Rule 264

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

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 1528

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, q,
n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

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 {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx &=\int \left (\frac {1}{x^2 \left (1+x^4\right )^{3/4}}-\frac {2 x^6}{\left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx\right )+\int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-2 \int \left (\frac {i \left (-3+i \sqrt {3}\right ) x^2}{\sqrt {3} \left (-3+i \sqrt {3}-6 x^4\right ) \left (1+x^4\right )^{3/4}}-\frac {i \left (3+i \sqrt {3}\right ) x^2}{\sqrt {3} \left (1+x^4\right )^{3/4} \left (3+i \sqrt {3}+6 x^4\right )}\right ) \, dx\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (3+i \sqrt {3}+6 x^4\right )} \, dx+\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {x^2}{\left (-3+i \sqrt {3}-6 x^4\right ) \left (1+x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-\left (2 \left (1-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{3+i \sqrt {3}-\left (-3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (2 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{-3+i \sqrt {3}-\left (3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-\frac {\left (i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3 i+\sqrt {3}}-\sqrt {-3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {-3 i+\sqrt {3}}}+\frac {\left (i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3 i+\sqrt {3}}+\sqrt {-3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {-3 i+\sqrt {3}}}+\frac {\left (i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-3 i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {3 i+\sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-3 i+\sqrt {3}}+\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {3 i+\sqrt {3}}}\\ &=-\frac {\sqrt [4]{1+x^4}}{x}-\frac {1}{2} \left (1+\frac {i}{\sqrt {3}}\right ) \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )-\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}}}+\frac {1}{2} \left (1+\frac {i}{\sqrt {3}}\right ) \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )+\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}}}\\ \end {align*}

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Mathematica [F]  time = 7.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 + 3*x^4 + x^8)/(x^2*(1 + x^4)^(3/4)*(1 + 3*x^4 + 3*x^8)),x]

[Out]

Integrate[(1 + 3*x^4 + x^8)/(x^2*(1 + x^4)^(3/4)*(1 + 3*x^4 + 3*x^8)), x]

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IntegrateAlgebraic [A]  time = 0.51, size = 115, normalized size = 1.22 \begin {gather*} -\frac {\sqrt [4]{1+x^4}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + 3*x^4 + x^8)/(x^2*(1 + x^4)^(3/4)*(1 + 3*x^4 + 3*x^8)),x]

[Out]

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

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fricas [B]  time = 19.50, size = 227, normalized size = 2.41 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, {\left (\sqrt {3} {\left (3 \, x^{5} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} - \sqrt {3} {\left (3 \, x^{7} + 4 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}\right )}}{21 \, x^{8} + 21 \, x^{4} - 1}\right ) - \sqrt {3} x \log \left (-\frac {441 \, x^{16} + 882 \, x^{12} + 543 \, x^{8} + 102 \, x^{4} + 4 \, \sqrt {3} {\left (63 \, x^{13} + 78 \, x^{9} + 24 \, x^{5} + x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, \sqrt {3} {\left (63 \, x^{15} + 111 \, x^{11} + 57 \, x^{7} + 8 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 24 \, {\left (18 \, x^{14} + 27 \, x^{10} + 11 \, x^{6} + x^{2}\right )} \sqrt {x^{4} + 1} + 1}{9 \, x^{16} + 18 \, x^{12} + 15 \, x^{8} + 6 \, x^{4} + 1}\right ) + 12 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{12 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+3*x^4+1)/x^2/(x^4+1)^(3/4)/(3*x^8+3*x^4+1),x, algorithm="fricas")

[Out]

-1/12*(2*sqrt(3)*x*arctan(2*(sqrt(3)*(3*x^5 - x)*(x^4 + 1)^(3/4) - sqrt(3)*(3*x^7 + 4*x^3)*(x^4 + 1)^(1/4))/(2
1*x^8 + 21*x^4 - 1)) - sqrt(3)*x*log(-(441*x^16 + 882*x^12 + 543*x^8 + 102*x^4 + 4*sqrt(3)*(63*x^13 + 78*x^9 +
24*x^5 + x)*(x^4 + 1)^(3/4) + 4*sqrt(3)*(63*x^15 + 111*x^11 + 57*x^7 + 8*x^3)*(x^4 + 1)^(1/4) + 24*(18*x^14 +
27*x^10 + 11*x^6 + x^2)*sqrt(x^4 + 1) + 1)/(9*x^16 + 18*x^12 + 15*x^8 + 6*x^4 + 1)) + 12*(x^4 + 1)^(1/4))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} + 3 \, x^{4} + 1}{{\left (3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+3*x^4+1)/x^2/(x^4+1)^(3/4)/(3*x^8+3*x^4+1),x, algorithm="giac")

[Out]

integrate((x^8 + 3*x^4 + 1)/((3*x^8 + 3*x^4 + 1)*(x^4 + 1)^(3/4)*x^2), x)

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maple [C]  time = 30.33, size = 287, normalized size = 3.05

method	result	size
trager	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}+\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {12 \RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {x^{4}+1}\, x^{6}+9 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{8}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+18 \left (x^{4}+1\right )^{\frac {1}{4}} x^{7}+6 \RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {x^{4}+1}\, x^{2}+9 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{4}+6 \left (x^{4}+1\right )^{\frac {3}{4}} x +12 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}-3\right )}{3 x^{8}+3 x^{4}+1}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {12 \RootOf \left (\textit {\_Z}^{2}+3\right ) \sqrt {x^{4}+1}\, x^{6}-9 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{8}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+18 \left (x^{4}+1\right )^{\frac {1}{4}} x^{7}+6 \RootOf \left (\textit {\_Z}^{2}+3\right ) \sqrt {x^{4}+1}\, x^{2}-9 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-6 \left (x^{4}+1\right )^{\frac {3}{4}} x +12 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+3\right )}{3 x^{8}+3 x^{4}+1}\right )}{6}\)	\(287\)
risch	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-9 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{16}-18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{13}+12 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{10}-27 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{12}+18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{7}-42 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+18 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-28 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{8}+12 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{3}-30 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+6 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}-11 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-6 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (x^{4}+1\right )^{2} \left (3 x^{8}+3 x^{4}+1\right )}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {9 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{16}+18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{13}+12 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{10}+27 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{12}+18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{7}+42 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+18 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{6}+28 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{8}+12 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{3}+30 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+6 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+11 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{4}+6 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x +\RootOf \left (\textit {\_Z}^{2}-3\right )}{\left (x^{4}+1\right )^{2} \left (3 x^{8}+3 x^{4}+1\right )}\right )}{6}\right ) \left (\left (x^{4}+1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{4}+1\right )^{\frac {3}{4}}}\)	\(626\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8+3*x^4+1)/x^2/(x^4+1)^(3/4)/(3*x^8+3*x^4+1),x,method=_RETURNVERBOSE)

[Out]

-(x^4+1)^(1/4)/x+1/6*RootOf(_Z^2-3)*ln((12*RootOf(_Z^2-3)*(x^4+1)^(1/2)*x^6+9*RootOf(_Z^2-3)*x^8+18*(x^4+1)^(3
/4)*x^5+18*(x^4+1)^(1/4)*x^7+6*RootOf(_Z^2-3)*(x^4+1)^(1/2)*x^2+9*RootOf(_Z^2-3)*x^4+6*(x^4+1)^(3/4)*x+12*x^3*
(x^4+1)^(1/4)+RootOf(_Z^2-3))/(3*x^8+3*x^4+1))-1/6*RootOf(_Z^2+3)*ln(-(12*RootOf(_Z^2+3)*(x^4+1)^(1/2)*x^6-9*R
ootOf(_Z^2+3)*x^8-18*(x^4+1)^(3/4)*x^5+18*(x^4+1)^(1/4)*x^7+6*RootOf(_Z^2+3)*(x^4+1)^(1/2)*x^2-9*RootOf(_Z^2+3
)*x^4-6*(x^4+1)^(3/4)*x+12*x^3*(x^4+1)^(1/4)-RootOf(_Z^2+3))/(3*x^8+3*x^4+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} + 3 \, x^{4} + 1}{{\left (3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+3*x^4+1)/x^2/(x^4+1)^(3/4)/(3*x^8+3*x^4+1),x, algorithm="maxima")

[Out]

integrate((x^8 + 3*x^4 + 1)/((3*x^8 + 3*x^4 + 1)*(x^4 + 1)^(3/4)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^8+3\,x^4+1}{x^2\,{\left (x^4+1\right )}^{3/4}\,\left (3\,x^8+3\,x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4 + x^8 + 1)/(x^2*(x^4 + 1)^(3/4)*(3*x^4 + 3*x^8 + 1)),x)

[Out]

int((3*x^4 + x^8 + 1)/(x^2*(x^4 + 1)^(3/4)*(3*x^4 + 3*x^8 + 1)), x)

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sympy [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((x**8+3*x**4+1)/x**2/(x**4+1)**(3/4)/(3*x**8+3*x**4+1),x)

[Out]

Timed out

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