3.14.35 \(\int \frac {-1+x^4}{(1+x^2+x^4) \sqrt [4]{x^2+x^6}} \, dx\)

Optimal. Leaf size=96 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+x^2}}{\sqrt {x^6+x^2}-x^2}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^6+x^2}}{\sqrt {2}}}{x \sqrt [4]{x^6+x^2}}\right )}{\sqrt {2}} \]

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Rubi [C]  time = 0.88, antiderivative size = 319, normalized size of antiderivative = 3.32, number of steps used = 15, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2056, 6715, 6728, 245, 1438, 429, 510} \begin {gather*} -\frac {2 \sqrt [4]{x^4+1} x F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 x^4}{1-i \sqrt {3}},-x^4\right )}{\sqrt [4]{x^6+x^2}}-\frac {2 \sqrt [4]{x^4+1} x F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 x^4}{1+i \sqrt {3}},-x^4\right )}{\sqrt [4]{x^6+x^2}}-\frac {2 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+1} x^3 F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{5 \left (\sqrt {3}+i\right ) \sqrt [4]{x^6+x^2}}-\frac {2 \left (\sqrt {3}+i\right ) \sqrt [4]{x^4+1} x^3 F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{5 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^6+x^2}}+\frac {2 \sqrt [4]{x^4+1} x \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-1 + x^4)/((1 + x^2 + x^4)*(x^2 + x^6)^(1/4)),x]

[Out]

(-2*x*(1 + x^4)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8, (-2*x^4)/(1 - I*Sqrt[3]), -x^4])/(x^2 + x^6)^(1/4) - (2*x*(1
+ x^4)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8, (-2*x^4)/(1 + I*Sqrt[3]), -x^4])/(x^2 + x^6)^(1/4) - (2*(I - Sqrt[3])*
x^3*(1 + x^4)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, -x^4, (-2*x^4)/(1 - I*Sqrt[3])])/(5*(I + Sqrt[3])*(x^2 + x^6)^
(1/4)) - (2*(I + Sqrt[3])*x^3*(1 + x^4)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, -x^4, (-2*x^4)/(1 + I*Sqrt[3])])/(5*
(I - Sqrt[3])*(x^2 + x^6)^(1/4)) + (2*x*(1 + x^4)^(1/4)*Hypergeometric2F1[1/8, 1/4, 9/8, -x^4])/(x^2 + x^6)^(1
/4)

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 1438

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {-1+x^4}{\sqrt {x} \sqrt [4]{1+x^4} \left (1+x^2+x^4\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {-1+x^8}{\sqrt [4]{1+x^8} \left (1+x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{1+x^8}}-\frac {2+x^4}{\sqrt [4]{1+x^8} \left (1+x^4+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {2+x^4}{\sqrt [4]{1+x^8} \left (1+x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1-i \sqrt {3}}{\left (1-i \sqrt {3}+2 x^4\right ) \sqrt [4]{1+x^8}}+\frac {1+i \sqrt {3}}{\left (1+i \sqrt {3}+2 x^4\right ) \sqrt [4]{1+x^8}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x^4\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x^4\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {i+\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^8\right ) \sqrt [4]{1+x^8}}+\frac {x^4}{\sqrt [4]{1+x^8} \left (1+i \sqrt {3}+2 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-i+\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^8\right ) \sqrt [4]{1+x^8}}+\frac {x^4}{\sqrt [4]{1+x^8} \left (1-i \sqrt {3}+2 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{1+x^8} \left (1+i \sqrt {3}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{1+x^8} \left (1-i \sqrt {3}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {2 x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 x^4}{1-i \sqrt {3}},-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {2 x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 x^4}{1+i \sqrt {3}},-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {2 \left (i-\sqrt {3}\right ) x^3 \sqrt [4]{1+x^4} F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{5 \left (i+\sqrt {3}\right ) \sqrt [4]{x^2+x^6}}-\frac {2 \left (i+\sqrt {3}\right ) x^3 \sqrt [4]{1+x^4} F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{5 \left (i-\sqrt {3}\right ) \sqrt [4]{x^2+x^6}}+\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-1 + x^4)/((1 + x^2 + x^4)*(x^2 + x^6)^(1/4)),x]

[Out]

Integrate[(-1 + x^4)/((1 + x^2 + x^4)*(x^2 + x^6)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 0.00, size = 96, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^2+x^6}}{-x^2+\sqrt {x^2+x^6}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2}}}{x \sqrt [4]{x^2+x^6}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^4)/((1 + x^2 + x^4)*(x^2 + x^6)^(1/4)),x]

[Out]

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

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fricas [B]  time = 57.21, size = 678, normalized size = 7.06 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {x^{9} + 2 \, x^{7} + 3 \, x^{5} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 3 \, x^{2} + 1\right )} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} + x^{2}} {\left (x^{5} + x^{3} + x\right )} + {\left (16 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} + 2 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 3 \, x^{3} + x\right )} + \sqrt {2} {\left (x^{9} - 8 \, x^{7} + x^{5} - 8 \, x^{3} + x\right )} + 4 \, {\left (x^{6} + x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + x^{3} + x}} + x}{x^{9} - 14 \, x^{7} + 3 \, x^{5} - 14 \, x^{3} + x}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {x^{9} + 2 \, x^{7} + 3 \, x^{5} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} + x^{2}} {\left (x^{5} + x^{3} + x\right )} + {\left (16 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} - 2 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 3 \, x^{3} + x\right )} - \sqrt {2} {\left (x^{9} - 8 \, x^{7} + x^{5} - 8 \, x^{3} + x\right )} + 4 \, {\left (x^{6} + x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + x^{3} + x}} + x}{x^{9} - 14 \, x^{7} + 3 \, x^{5} - 14 \, x^{3} + x}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + x^{3} + x}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + x^{3} + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x, algorithm="fricas")

[Out]

-1/2*sqrt(2)*arctan((x^9 + 2*x^7 + 3*x^5 + 2*x^3 + 2*sqrt(2)*(x^6 + x^2)^(3/4)*(x^4 - 3*x^2 + 1) + 2*sqrt(2)*(
3*x^6 - x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 4*sqrt(x^6 + x^2)*(x^5 + x^3 + x) + (16*(x^6 + x^2)^(3/4)*x^2 + 2*sqr
t(2)*sqrt(x^6 + x^2)*(x^5 - 3*x^3 + x) + sqrt(2)*(x^9 - 8*x^7 + x^5 - 8*x^3 + x) + 4*(x^6 + x^4 + x^2)*(x^6 +
x^2)^(1/4))*sqrt((x^5 + x^3 + 2*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(x^6 + x^2)*x + 2*sqrt(2)*(x^6 + x^2)^(3
/4) + x)/(x^5 + x^3 + x)) + x)/(x^9 - 14*x^7 + 3*x^5 - 14*x^3 + x)) + 1/2*sqrt(2)*arctan((x^9 + 2*x^7 + 3*x^5
+ 2*x^3 - 2*sqrt(2)*(x^6 + x^2)^(3/4)*(x^4 - 3*x^2 + 1) - 2*sqrt(2)*(3*x^6 - x^4 + 3*x^2)*(x^6 + x^2)^(1/4) +
4*sqrt(x^6 + x^2)*(x^5 + x^3 + x) + (16*(x^6 + x^2)^(3/4)*x^2 - 2*sqrt(2)*sqrt(x^6 + x^2)*(x^5 - 3*x^3 + x) -
sqrt(2)*(x^9 - 8*x^7 + x^5 - 8*x^3 + x) + 4*(x^6 + x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt((x^5 + x^3 - 2*sqrt(2)*(
x^6 + x^2)^(1/4)*x^2 + 4*sqrt(x^6 + x^2)*x - 2*sqrt(2)*(x^6 + x^2)^(3/4) + x)/(x^5 + x^3 + x)) + x)/(x^9 - 14*
x^7 + 3*x^5 - 14*x^3 + x)) - 1/8*sqrt(2)*log(4*(x^5 + x^3 + 2*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 4*sqrt(x^6 + x^2
)*x + 2*sqrt(2)*(x^6 + x^2)^(3/4) + x)/(x^5 + x^3 + x)) + 1/8*sqrt(2)*log(4*(x^5 + x^3 - 2*sqrt(2)*(x^6 + x^2)
^(1/4)*x^2 + 4*sqrt(x^6 + x^2)*x - 2*sqrt(2)*(x^6 + x^2)^(3/4) + x)/(x^5 + x^3 + x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 - 1)/((x^6 + x^2)^(1/4)*(x^4 + x^2 + 1)), x)

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maple [C]  time = 5.00, size = 238, normalized size = 2.48

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x +2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right ) x \left (x^{2}-x +1\right )}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-2 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x +\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right ) x \left (x^{2}-x +1\right )}\right )}{2}\)	\(238\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/2*RootOf(_Z^4+1)^3*ln(-(RootOf(_Z^4+1)^3*x^5-RootOf(_Z^4+1)^3*x^3+2*(x^6+x^2)^(1/4)*RootOf(_Z^4+1)^2*x^2-2*
(x^6+x^2)^(1/2)*RootOf(_Z^4+1)*x+RootOf(_Z^4+1)^3*x+2*(x^6+x^2)^(3/4))/(x^2+x+1)/x/(x^2-x+1))-1/2*RootOf(_Z^4+
1)*ln(-(-2*(x^6+x^2)^(1/2)*RootOf(_Z^4+1)^3*x+RootOf(_Z^4+1)*x^5-2*(x^6+x^2)^(1/4)*RootOf(_Z^4+1)^2*x^2-RootOf
(_Z^4+1)*x^3+2*(x^6+x^2)^(3/4)+RootOf(_Z^4+1)*x)/(x^2+x+1)/x/(x^2-x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/(x^4+x^2+1)/(x^6+x^2)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^4 - 1)/((x^6 + x^2)^(1/4)*(x^4 + x^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4-1}{{\left (x^6+x^2\right )}^{1/4}\,\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^2 + x^6)^(1/4)*(x^2 + x^4 + 1)),x)

[Out]

int((x^4 - 1)/((x^2 + x^6)^(1/4)*(x^2 + x^4 + 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 [4]{x^{2} \left (x^{4} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)/(x**4+x**2+1)/(x**6+x**2)**(1/4),x)

[Out]

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

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