3.18.40 \(\int \frac {\sqrt [4]{x^2+x^4} (-1-x^4+x^8)}{-1+x^4} \, dx\)

Optimal. Leaf size=117 \[ -\frac {7}{128} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^2}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^2}}\right )}{2^{3/4}}+\frac {7}{128} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^2}}\right )}{2^{3/4}}+\frac {1}{192} \sqrt [4]{x^4+x^2} \left (32 x^5+4 x^3-7 x\right ) \]

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Rubi [B]  time = 0.58, antiderivative size = 248, normalized size of antiderivative = 2.12, number of steps used = 24, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2056, 1586, 6725, 321, 329, 331, 298, 203, 206, 466, 494} \begin {gather*} -\frac {7}{192} \sqrt [4]{x^4+x^2} x-\frac {7 \sqrt [4]{x^4+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{128 \sqrt [4]{x^2+1} \sqrt {x}}-\frac {\sqrt [4]{x^4+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2^{3/4} \sqrt [4]{x^2+1} \sqrt {x}}+\frac {7 \sqrt [4]{x^4+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{128 \sqrt [4]{x^2+1} \sqrt {x}}+\frac {\sqrt [4]{x^4+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2^{3/4} \sqrt [4]{x^2+1} \sqrt {x}}+\frac {1}{6} \sqrt [4]{x^4+x^2} x^5+\frac {1}{48} \sqrt [4]{x^4+x^2} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*x*(x^2 + x^4)^(1/4))/192 + (x^3*(x^2 + x^4)^(1/4))/48 + (x^5*(x^2 + x^4)^(1/4))/6 - (7*(x^2 + x^4)^(1/4)*A
rcTan[Sqrt[x]/(1 + x^2)^(1/4)])/(128*Sqrt[x]*(1 + x^2)^(1/4)) - ((x^2 + x^4)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(1
+ x^2)^(1/4)])/(2^(3/4)*Sqrt[x]*(1 + x^2)^(1/4)) + (7*(x^2 + x^4)^(1/4)*ArcTanh[Sqrt[x]/(1 + x^2)^(1/4)])/(12
8*Sqrt[x]*(1 + x^2)^(1/4)) + ((x^2 + x^4)^(1/4)*ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)])/(2^(3/4)*Sqrt[x]*(
1 + x^2)^(1/4))

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 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 321

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*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 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 [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx &=\frac {\sqrt [4]{x^2+x^4} \int \frac {\sqrt {x} \sqrt [4]{1+x^2} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \int \frac {\sqrt {x} \left (-1-x^4+x^8\right )}{\left (-1+x^2\right ) \left (1+x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \int \left (\frac {x^{9/2}}{\left (1+x^2\right )^{3/4}}+\frac {x^{13/2}}{\left (1+x^2\right )^{3/4}}-\frac {\sqrt {x}}{\left (-1+x^2\right ) \left (1+x^2\right )^{3/4}}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \int \frac {x^{9/2}}{\left (1+x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \int \frac {x^{13/2}}{\left (1+x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \int \frac {\sqrt {x}}{\left (-1+x^2\right ) \left (1+x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {\left (7 \sqrt [4]{x^2+x^4}\right ) \int \frac {x^{5/2}}{\left (1+x^2\right )^{3/4}} \, dx}{8 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (11 \sqrt [4]{x^2+x^4}\right ) \int \frac {x^{9/2}}{\left (1+x^2\right )^{3/4}} \, dx}{12 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (2 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}+\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \int \frac {\sqrt {x}}{\left (1+x^2\right )^{3/4}} \, dx}{32 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \int \frac {x^{5/2}}{\left (1+x^2\right )^{3/4}} \, dx}{96 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (2 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \int \frac {\sqrt {x}}{\left (1+x^2\right )^{3/4}} \, dx}{128 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{16 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{64 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{16 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (21 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{64 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {21 \sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {21 \sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{128 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (77 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{128 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {7}{192} x \sqrt [4]{x^2+x^4}+\frac {1}{48} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{x^2+x^4}-\frac {7 \sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{128 \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {7 \sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{128 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.59, size = 117, normalized size = 1.00 \begin {gather*} \frac {1}{384} x \sqrt [4]{x^4+x^2} \left (64 x^4+8 x^2+\frac {3 \left (\frac {1}{x^2}+1\right )^{3/4} \left (7 \tan ^{-1}\left (\sqrt [4]{\frac {1}{x^2}+1}\right )+64 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{2}}\right )+7 \tanh ^{-1}\left (\sqrt [4]{\frac {1}{x^2}+1}\right )+64 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{2}}\right )\right )}{x^2+1}-14\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(x^2 + x^4)^(1/4)*(-14 + 8*x^2 + 64*x^4 + (3*(1 + x^(-2))^(3/4)*(7*ArcTan[(1 + x^(-2))^(1/4)] + 64*2^(1/4)*
ArcTan[(1 + x^(-2))^(1/4)/2^(1/4)] + 7*ArcTanh[(1 + x^(-2))^(1/4)] + 64*2^(1/4)*ArcTanh[(1 + x^(-2))^(1/4)/2^(
1/4)]))/(1 + x^2)))/384

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IntegrateAlgebraic [A]  time = 0.50, size = 117, normalized size = 1.00 \begin {gather*} \frac {1}{192} \sqrt [4]{x^2+x^4} \left (-7 x+4 x^3+32 x^5\right )-\frac {7}{128} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^2+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2^{3/4}}+\frac {7}{128} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^2+x^4}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x^2 + x^4)^(1/4)*(-7*x + 4*x^3 + 32*x^5))/192 - (7*ArcTan[x/(x^2 + x^4)^(1/4)])/128 - ArcTan[(2^(1/4)*x)/(x^
2 + x^4)^(1/4)]/2^(3/4) + (7*ArcTanh[x/(x^2 + x^4)^(1/4)])/128 + ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/2^(3/4
)

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fricas [B]  time = 7.46, size = 330, normalized size = 2.82 \begin {gather*} \frac {1}{8} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {16 \cdot 8^{\frac {1}{4}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (8^{\frac {3}{4}} {\left (3 \, x^{3} + x\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x\right )} + 4 \cdot 8^{\frac {3}{4}} {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{8 \, {\left (x^{3} - x\right )}}\right ) + \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x + 8^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x - 8^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + \frac {1}{192} \, {\left (32 \, x^{5} + 4 \, x^{3} - 7 \, x\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} - \frac {7}{256} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) + \frac {7}{256} \, \log \left (\frac {2 \, x^{3} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} + x^{2}} x + x + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*8^(3/4)*arctan(1/8*(16*8^(1/4)*(x^4 + x^2)^(1/4)*x^2 + 2^(3/4)*(8^(3/4)*(3*x^3 + x) + 8*8^(1/4)*sqrt(x^4 +
x^2)*x) + 4*8^(3/4)*(x^4 + x^2)^(3/4))/(x^3 - x)) + 1/32*8^(3/4)*log((4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 + 8^(3/
4)*sqrt(x^4 + x^2)*x + 8^(1/4)*(3*x^3 + x) + 4*(x^4 + x^2)^(3/4))/(x^3 - x)) - 1/32*8^(3/4)*log((4*sqrt(2)*(x^
4 + x^2)^(1/4)*x^2 - 8^(3/4)*sqrt(x^4 + x^2)*x - 8^(1/4)*(3*x^3 + x) + 4*(x^4 + x^2)^(3/4))/(x^3 - x)) + 1/192
*(32*x^5 + 4*x^3 - 7*x)*(x^4 + x^2)^(1/4) - 7/256*arctan(2*((x^4 + x^2)^(1/4)*x^2 + (x^4 + x^2)^(3/4))/x) + 7/
256*log((2*x^3 + 2*(x^4 + x^2)^(1/4)*x^2 + 2*sqrt(x^4 + x^2)*x + x + 2*(x^4 + x^2)^(3/4))/x)

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giac [A]  time = 0.20, size = 123, normalized size = 1.05 \begin {gather*} -\frac {1}{192} \, {\left (7 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {9}{4}} - 18 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} - 21 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right )} x^{6} + \frac {1}{2} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {7}{128} \, \arctan \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {7}{256} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {7}{256} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(7*(1/x^2 + 1)^(9/4) - 18*(1/x^2 + 1)^(5/4) - 21*(1/x^2 + 1)^(1/4))*x^6 + 1/2*2^(1/4)*arctan(1/2*2^(3/4
)*(1/x^2 + 1)^(1/4)) + 1/4*2^(1/4)*log(2^(1/4) + (1/x^2 + 1)^(1/4)) - 1/4*2^(1/4)*log(abs(-2^(1/4) + (1/x^2 +
1)^(1/4))) + 7/128*arctan((1/x^2 + 1)^(1/4)) + 7/256*log((1/x^2 + 1)^(1/4) + 1) - 7/256*log((1/x^2 + 1)^(1/4)
- 1)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+x^{2}\right )^{\frac {1}{4}} \left (x^{8}-x^{4}-1\right )}{x^{4}-1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**2*(x**2 + 1))**(1/4)*(x**8 - x**4 - 1)/((x - 1)*(x + 1)*(x**2 + 1)), x)

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