3.28.99 \(\int \frac {(1-x^4) \sqrt [4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}}{1+x^4} \, dx\)

Optimal. Leaf size=271 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12}+6 \text {$\#$1}^8-4 \text {$\#$1}^4+2\& ,\frac {\text {$\#$1}^9 \left (-\log \left (x^2-1\right )\right )+\text {$\#$1}^9 \log \left (-\text {$\#$1} x^2+\text {$\#$1}+\sqrt [4]{-x^9+x^8+4 x^7-4 x^6-6 x^5+6 x^4+4 x^3-4 x^2-x+1}\right )+2 \text {$\#$1}^5 \log \left (x^2-1\right )-2 \text {$\#$1}^5 \log \left (-\text {$\#$1} x^2+\text {$\#$1}+\sqrt [4]{-x^9+x^8+4 x^7-4 x^6-6 x^5+6 x^4+4 x^3-4 x^2-x+1}\right )}{\text {$\#$1}^{12}-3 \text {$\#$1}^8+3 \text {$\#$1}^4-1}\& \right ]-\frac {4 \sqrt [4]{-x^9+x^8+4 x^7-4 x^6-6 x^5+6 x^4+4 x^3-4 x^2-x+1} \left (9 x^2+8 x-17\right )}{117 (x+1)} \]

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Rubi [C]  time = 2.03, antiderivative size = 597, normalized size of antiderivative = 2.20, number of steps used = 27, number of rules used = 9, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.145, Rules used = {6688, 6719, 1731, 6740, 194, 212, 206, 203, 1972} \begin {gather*} -\frac {4 \sqrt [4]{(1-x)^5 (x+1)^4} (1-x)^2}{13 (x+1)}+\frac {8 \sqrt [4]{(1-x)^5 (x+1)^4} (1-x)}{9 (x+1)}+\frac {\sqrt [4]{-4-(2+2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{-1-\sqrt [4]{-1}}}\right )}{(x-1)^{5/4} (x+1)}-\frac {\sqrt [4]{-4+(2+2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{\sqrt [4]{-1}-1}}\right )}{(x-1)^{5/4} (x+1)}-\frac {\sqrt [4]{-4+(2-2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{-1-(-1)^{3/4}}}\right )}{(x-1)^{5/4} (x+1)}+\frac {\sqrt [4]{-4-(2-2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{(-1)^{3/4}-1}}\right )}{(x-1)^{5/4} (x+1)}+\frac {\sqrt [4]{-4-(2+2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{-1-\sqrt [4]{-1}}}\right )}{(x-1)^{5/4} (x+1)}-\frac {\sqrt [4]{-4+(2+2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{\sqrt [4]{-1}-1}}\right )}{(x-1)^{5/4} (x+1)}-\frac {\sqrt [4]{-4+(2-2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{-1-(-1)^{3/4}}}\right )}{(x-1)^{5/4} (x+1)}+\frac {\sqrt [4]{-4-(2-2 i) \sqrt {2}} \sqrt [4]{(1-x)^5 (x+1)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{(-1)^{3/4}-1}}\right )}{(x-1)^{5/4} (x+1)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 - x^4)*(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4))/(1 + x^4),x]

[Out]

(8*(1 - x)*((1 - x)^5*(1 + x)^4)^(1/4))/(9*(1 + x)) - (4*(1 - x)^2*((1 - x)^5*(1 + x)^4)^(1/4))/(13*(1 + x)) +
((-4 - (2 + 2*I)*Sqrt[2])^(1/4)*((1 - x)^5*(1 + x)^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/(-1 - (-1)^(1/4))^(1/4)])/(
(-1 + x)^(5/4)*(1 + x)) - ((-4 + (2 + 2*I)*Sqrt[2])^(1/4)*((1 - x)^5*(1 + x)^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/(-
1 + (-1)^(1/4))^(1/4)])/((-1 + x)^(5/4)*(1 + x)) - ((-4 + (2 - 2*I)*Sqrt[2])^(1/4)*((1 - x)^5*(1 + x)^4)^(1/4)
*ArcTan[(-1 + x)^(1/4)/(-1 - (-1)^(3/4))^(1/4)])/((-1 + x)^(5/4)*(1 + x)) + ((-4 - (2 - 2*I)*Sqrt[2])^(1/4)*((
1 - x)^5*(1 + x)^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/(-1 + (-1)^(3/4))^(1/4)])/((-1 + x)^(5/4)*(1 + x)) + ((-4 - (2
+ 2*I)*Sqrt[2])^(1/4)*((1 - x)^5*(1 + x)^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/(-1 - (-1)^(1/4))^(1/4)])/((-1 + x)^
(5/4)*(1 + x)) - ((-4 + (2 + 2*I)*Sqrt[2])^(1/4)*((1 - x)^5*(1 + x)^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/(-1 + (-1)
^(1/4))^(1/4)])/((-1 + x)^(5/4)*(1 + x)) - ((-4 + (2 - 2*I)*Sqrt[2])^(1/4)*((1 - x)^5*(1 + x)^4)^(1/4)*ArcTanh
[(-1 + x)^(1/4)/(-1 - (-1)^(3/4))^(1/4)])/((-1 + x)^(5/4)*(1 + x)) + ((-4 - (2 - 2*I)*Sqrt[2])^(1/4)*((1 - x)^
5*(1 + x)^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/(-1 + (-1)^(3/4))^(1/4)])/((-1 + x)^(5/4)*(1 + x))

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 212

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

Rule 1731

Int[(Px_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e
*x, x]*(d + e*x)^(q + 1)*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRe
mainder[Px, d + e*x, x], 0]

Rule 1972

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[u, x] &&  !BinomialMatchQ[
u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6740

Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x
] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]

Rubi steps

\begin {align*} \int \frac {\left (1-x^4\right ) \sqrt [4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}}{1+x^4} \, dx &=\int \frac {\sqrt [4]{-(-1+x)^5 (1+x)^4} \left (1-x^4\right )}{1+x^4} \, dx\\ &=\frac {\sqrt [4]{-(-1+x)^5 (1+x)^4} \int \frac {(-1+x)^{5/4} (1+x) \left (1-x^4\right )}{1+x^4} \, dx}{(-1+x)^{5/4} (1+x)}\\ &=\frac {\sqrt [4]{-(-1+x)^5 (1+x)^4} \int \frac {(-1+x)^{9/4} \left (-1-2 x-2 x^2-2 x^3-x^4\right )}{1+x^4} \, dx}{(-1+x)^{5/4} (1+x)}\\ &=\frac {\sqrt [4]{-(-1+x)^5 (1+x)^4} \int \frac {(-1+x)^{9/4} (1+x)^2 \left (-1-x^2\right )}{1+x^4} \, dx}{(-1+x)^{5/4} (1+x)}\\ &=\frac {\left (4 \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \left (2+x^4\right )^2 \left (-1-\left (1+x^4\right )^2\right )}{1+\left (1+x^4\right )^4} \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}\\ &=\frac {\left (4 \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \left (x^4+\left (1+x^4\right )^2-\left (1+x^4\right )^3+\frac {2 \left (-x^4-\left (1+x^4\right )^2+\left (1+x^4\right )^3\right )}{1+\left (1+x^4\right )^4}\right ) \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}\\ &=\frac {4 \sqrt [4]{(1-x)^5 (1+x)^4}}{5 (1+x)}+\frac {\left (4 \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \left (1+x^4\right )^2 \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\left (4 \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \left (1+x^4\right )^3 \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\left (8 \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {-x^4-\left (1+x^4\right )^2+\left (1+x^4\right )^3}{1+\left (1+x^4\right )^4} \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}\\ &=\frac {4 \sqrt [4]{(1-x)^5 (1+x)^4}}{5 (1+x)}+\frac {\left (4 \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \left (1+2 x^4+x^8\right ) \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\left (4 \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \left (1+3 x^4+3 x^8+x^{12}\right ) \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\left (8 \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \left (\frac {(-1-i)+\sqrt [4]{-1}-(-1)^{3/4}}{4 \left (-1+\sqrt [4]{-1}-x^4\right )}+\frac {(1+i)+\sqrt [4]{-1}-(-1)^{3/4}}{4 \left (1+\sqrt [4]{-1}+x^4\right )}+\frac {(-1-i)+\sqrt [4]{-1}+(-1)^{3/4}}{4 \left (\sqrt [4]{-1}-i \left (1+x^4\right )\right )}+\frac {(1+i)+\sqrt [4]{-1}+(-1)^{3/4}}{4 \left (\sqrt [4]{-1}+i \left (1+x^4\right )\right )}\right ) \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}\\ &=\frac {8 (1-x) \sqrt [4]{(1-x)^5 (1+x)^4}}{9 (1+x)}-\frac {4 (1-x)^2 \sqrt [4]{(1-x)^5 (1+x)^4}}{13 (1+x)}+\frac {\left (2 \left ((1+i)+i \sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+i \left (1+x^4\right )} \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\left (2 \left ((-1-i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [4]{-1}-x^4} \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\left (2 i \left ((-1+i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-i \left (1+x^4\right )} \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\left (2 \left ((1+i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{-1}+x^4} \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}\\ &=\frac {8 (1-x) \sqrt [4]{(1-x)^5 (1+x)^4}}{9 (1+x)}-\frac {4 (1-x)^2 \sqrt [4]{(1-x)^5 (1+x)^4}}{13 (1+x)}+\frac {\left (2 \left ((1+i)+i \sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{i+\sqrt [4]{-1}+i x^4} \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\left (\left ((-1-i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt [4]{-1}}-x^2} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt {-1+\sqrt [4]{-1}} (-1+x)^{5/4} (1+x)}+\frac {\left (\left ((-1-i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt [4]{-1}}+x^2} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt {-1+\sqrt [4]{-1}} (-1+x)^{5/4} (1+x)}+\frac {\left (2 i \left ((-1+i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-i+\sqrt [4]{-1}-i x^4} \, dx,x,\sqrt [4]{-1+x}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\left (\left ((1+i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-\sqrt [4]{-1}}-x^2} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt {-1-\sqrt [4]{-1}} (-1+x)^{5/4} (1+x)}-\frac {\left (\left ((1+i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-\sqrt [4]{-1}}+x^2} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt {-1-\sqrt [4]{-1}} (-1+x)^{5/4} (1+x)}\\ &=\frac {8 (1-x) \sqrt [4]{(1-x)^5 (1+x)^4}}{9 (1+x)}-\frac {4 (1-x)^2 \sqrt [4]{(1-x)^5 (1+x)^4}}{13 (1+x)}+\frac {\sqrt {2} \sqrt [4]{-1-\sqrt [4]{-1}} \sqrt [4]{(1-x)^5 (1+x)^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1-\sqrt [4]{-1}}}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\sqrt {2} \sqrt [4]{-1+\sqrt [4]{-1}} \sqrt [4]{(1-x)^5 (1+x)^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1+\sqrt [4]{-1}}}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\sqrt {2} \sqrt [4]{-1-\sqrt [4]{-1}} \sqrt [4]{(1-x)^5 (1+x)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1-\sqrt [4]{-1}}}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\sqrt {2} \sqrt [4]{-1+\sqrt [4]{-1}} \sqrt [4]{(1-x)^5 (1+x)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1+\sqrt [4]{-1}}}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\left (i \left ((1+i)+i \sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+(-1)^{3/4}}-x^2} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt {-1+(-1)^{3/4}} (-1+x)^{5/4} (1+x)}+\frac {\left (i \left ((1+i)+i \sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+(-1)^{3/4}}+x^2} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt {-1+(-1)^{3/4}} (-1+x)^{5/4} (1+x)}+\frac {\left (\left ((-1+i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-(-1)^{3/4}}-x^2} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt {-1-(-1)^{3/4}} (-1+x)^{5/4} (1+x)}+\frac {\left (\left ((-1+i)+\sqrt {2}\right ) \sqrt [4]{-(-1+x)^5 (1+x)^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-(-1)^{3/4}}+x^2} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt {-1-(-1)^{3/4}} (-1+x)^{5/4} (1+x)}\\ &=\frac {8 (1-x) \sqrt [4]{(1-x)^5 (1+x)^4}}{9 (1+x)}-\frac {4 (1-x)^2 \sqrt [4]{(1-x)^5 (1+x)^4}}{13 (1+x)}+\frac {\sqrt {2} \sqrt [4]{-1-\sqrt [4]{-1}} \sqrt [4]{(1-x)^5 (1+x)^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1-\sqrt [4]{-1}}}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\sqrt {2} \sqrt [4]{-1+\sqrt [4]{-1}} \sqrt [4]{(1-x)^5 (1+x)^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1+\sqrt [4]{-1}}}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\sqrt {2} \sqrt [4]{-1-(-1)^{3/4}} \sqrt [4]{(1-x)^5 (1+x)^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1-(-1)^{3/4}}}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\sqrt {2} \sqrt [4]{-1+(-1)^{3/4}} \sqrt [4]{(1-x)^5 (1+x)^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1+(-1)^{3/4}}}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\sqrt {2} \sqrt [4]{-1-\sqrt [4]{-1}} \sqrt [4]{(1-x)^5 (1+x)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1-\sqrt [4]{-1}}}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\sqrt {2} \sqrt [4]{-1+\sqrt [4]{-1}} \sqrt [4]{(1-x)^5 (1+x)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1+\sqrt [4]{-1}}}\right )}{(-1+x)^{5/4} (1+x)}-\frac {\sqrt {2} \sqrt [4]{-1-(-1)^{3/4}} \sqrt [4]{(1-x)^5 (1+x)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1-(-1)^{3/4}}}\right )}{(-1+x)^{5/4} (1+x)}+\frac {\sqrt {2} \sqrt [4]{-1+(-1)^{3/4}} \sqrt [4]{(1-x)^5 (1+x)^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{-1+(-1)^{3/4}}}\right )}{(-1+x)^{5/4} (1+x)}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 135, normalized size = 0.50 \begin {gather*} -\frac {\sqrt [4]{-(x-1)^5 (x+1)^4} \left (8 (x-1)^{9/4} (9 x+17)-117 \text {RootSum}\left [\text {$\#$1}^{16}+4 \text {$\#$1}^{12}+6 \text {$\#$1}^8+4 \text {$\#$1}^4+2\&,\frac {\text {$\#$1}^9 \log \left (\sqrt [4]{x-1}-\text {$\#$1}\right )+2 \text {$\#$1}^5 \log \left (\sqrt [4]{x-1}-\text {$\#$1}\right )}{\text {$\#$1}^{12}+3 \text {$\#$1}^8+3 \text {$\#$1}^4+1}\&\right ]\right )}{234 (x-1)^{5/4} (x+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - x^4)*(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4))/(1 + x^4),x]

[Out]

-1/234*((-((-1 + x)^5*(1 + x)^4))^(1/4)*(8*(-1 + x)^(9/4)*(17 + 9*x) - 117*RootSum[2 + 4*#1^4 + 6*#1^8 + 4*#1^
12 + #1^16 & , (2*Log[(-1 + x)^(1/4) - #1]*#1^5 + Log[(-1 + x)^(1/4) - #1]*#1^9)/(1 + 3*#1^4 + 3*#1^8 + #1^12)
& ]))/((-1 + x)^(5/4)*(1 + x))

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IntegrateAlgebraic [A]  time = 0.61, size = 271, normalized size = 1.00 \begin {gather*} -\frac {4 \left (-17+8 x+9 x^2\right ) \sqrt [4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}}{117 (1+x)}-\frac {1}{2} \text {RootSum}\left [2-4 \text {$\#$1}^4+6 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {2 \log \left (-1+x^2\right ) \text {$\#$1}^5-2 \log \left (\sqrt [4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}+\text {$\#$1}-x^2 \text {$\#$1}\right ) \text {$\#$1}^5-\log \left (-1+x^2\right ) \text {$\#$1}^9+\log \left (\sqrt [4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}+\text {$\#$1}-x^2 \text {$\#$1}\right ) \text {$\#$1}^9}{-1+3 \text {$\#$1}^4-3 \text {$\#$1}^8+\text {$\#$1}^{12}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - x^4)*(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4))/(1 +
x^4),x]

[Out]

(-4*(-17 + 8*x + 9*x^2)*(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4))/(117*(1 + x
)) - RootSum[2 - 4*#1^4 + 6*#1^8 - 4*#1^12 + #1^16 & , (2*Log[-1 + x^2]*#1^5 - 2*Log[(1 - x - 4*x^2 + 4*x^3 +
6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4) + #1 - x^2*#1]*#1^5 - Log[-1 + x^2]*#1^9 + Log[(1 - x - 4*x^2
+ 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4) + #1 - x^2*#1]*#1^9)/(-1 + 3*#1^4 - 3*#1^8 + #1^12
) & ]/2

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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((-x^4+1)*(-x^9+x^8+4*x^7-4*x^6-6*x^5+6*x^4+4*x^3-4*x^2-x+1)^(1/4)/(x^4+1),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 (-x^{9} + x^{8} + 4 \, x^{7} - 4 \, x^{6} - 6 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} - 4 \, x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)*(-x^9+x^8+4*x^7-4*x^6-6*x^5+6*x^4+4*x^3-4*x^2-x+1)^(1/4)/(x^4+1),x, algorithm="giac")

[Out]

integrate(-(-x^9 + x^8 + 4*x^7 - 4*x^6 - 6*x^5 + 6*x^4 + 4*x^3 - 4*x^2 - x + 1)^(1/4)*(x^4 - 1)/(x^4 + 1), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-x^{4}+1\right ) \left (-x^{9}+x^{8}+4 x^{7}-4 x^{6}-6 x^{5}+6 x^{4}+4 x^{3}-4 x^{2}-x +1\right )^{\frac {1}{4}}}{x^{4}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+1)*(-x^9+x^8+4*x^7-4*x^6-6*x^5+6*x^4+4*x^3-4*x^2-x+1)^(1/4)/(x^4+1),x)

[Out]

int((-x^4+1)*(-x^9+x^8+4*x^7-4*x^6-6*x^5+6*x^4+4*x^3-4*x^2-x+1)^(1/4)/(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^{9} + x^{8} + 4 \, x^{7} - 4 \, x^{6} - 6 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} - 4 \, x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)*(-x^9+x^8+4*x^7-4*x^6-6*x^5+6*x^4+4*x^3-4*x^2-x+1)^(1/4)/(x^4+1),x, algorithm="maxima")

[Out]

-integrate((-x^9 + x^8 + 4*x^7 - 4*x^6 - 6*x^5 + 6*x^4 + 4*x^3 - 4*x^2 - x + 1)^(1/4)*(x^4 - 1)/(x^4 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (x^4-1\right )\,{\left (-x^9+x^8+4\,x^7-4\,x^6-6\,x^5+6\,x^4+4\,x^3-4\,x^2-x+1\right )}^{1/4}}{x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^4 - 1)*(4*x^3 - 4*x^2 - x + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9 + 1)^(1/4))/(x^4 + 1),x)

[Out]

-int(((x^4 - 1)*(4*x^3 - 4*x^2 - x + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9 + 1)^(1/4))/(x^4 + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {\sqrt [4]{- x^{9} + x^{8} + 4 x^{7} - 4 x^{6} - 6 x^{5} + 6 x^{4} + 4 x^{3} - 4 x^{2} - x + 1}}{x^{4} + 1}\right )\, dx - \int \frac {x^{4} \sqrt [4]{- x^{9} + x^{8} + 4 x^{7} - 4 x^{6} - 6 x^{5} + 6 x^{4} + 4 x^{3} - 4 x^{2} - x + 1}}{x^{4} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4+1)*(-x**9+x**8+4*x**7-4*x**6-6*x**5+6*x**4+4*x**3-4*x**2-x+1)**(1/4)/(x**4+1),x)

[Out]

-Integral(-(-x**9 + x**8 + 4*x**7 - 4*x**6 - 6*x**5 + 6*x**4 + 4*x**3 - 4*x**2 - x + 1)**(1/4)/(x**4 + 1), x)
- Integral(x**4*(-x**9 + x**8 + 4*x**7 - 4*x**6 - 6*x**5 + 6*x**4 + 4*x**3 - 4*x**2 - x + 1)**(1/4)/(x**4 + 1)
, x)

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