3.24.6 \(\int \frac {(1+x^4) \sqrt [4]{-x^3+x^4}}{x^4 (-1+x^4)} \, dx\) [2306]

Optimal. Leaf size=177 \[ -\frac {4 \left (-5+x+4 x^2\right ) \sqrt [4]{-x^3+x^4}}{45 x^3}+\sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 1.08, antiderivative size = 608, normalized size of antiderivative = 3.44, number of steps used = 36, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2081, 1600, 6865, 6874, 277, 270, 508, 304, 209, 212, 6857, 1543} \begin {gather*} -\frac {(1-i)^{5/4} \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{(1-i)^{3/4} \sqrt [4]{x-1} x^{3/4}}-\frac {(1+i)^{5/4} \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{(1+i)^{3/4} \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{2} \sqrt [4]{x^4-x^3} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {16 \sqrt [4]{x^4-x^3}}{45 x}+\frac {4 \sqrt [4]{x^4-x^3}}{9 x^3}+\frac {(1-i)^{5/4} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{(1-i)^{3/4} \sqrt [4]{x-1} x^{3/4}}+\frac {(1+i)^{5/4} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{(1+i)^{3/4} \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {4 \sqrt [4]{x^4-x^3}}{45 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-x^3 + x^4)^(1/4))/(9*x^3) - (4*(-x^3 + x^4)^(1/4))/(45*x^2) - (16*(-x^3 + x^4)^(1/4))/(45*x) - ((-x^3 + x
^4)^(1/4)*ArcTan[((1 - I)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((1 - I)^(3/4)*(-1 + x)^(1/4)*x^(3/4)) - ((1 - I)^(5
/4)*(-x^3 + x^4)^(1/4)*ArcTan[((1 - I)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(2*(-1 + x)^(1/4)*x^(3/4)) - ((-x^3 + x
^4)^(1/4)*ArcTan[((1 + I)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((1 + I)^(3/4)*(-1 + x)^(1/4)*x^(3/4)) - ((1 + I)^(5
/4)*(-x^3 + x^4)^(1/4)*ArcTan[((1 + I)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(2*(-1 + x)^(1/4)*x^(3/4)) + (2^(1/4)*(
-x^3 + x^4)^(1/4)*ArcTan[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((-1 + x)^(1/4)*x^(3/4)) + ((-x^3 + x^4)^(1/4)*Arc
Tanh[((1 - I)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((1 - I)^(3/4)*(-1 + x)^(1/4)*x^(3/4)) + ((1 - I)^(5/4)*(-x^3 +
x^4)^(1/4)*ArcTanh[((1 - I)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(2*(-1 + x)^(1/4)*x^(3/4)) + ((-x^3 + x^4)^(1/4)*A
rcTanh[((1 + I)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((1 + I)^(3/4)*(-1 + x)^(1/4)*x^(3/4)) + ((1 + I)^(5/4)*(-x^3
+ x^4)^(1/4)*ArcTanh[((1 + I)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(2*(-1 + x)^(1/4)*x^(3/4)) - (2^(1/4)*(-x^3 + x^
4)^(1/4)*ArcTanh[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((-1 + x)^(1/4)*x^(3/4))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 270

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 277

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

Rule 304

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 508

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 1543

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

Rule 1600

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 2081

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 6857

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]

Rule 6865

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6874

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

Rubi steps

\begin {align*} \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} \left (1+x^4\right )}{x^{13/4} \left (-1+x^4\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {1+x^4}{(-1+x)^{3/4} x^{13/4} \left (1+x+x^2+x^3\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1+x^{16}}{x^{10} \left (-1+x^4\right )^{3/4} \left (1+x^4+x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{x^{10} \left (-1+x^4\right )^{3/4}}-\frac {1}{x^6 \left (-1+x^4\right )^{3/4}}-\frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+x^4\right )}+\frac {x^2 \left (1+x^4\right )}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (1+x^4\right )}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{5 x^2}-\frac {\left (16 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{5 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (32 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{9 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )}+\frac {x^6}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{5 x}+\frac {\left (128 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{45 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {i x^2}{2 \left (-1+x^4\right )^{3/4} \left (-i+x^4\right )}+\frac {i x^2}{2 \left (-1+x^4\right )^{3/4} \left (i+x^4\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{2 \left (-1+x^4\right )^{3/4} \left (-i+x^4\right )}+\frac {x^2}{2 \left (-1+x^4\right )^{3/4} \left (i+x^4\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-i+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (i+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-i+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (i+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{i-(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-i-(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{i-(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-i-(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (i (1-i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (i (1-i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left ((1-i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((1-i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (i (1+i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (i (1+i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left ((1+i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((1+i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}-\frac {\sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{(1-i)^{3/4} \sqrt [4]{-1+x} x^{3/4}}-\frac {(1-i)^{5/4} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{(1+i)^{3/4} \sqrt [4]{-1+x} x^{3/4}}-\frac {(1+i)^{5/4} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{(1-i)^{3/4} \sqrt [4]{-1+x} x^{3/4}}+\frac {(1-i)^{5/4} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{(1+i)^{3/4} \sqrt [4]{-1+x} x^{3/4}}+\frac {(1+i)^{5/4} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 195, normalized size = 1.10 \begin {gather*} \frac {(-1+x)^{3/4} \left (-8 \left (4 \sqrt [4]{-1+x} \left (-5+x+4 x^2\right )-45 \sqrt [4]{2} x^{9/4} \text {ArcTan}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+45 \sqrt [4]{2} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )+45 x^{9/4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{360 \left ((-1+x) x^3\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x)^(3/4)*(-8*(4*(-1 + x)^(1/4)*(-5 + x + 4*x^2) - 45*2^(1/4)*x^(9/4)*ArcTan[2^(1/4)/((-1 + x)/x)^(1/4)]
+ 45*2^(1/4)*x^(9/4)*ArcTanh[2^(1/4)/((-1 + x)/x)^(1/4)]) + 45*x^(9/4)*RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[
x] + 8*Log[(-1 + x)^(1/4) - x^(1/4)*#1] + Log[x]*#1^4 - 4*Log[(-1 + x)^(1/4) - x^(1/4)*#1]*#1^4)/(-#1^3 + #1^7
) & ]))/(360*((-1 + x)*x^3)^(3/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 57.00, size = 4004, normalized size = 22.62

method	result	size
trager	\(\text {Expression too large to display}\)	\(4004\)
risch	\(\text {Expression too large to display}\)	\(8065\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/45*(4*x^2+x-5)/x^3*(x^4-x^3)^(1/4)-1/2*RootOf(_Z^4-2)*ln((3*RootOf(_Z^4-2)^3*x^3-RootOf(_Z^4-2)^3*x^2+4*Roo
tOf(_Z^4-2)^2*(x^4-x^3)^(1/4)*x^2+4*(x^4-x^3)^(1/2)*RootOf(_Z^4-2)*x+4*(x^4-x^3)^(3/4))/x^2/(1+x))-1/2*RootOf(
_Z^2+RootOf(_Z^4-2)^2)*ln(-(3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^3+4*RootOf(_Z^4-2)^2*(x^4-x^3)^
(1/4)*x^2-RootOf(_Z^4-2)^2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x^2-4*(x^4-x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x
-4*(x^4-x^3)^(3/4))/x^2/(1+x))+1/4*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*ln((23*Roo
tOf(_Z^2+RootOf(_Z^4-2)^2)^2*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3*RootOf(_Z^4-2)
^6*x^3-46*RootOf(_Z^2+RootOf(_Z^4-2)^2)^2*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3*R
ootOf(_Z^4-2)^6*x^2-346*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_
Z^2+RootOf(_Z^4-2)^2)-16)^3*x^3-1008*RootOf(_Z^4-2)^3*(x^4-x^3)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z
^2+RootOf(_Z^4-2)^2)-16)^2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x^2+94*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)
*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3*x^2-2016*(x^4-x^3)^(1/2)*RootOf(_Z^4-2)^3*
RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*x-4032*RootOf(_
Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*(x^4-x^3)^(3/4)-352*x^3*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+Root
Of(_Z^4-2)^2)-16)^3-512*(x^4-x^3)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^2*x^2
+132*x^2*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3-1024*(x^4-x^3)^(1/2)*RootOf(_Z^4-8
*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*x-2048*(x^4-x^3)^(3/4))/(RootOf(_Z^2+RootOf(_Z^4-2)^2)*Roo
tOf(_Z^4-2)^3*x-2*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)+4*x+2)/x^2)-1/8*RootOf(_Z^4-2)^3*RootOf(_Z^2+
RootOf(_Z^4-2)^2)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*ln((11*RootOf(_Z^2+RootOf(_
Z^4-2)^2)^2*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3*RootOf(_Z^4-2)^6*x^3-22*RootOf(
_Z^2+RootOf(_Z^4-2)^2)^2*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3*RootOf(_Z^4-2)^6*x
^2-222*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2
)^2)-16)^3*x^3+1008*RootOf(_Z^4-2)^3*(x^4-x^3)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)
^2)-16)^2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x^2+158*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-8*R
ootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3*x^2+512*(x^4-x^3)^(1/2)*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootO
f(_Z^4-2)^2)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*x-4032*RootOf(_Z^2+RootOf(_Z^4-2
)^2)*RootOf(_Z^4-2)^3*(x^4-x^3)^(3/4)+736*x^3*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)
^3+512*(x^4-x^3)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^2*x^2-276*x^2*RootOf(_
Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3-4032*(x^4-x^3)^(1/2)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3
*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*x-2048*(x^4-x^3)^(3/4))/(RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*x-2
*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)+4*x+2)/x^2)-1/32*ln((9*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z
^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*RootOf(_Z^4-2)^6*x^3-37*RootOf(_Z^2+RootOf(_Z^4-2)^2)^
2*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*RootOf(_Z^4-2)^5*x^3-18*RootOf(_Z^2+RootOf(
_Z^4-2)^2)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*RootOf(_Z^4-2)^6*x^2+74*RootOf(_Z^
2+RootOf(_Z^4-2)^2)^2*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*RootOf(_Z^4-2)^5*x^2-81
6*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*(x^4-x^3)^(3/4)-36*x^3*RootOf(_Z^4-2)^3*RootOf(_Z^4-8*RootOf(
_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)-332*RootOf(_Z^4-2)^2*(x^4-x^3)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^4-2)^
3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^2*x^2+148*x^3*RootOf(_Z^4-2)^2*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+
RootOf(_Z^4-2)^2)-16)*RootOf(_Z^2+RootOf(_Z^4-2)^2)-204*(x^4-x^3)^(1/2)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(
_Z^2+RootOf(_Z^4-2)^2)-16)^3*RootOf(_Z^4-2)*x+76*RootOf(_Z^4-2)*(x^4-x^3)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^4-2)^3
*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x^2-128*(x^4-x^3)^(1/2)*RootOf(_Z^4-8*RootO
f(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x-18*x^2*RootOf(_Z^4-2)^3*RootOf
(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)+74*x^2*RootOf(_Z^4-2)^2*RootOf(_Z^4-8*RootOf(_Z^4-2
)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)*RootOf(_Z^2+RootOf(_Z^4-2)^2)+1024*(x^4-x^3)^(3/4))/(RootOf(_Z^2+RootOf(
_Z^4-2)^2)*RootOf(_Z^4-2)^3*x-2*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-4*x-2)/x^2)*RootOf(_Z^4-8*RootO
f(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^4-2)^2)-16)^3*RootOf(_Z^4-2)-1/32*ln((9*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf
(_Z^4-8*RootOf(_Z^4-2)^3*RootOf(_Z^2+RootOf(_Z^...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.43, size = 2131, normalized size = 12.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/720*(180*2^(5/8)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*x^3*sqrt(-2*sqrt(2) + 4)*arct
an(1/4*(2^(3/8)*(sqrt(2)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) - sqrt(2)*(sqrt(2)*x + x))*sqrt(2*(2*sqrt(2) + 3
)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt((4*2^(1/4)*x^2 + 2^(1/8)*((x^4 - x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqr
t(-2*sqrt(2) + 4) - 4*(x^4 - x^3)^(1/4)*x)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sq
rt(x^4 - x^3))/x^2) - 2*2^(3/8)*(sqrt(2)*(x^4 - x^3)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) - sqrt(2)*(x^4 -
x^3)^(1/4)*(sqrt(2) + 1))*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) - 4*sqrt(2)*x + 4*(sqr
t(2)*x + x)*sqrt(-2*sqrt(2) + 4) - 4*x)/x) + 180*2^(5/8)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(
2) + 8)*x^3*sqrt(-2*sqrt(2) + 4)*arctan(1/4*(2^(3/8)*(sqrt(2)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) - sqrt(2)*(
sqrt(2)*x + x))*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt((4*2^(1/4)*x^2 - 2^(1/8)*((x
^4 - x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 4*(x^4 - x^3)^(1/4)*x)*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2
*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 - x^3))/x^2) - 2*2^(3/8)*(sqrt(2)*(x^4 - x^3)^(1/4)*(sqrt(2) + 1)*
sqrt(-2*sqrt(2) + 4) - sqrt(2)*(x^4 - x^3)^(1/4)*(sqrt(2) + 1))*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) +
4*sqrt(2) + 8) + 4*sqrt(2)*x - 4*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) + 4*x)/x) + 180*2^(5/8)*sqrt(-2*(2*sqrt(
2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*x^3*sqrt(-2*sqrt(2) + 4)*arctan(1/4*(2^(3/8)*(sqrt(2)*(sqrt(2)*x
+ x)*sqrt(-2*sqrt(2) + 4) + sqrt(2)*(sqrt(2)*x + x))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2)
+ 8)*sqrt((4*2^(1/4)*x^2 + 2^(1/8)*((x^4 - x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4*(x^4 - x^3)^
(1/4)*x)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 - x^3))/x^2) - 2*2^(3/8)*(
sqrt(2)*(x^4 - x^3)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + sqrt(2)*(x^4 - x^3)^(1/4)*(sqrt(2) + 1))*sqrt(-
2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) - 4*sqrt(2)*x - 4*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4)
- 4*x)/x) + 180*2^(5/8)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*x^3*sqrt(-2*sqrt(2) + 4
)*arctan(1/4*(2^(3/8)*(sqrt(2)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) + sqrt(2)*(sqrt(2)*x + x))*sqrt(-2*(2*sqrt
(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*sqrt((4*2^(1/4)*x^2 - 2^(1/8)*((x^4 - x^3)^(1/4)*(sqrt(2)*x + 2
*x)*sqrt(-2*sqrt(2) + 4) + 4*(x^4 - x^3)^(1/4)*x)*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8
) + 4*sqrt(x^4 - x^3))/x^2) - 2*2^(3/8)*(sqrt(2)*(x^4 - x^3)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + sqrt(2
)*(x^4 - x^3)^(1/4)*(sqrt(2) + 1))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(2)*x
+ 4*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4) + 4*x)/x) - 360*8^(3/4)*x^3*arctan(1/2*(8^(1/4)*x*sqrt((sqrt(2)*x^2
+ sqrt(x^4 - x^3))/x^2) - 8^(1/4)*(x^4 - x^3)^(1/4))/x) + 90*8^(3/4)*x^3*log((8^(3/4)*x + 4*(x^4 - x^3)^(1/4))
/x) - 90*8^(3/4)*x^3*log(-(8^(3/4)*x - 4*(x^4 - x^3)^(1/4))/x) + 45*2^(1/8)*(4*x^3 - (sqrt(2)*x^3 + 2*x^3)*sqr
t(-2*sqrt(2) + 4))*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*log(1/4*(4*2^(1/4)*x^2 + 2^(1/
8)*((x^4 - x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 4*(x^4 - x^3)^(1/4)*x)*sqrt(2*(2*sqrt(2) + 3)*s
qrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 - x^3))/x^2) - 45*2^(1/8)*(4*x^3 - (sqrt(2)*x^3 + 2*x^3)*sqr
t(-2*sqrt(2) + 4))*sqrt(2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*log(1/4*(4*2^(1/4)*x^2 - 2^(1/
8)*((x^4 - x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 4*(x^4 - x^3)^(1/4)*x)*sqrt(2*(2*sqrt(2) + 3)*s
qrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 - x^3))/x^2) - 45*2^(1/8)*(4*x^3 + (sqrt(2)*x^3 + 2*x^3)*sqr
t(-2*sqrt(2) + 4))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*log(1/4*(4*2^(1/4)*x^2 + 2^(1
/8)*((x^4 - x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4*(x^4 - x^3)^(1/4)*x)*sqrt(-2*(2*sqrt(2) + 3)
*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 - x^3))/x^2) + 45*2^(1/8)*(4*x^3 + (sqrt(2)*x^3 + 2*x^3)*s
qrt(-2*sqrt(2) + 4))*sqrt(-2*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8)*log(1/4*(4*2^(1/4)*x^2 - 2^
(1/8)*((x^4 - x^3)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4*(x^4 - x^3)^(1/4)*x)*sqrt(-2*(2*sqrt(2) +
3)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 8) + 4*sqrt(x^4 - x^3))/x^2) + 64*(x^4 - x^3)^(1/4)*(4*x^2 + x - 5))/x^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x^{4} + 1\right )}{x^{4} \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+1)*(x**4-x**3)**(1/4)/x**4/(x**4-1),x)

[Out]

Integral((x**3*(x - 1))**(1/4)*(x**4 + 1)/(x**4*(x - 1)*(x + 1)*(x**2 + 1)), x)

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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.45, size = 272, normalized size = 1.54 \begin {gather*} -\frac {4}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \frac {4}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \left (-\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (i \, \left (73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} - \left (65536 i - 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \left (-\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (i \, \left (73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} + \left (65536 i - 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - i \, \left (\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (i \, \left (-73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} - \left (65536 i - 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + i \, \left (\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (i \, \left (-73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} + \left (65536 i - 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (-i \, \left (-73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} + \left (65536 i + 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (-i \, \left (-73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} - \left (65536 i + 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2 i \, \left (-\frac {1}{256} i + \frac {1}{256}\right )^{\frac {1}{4}} \log \left (-i \, \left (85070591730234615865843651857942052864 i - 85070591730234615865843651857942052864\right )^{\frac {1}{4}} + \left (2147483648 i + 2147483648\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2 i \, \left (-\frac {1}{256} i + \frac {1}{256}\right )^{\frac {1}{4}} \log \left (-i \, \left (85070591730234615865843651857942052864 i - 85070591730234615865843651857942052864\right )^{\frac {1}{4}} - \left (2147483648 i + 2147483648\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*(1/x - 1)^2*(-1/x + 1)^(1/4) + 4/5*(-1/x + 1)^(5/4) + 2^(1/4)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) + 1/2*
2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) + (-1/16*I + 1/16)^(1/4)*log(I*(73786976294838206464*I - 7378697629483
8206464)^(1/4) - (65536*I - 65536)*(-1/x + 1)^(1/4)) - (-1/16*I + 1/16)^(1/4)*log(I*(73786976294838206464*I -
73786976294838206464)^(1/4) + (65536*I - 65536)*(-1/x + 1)^(1/4)) - I*(1/16*I + 1/16)^(1/4)*log(I*(-7378697629
4838206464*I - 73786976294838206464)^(1/4) - (65536*I - 65536)*(-1/x + 1)^(1/4)) + I*(1/16*I + 1/16)^(1/4)*log
(I*(-73786976294838206464*I - 73786976294838206464)^(1/4) + (65536*I - 65536)*(-1/x + 1)^(1/4)) + (1/16*I + 1/
16)^(1/4)*log(-I*(-73786976294838206464*I - 73786976294838206464)^(1/4) + (65536*I + 65536)*(-1/x + 1)^(1/4))
- (1/16*I + 1/16)^(1/4)*log(-I*(-73786976294838206464*I - 73786976294838206464)^(1/4) - (65536*I + 65536)*(-1/
x + 1)^(1/4)) + 2*I*(-1/256*I + 1/256)^(1/4)*log(-I*(85070591730234615865843651857942052864*I - 85070591730234
615865843651857942052864)^(1/4) + (2147483648*I + 2147483648)*(-1/x + 1)^(1/4)) - 2*I*(-1/256*I + 1/256)^(1/4)
*log(-I*(85070591730234615865843651857942052864*I - 85070591730234615865843651857942052864)^(1/4) - (214748364
8*I + 2147483648)*(-1/x + 1)^(1/4)) - 1/2*2^(1/4)*log(abs(-2^(1/4) + (-1/x + 1)^(1/4)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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