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

Optimal. Leaf size=156 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4-x^3}-\text {$\#$1} x\right )-2 \log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]-\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {4}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^3}}\right ) \]

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Rubi [C]  time = 0.87, antiderivative size = 746, normalized size of antiderivative = 4.78, number of steps used = 54, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2056, 6725, 105, 50, 63, 240, 212, 206, 203, 93, 298} \begin {gather*} \frac {1}{3} (-1)^{2/3} \sqrt [4]{x^4-x^3}-\frac {1}{3} \sqrt [3]{-1} \sqrt [4]{x^4-x^3}+\frac {1}{3} \sqrt [4]{x^4-x^3}-\frac {(-1)^{2/3} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [3]{-1} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}-\frac {4 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{1+i \sqrt {3}} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {(-1)^{2/3} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [3]{-1} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{6 \sqrt [4]{x-1} x^{3/4}}+\frac {4 \sqrt [4]{2} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{1-i \sqrt {3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1-\sqrt [3]{-1}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{1+i \sqrt {3}} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1+(-1)^{2/3}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{3 \sqrt [4]{x-1} x^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-x^3 + x^4)^(1/4)/3 - ((-1)^(1/3)*(-x^3 + x^4)^(1/4))/3 + ((-1)^(2/3)*(-x^3 + x^4)^(1/4))/3 - ((-x^3 + x^4)^(
1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(6*(-1 + x)^(1/4)*x^(3/4)) + ((-1)^(1/3)*(-x^3 + x^4)^(1/4)*ArcTan[(-1 +
x)^(1/4)/x^(1/4)])/(6*(-1 + x)^(1/4)*x^(3/4)) - ((-1)^(2/3)*(-x^3 + x^4)^(1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])
/(6*(-1 + x)^(1/4)*x^(3/4)) - (4*2^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 +
x)^(1/4)*x^(3/4)) + (2^(3/4)*(1 - I*Sqrt[3])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[((1 - (-1)^(1/3))^(1/4)*x^(1/4))
/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) + (2^(3/4)*(1 + I*Sqrt[3])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTan[((1 +
(-1)^(2/3))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4)) - ((-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^
(1/4)/x^(1/4)])/(6*(-1 + x)^(1/4)*x^(3/4)) + ((-1)^(1/3)*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(
6*(-1 + x)^(1/4)*x^(3/4)) - ((-1)^(2/3)*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^(1/4)/x^(1/4)])/(6*(-1 + x)^(1/4)*
x^(3/4)) + (4*2^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[(2^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4))
- (2^(3/4)*(1 - I*Sqrt[3])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[((1 - (-1)^(1/3))^(1/4)*x^(1/4))/(-1 + x)^(1/4)])
/(3*(-1 + x)^(1/4)*x^(3/4)) - (2^(3/4)*(1 + I*Sqrt[3])^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[((1 + (-1)^(2/3))^(1/4
)*x^(1/4))/(-1 + x)^(1/4)])/(3*(-1 + x)^(1/4)*x^(3/4))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

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

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Mathematica [C]  time = 0.18, size = 101, normalized size = 0.65 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (2 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {x-1}{2 x}\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {2 i (x-1)}{\left (-i+\sqrt {3}\right ) x}\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {2 i (x-1)}{\left (i+\sqrt {3}\right ) x}\right )\right )}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-1 + x)*x^3)^(1/4)*(2*Hypergeometric2F1[1/4, 1, 5/4, (-1 + x)/(2*x)] - Hypergeometric2F1[1/4, 1, 5/4, ((-
2*I)*(-1 + x))/((-I + Sqrt[3])*x)] - Hypergeometric2F1[1/4, 1, 5/4, ((2*I)*(-1 + x))/((I + Sqrt[3])*x)]))/(3*x
)

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IntegrateAlgebraic [A]  time = 0.41, size = 156, normalized size = 1.00 \begin {gather*} -\frac {4}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {4}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.56, size = 881, normalized size = 5.65

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(2*(2*x^2 + (x^4 - x^3)^(1/4)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) +
8) + 2*sqrt(x^4 - x^3))/x^2) + 1/12*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(2*(2*x^2 - (x^4 - x^3)^(1/4)*(sqrt(
3)*x + 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 - x^3))/x^2) - 1/6*sqrt(sqrt(3) + 2)*(sqrt(3) - 2)*log(4*(x^2 +
(x^4 - x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^4 - x^3))/x^2) + 1/6*sqrt(sqrt(3) + 2)*(sqrt(3)
- 2)*log(4*(x^2 - (x^4 - x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^4 - x^3))/x^2) + 1/3*sqrt(-4
*sqrt(3) + 8)*arctan(1/2*(sqrt(2)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8)*sqrt((2*x^2 + (x^4 - x^3)^(1/4)*(sqrt
(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 - x^3))/x^2) - 2*(x^4 - x^3)^(1/4)*(sqrt(3) + 2)*sqrt(-4*sqrt(3
) + 8) - 2*sqrt(3)*x - 4*x)/x) + 1/3*sqrt(-4*sqrt(3) + 8)*arctan(1/2*(sqrt(2)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3
) + 8)*sqrt((2*x^2 - (x^4 - x^3)^(1/4)*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(x^4 - x^3))/x^2) - 2*(x
^4 - x^3)^(1/4)*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8) + 2*sqrt(3)*x + 4*x)/x) - 2/3*sqrt(sqrt(3) + 2)*arctan((2*(
sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2)*sqrt((x^2 + (x^4 - x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2) + sqrt(x^
4 - x^3))/x^2) - 2*(x^4 - x^3)^(1/4)*sqrt(sqrt(3) + 2)*(sqrt(3) - 2) + sqrt(3)*x - 2*x)/x) - 2/3*sqrt(sqrt(3)
+ 2)*arctan((2*(sqrt(3)*x - 2*x)*sqrt(sqrt(3) + 2)*sqrt((x^2 - (x^4 - x^3)^(1/4)*(sqrt(3)*x - 2*x)*sqrt(sqrt(3
) + 2) + sqrt(x^4 - x^3))/x^2) - 2*(x^4 - x^3)^(1/4)*sqrt(sqrt(3) + 2)*(sqrt(3) - 2) - sqrt(3)*x + 2*x)/x) - 8
/3*2^(1/4)*arctan(1/2*(2^(3/4)*x*sqrt((sqrt(2)*x^2 + sqrt(x^4 - x^3))/x^2) - 2^(3/4)*(x^4 - x^3)^(1/4))/x) + 2
/3*2^(1/4)*log((2^(1/4)*x + (x^4 - x^3)^(1/4))/x) - 2/3*2^(1/4)*log(-(2^(1/4)*x - (x^4 - x^3)^(1/4))/x)

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giac [B]  time = 1.07, size = 392, normalized size = 2.51 \begin {gather*} \frac {1}{6} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{6} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{6} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{6} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{3} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {2}{3} \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((-1+x)*(x^4-x^3)^(1/4)/x/(x^3+1),x, algorithm="giac")

[Out]

1/6*(sqrt(6) + sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) + 1/6*(sqrt(6) +
sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) + sqrt(2))) + 1/6*(sqrt(6) - sqrt(2))*arcta
n((sqrt(6) + sqrt(2) + 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) + 1/6*(sqrt(6) - sqrt(2))*arctan(-(sqrt(6) + s
qrt(2) - 4*(-1/x + 1)^(1/4))/(sqrt(6) - sqrt(2))) + 1/12*(sqrt(6) + sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*(-1/x
+ 1)^(1/4) + sqrt(-1/x + 1) + 1) - 1/12*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6) + sqrt(2))*(-1/x + 1)^(1/4) + s
qrt(-1/x + 1) + 1) + 1/12*(sqrt(6) - sqrt(2))*log(1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) +
1) - 1/12*(sqrt(6) - sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*(-1/x + 1)^(1/4) + sqrt(-1/x + 1) + 1) - 1/3*8^(3/4
)*arctan(1/2*2^(3/4)*(-1/x + 1)^(1/4)) - 2/3*2^(1/4)*log(2^(1/4) + (-1/x + 1)^(1/4)) + 2/3*2^(1/4)*log(abs(-2^
(1/4) + (-1/x + 1)^(1/4)))

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maple [B]  time = 9.04, size = 3100, normalized size = 19.87

method	result	size
trager	\(\text {Expression too large to display}\)	\(3100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729*RootOf(6561*_Z^8-81*_Z^4+1)^7*ln(-(13122*x^3*RootOf(6561*_Z^8-81*_Z^4+1)^9-26244*RootOf(6561*_Z^8-81*_Z^4
+1)^9*x^2+11664*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z^4+1)^7*x-2430*RootOf(6561*_Z^8-81*_Z^4+1)^6*(x^4-x^3)^(
1/4)*x^2-1053*RootOf(6561*_Z^8-81*_Z^4+1)^5*x^3+270*(x^4-x^3)^(3/4)*RootOf(6561*_Z^8-81*_Z^4+1)^4+972*RootOf(6
561*_Z^8-81*_Z^4+1)^5*x^2-90*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z^4+1)^3*x-18*RootOf(6561*_Z^8-81*_Z^4+1)^2*
(x^4-x^3)^(1/4)*x^2+15*RootOf(6561*_Z^8-81*_Z^4+1)*x^3+2*(x^4-x^3)^(3/4)-9*RootOf(6561*_Z^8-81*_Z^4+1)*x^2)/(8
1*RootOf(6561*_Z^8-81*_Z^4+1)^4*x-162*RootOf(6561*_Z^8-81*_Z^4+1)^4+x+1)/x^2)-81*RootOf(6561*_Z^8-81*_Z^4+1)^5
*ln((-177147*RootOf(6561*_Z^8-81*_Z^4+1)^11*x^3+354294*RootOf(6561*_Z^8-81*_Z^4+1)^11*x^2-2187*RootOf(6561*_Z^
8-81*_Z^4+1)^7*x^3-11664*RootOf(6561*_Z^8-81*_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2-10935*RootOf(6561*_Z^8-81*_Z^4+1)^7
*x^2-1458*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z^4+1)^5*x+810*(x^4-x^3)^(3/4)*RootOf(6561*_Z^8-81*_Z^4+1)^4+32
4*RootOf(6561*_Z^8-81*_Z^4+1)^3*x^3+54*RootOf(6561*_Z^8-81*_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-81*RootOf(6561*_Z^8-8
1*_Z^4+1)^3*x^2-30*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z^4+1)*x-16*(x^4-x^3)^(3/4))/(81*RootOf(6561*_Z^8-81*_
Z^4+1)^4*x-162*RootOf(6561*_Z^8-81*_Z^4+1)^4-2*x+1)/x^2)+729*RootOf(6561*_Z^8-81*_Z^4+1)^7*ln((59049*x^3*RootO
f(6561*_Z^8-81*_Z^4+1)^9-118098*RootOf(6561*_Z^8-81*_Z^4+1)^9*x^2+34992*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z
^4+1)^7*x+11664*RootOf(6561*_Z^8-81*_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2+2430*RootOf(6561*_Z^8-81*_Z^4+1)^5*x^3+810*(
x^4-x^3)^(3/4)*RootOf(6561*_Z^8-81*_Z^4+1)^4+243*RootOf(6561*_Z^8-81*_Z^4+1)^5*x^2-162*(x^4-x^3)^(1/2)*RootOf(
6561*_Z^8-81*_Z^4+1)^3*x-54*RootOf(6561*_Z^8-81*_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-24*RootOf(6561*_Z^8-81*_Z^4+1)*x
^3-16*(x^4-x^3)^(3/4)+6*RootOf(6561*_Z^8-81*_Z^4+1)*x^2)/(81*RootOf(6561*_Z^8-81*_Z^4+1)^4*x-162*RootOf(6561*_
Z^8-81*_Z^4+1)^4-2*x+1)/x^2)-9*RootOf(6561*_Z^8-81*_Z^4+1)^3*ln((59049*x^3*RootOf(6561*_Z^8-81*_Z^4+1)^9-11809
8*RootOf(6561*_Z^8-81*_Z^4+1)^9*x^2+34992*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z^4+1)^7*x+11664*RootOf(6561*_Z
^8-81*_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2+2430*RootOf(6561*_Z^8-81*_Z^4+1)^5*x^3+810*(x^4-x^3)^(3/4)*RootOf(6561*_Z^
8-81*_Z^4+1)^4+243*RootOf(6561*_Z^8-81*_Z^4+1)^5*x^2-162*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z^4+1)^3*x-54*Ro
otOf(6561*_Z^8-81*_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-24*RootOf(6561*_Z^8-81*_Z^4+1)*x^3-16*(x^4-x^3)^(3/4)+6*RootOf
(6561*_Z^8-81*_Z^4+1)*x^2)/(81*RootOf(6561*_Z^8-81*_Z^4+1)^4*x-162*RootOf(6561*_Z^8-81*_Z^4+1)^4-2*x+1)/x^2)-R
ootOf(6561*_Z^8-81*_Z^4+1)*ln(-(-59049*RootOf(6561*_Z^8-81*_Z^4+1)^11*x^3+118098*RootOf(6561*_Z^8-81*_Z^4+1)^1
1*x^2+2187*RootOf(6561*_Z^8-81*_Z^4+1)^7*x^3+2430*RootOf(6561*_Z^8-81*_Z^4+1)^6*(x^4-x^3)^(1/4)*x^2+729*RootOf
(6561*_Z^8-81*_Z^4+1)^7*x^2+810*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z^4+1)^5*x+270*(x^4-x^3)^(3/4)*RootOf(656
1*_Z^8-81*_Z^4+1)^4+90*RootOf(6561*_Z^8-81*_Z^4+1)^3*x^3+18*RootOf(6561*_Z^8-81*_Z^4+1)^2*(x^4-x^3)^(1/4)*x^2-
54*RootOf(6561*_Z^8-81*_Z^4+1)^3*x^2+6*(x^4-x^3)^(1/2)*RootOf(6561*_Z^8-81*_Z^4+1)*x+2*(x^4-x^3)^(3/4))/(81*Ro
otOf(6561*_Z^8-81*_Z^4+1)^4*x-162*RootOf(6561*_Z^8-81*_Z^4+1)^4+x+1)/x^2)-1/3*RootOf(_Z^2+972*RootOf(6561*_Z^8
-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-81*_Z^4+1))*ln((729*RootOf(6561*_Z^8-81*_Z
^4+1)^5*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-81
*_Z^4+1))*x^3-1944*RootOf(6561*_Z^8-81*_Z^4+1)^5*(x^4-x^3)^(1/4)*x^2-243*RootOf(6561*_Z^8-81*_Z^4+1)^5*RootOf(
_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-81*_Z^4+1))*x^2-8
1*RootOf(6561*_Z^8-81*_Z^4+1)^3*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^
3-12*RootOf(6561*_Z^8-81*_Z^4+1))*x^3+216*RootOf(6561*_Z^8-81*_Z^4+1)^3*(x^4-x^3)^(1/4)*x^2+27*RootOf(6561*_Z^
8-81*_Z^4+1)^3*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*
_Z^8-81*_Z^4+1))*x^2-9*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*Root
Of(6561*_Z^8-81*_Z^4+1))*RootOf(6561*_Z^8-81*_Z^4+1)*x^3-4*(x^4-x^3)^(1/2)*RootOf(_Z^2+972*RootOf(6561*_Z^8-81
*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-81*_Z^4+1))*x+24*RootOf(6561*_Z^8-81*_Z^4+1)*
(x^4-x^3)^(1/4)*x^2+3*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootO
f(6561*_Z^8-81*_Z^4+1))*RootOf(6561*_Z^8-81*_Z^4+1)*x^2+8*(x^4-x^3)^(3/4))/x^2/(1+x))+243*RootOf(6561*_Z^8-81*
_Z^4+1)^6*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-
81*_Z^4+1))*ln((2916*(x^4-x^3)^(1/2)*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^
4+1)^3-12*RootOf(6561*_Z^8-81*_Z^4+1))*RootOf(6561*_Z^8-81*_Z^4+1)^6*x-729*RootOf(6561*_Z^8-81*_Z^4+1)^5*RootO
f(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-81*_Z^4+1))*x^3
+1944*RootOf(6561*_Z^8-81*_Z^4+1)^5*(x^4-x^3)^(1/4)*x^2+243*RootOf(6561*_Z^8-81*_Z^4+1)^5*RootOf(_Z^2+972*Root
Of(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-81*_Z^4+1))*x^2-81*RootOf(6561
*_Z^8-81*_Z^4+1)^3*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6
561*_Z^8-81*_Z^4+1))*x^3-216*RootOf(6561*_Z^8-81*_Z^4+1)^3*(x^4-x^3)^(1/4)*x^2+27*RootOf(6561*_Z^8-81*_Z^4+1)^
3*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-81*_Z^4+
1))*x^2+9*RootOf(_Z^2+972*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-
81*_Z^4+1))*RootOf(6561*_Z^8-81*_Z^4+1)*x^3-24*RootOf(6561*_Z^8-81*_Z^4+1)*(x^4-x^3)^(1/4)*x^2-3*RootOf(_Z^2+9
72*RootOf(6561*_Z^8-81*_Z^4+1)^5-108*RootOf(6561*_Z^8-81*_Z^4+1)^3-12*RootOf(6561*_Z^8-81*_Z^4+1))*RootOf(6561
*_Z^8-81*_Z^4+1)*x^2+8*(x^4-x^3)^(3/4))/x^2/(1+x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 - 1\right )}{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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