3.32.16 \(\int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx\)
Optimal. Leaf size=669 \[ \frac {\log \left (\sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}} \left (\sqrt [3]{1-a} x^2-\sqrt [3]{1-a}\right )+1\right )}{2 \sqrt [3]{1-a}}+\frac {\log \left (\left (\sqrt [3]{a+1} x^2-\sqrt [3]{a+1}\right ) \sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}}-1\right )}{2 \sqrt [3]{a+1}}-\frac {\log \left (\sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}} \left (\sqrt [3]{1-a}-\sqrt [3]{1-a} x^2\right )+\left ((1-a)^{2/3} x^4-2 (1-a)^{2/3} x^2+(1-a)^{2/3}\right ) \left (\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}\right )^{2/3}+1\right )}{4 \sqrt [3]{1-a}}-\frac {\log \left (\sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}} \left (\sqrt [3]{a+1} x^2-\sqrt [3]{a+1}\right )+\left ((a+1)^{2/3} x^4-2 (a+1)^{2/3} x^2+(a+1)^{2/3}\right ) \left (\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}\right )^{2/3}+1\right )}{4 \sqrt [3]{a+1}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{1-a} \left (2 x^2-2\right ) \sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1-a}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a+1} \left (2 x^2-2\right ) \sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a+1}} \]
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Rubi [A] time = 0.20, antiderivative size = 473, normalized size of antiderivative = 0.71,
number of steps used = 10, number of rules used = 6, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used
= {6688, 6718, 912, 105, 59, 91} \begin {gather*} \frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \log (1-x)}{4 \sqrt [3]{a+1} \sqrt [3]{x}}-\frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \log (x+1)}{4 \sqrt [3]{a-1} \sqrt [3]{x}}+\frac {3 \left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \log \left (\frac {\sqrt [3]{a x+1}}{\sqrt [3]{a-1}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a-1} \sqrt [3]{x}}-\frac {3 \left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \log \left (\frac {\sqrt [3]{a x+1}}{\sqrt [3]{a+1}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a+1} \sqrt [3]{x}}+\frac {\sqrt {3} \left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x+1}}{\sqrt {3} \sqrt [3]{a-1} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a-1} \sqrt [3]{x}}-\frac {\sqrt {3} \left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x+1}}{\sqrt {3} \sqrt [3]{a+1} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a+1} \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3),x]
[Out]
(Sqrt[3]*(1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*ArcTan[1/Sqrt[3] + (2*(1 + a*x)^(1/3))
/(Sqrt[3]*(-1 + a)^(1/3)*x^(1/3))])/(2*(-1 + a)^(1/3)*x^(1/3)) - (Sqrt[3]*(1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 -
x^2)^3)))^(1/3)*(1 - x^2)*ArcTan[1/Sqrt[3] + (2*(1 + a*x)^(1/3))/(Sqrt[3]*(1 + a)^(1/3)*x^(1/3))])/(2*(1 + a)
^(1/3)*x^(1/3)) + ((1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*Log[1 - x])/(4*(1 + a)^(1/3)
*x^(1/3)) - ((1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*Log[1 + x])/(4*(-1 + a)^(1/3)*x^(1
/3)) + (3*(1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*Log[-x^(1/3) + (1 + a*x)^(1/3)/(-1 +
a)^(1/3)])/(4*(-1 + a)^(1/3)*x^(1/3)) - (3*(1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*Log[
-x^(1/3) + (1 + a*x)^(1/3)/(1 + a)^(1/3)])/(4*(1 + a)^(1/3)*x^(1/3))
Rule 59
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]
Rule 91
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, 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 912
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] && !IntegerQ[m] && !IntegerQ[n]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6718
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n*z^q)^FracP
art[p])/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p])), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x] && !FreeQ[z, x]
Rubi steps
\begin {align*} \int \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx &=\int \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \, dx\\ &=\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{1+a x} \left (-1+x^2\right )} \, dx}{\sqrt [3]{x}}\\ &=\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \left (-\frac {\sqrt [3]{x}}{2 (1-x) \sqrt [3]{1+a x}}-\frac {\sqrt [3]{x}}{2 (1+x) \sqrt [3]{1+a x}}\right ) \, dx}{\sqrt [3]{x}}\\ &=-\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {\sqrt [3]{x}}{(1-x) \sqrt [3]{1+a x}} \, dx}{2 \sqrt [3]{x}}-\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {\sqrt [3]{x}}{(1+x) \sqrt [3]{1+a x}} \, dx}{2 \sqrt [3]{x}}\\ &=-\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {1}{(1-x) x^{2/3} \sqrt [3]{1+a x}} \, dx}{2 \sqrt [3]{x}}+\frac {\left (\sqrt [3]{1+a x} \sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right )\right ) \int \frac {1}{x^{2/3} (1+x) \sqrt [3]{1+a x}} \, dx}{2 \sqrt [3]{x}}\\ &=\frac {\sqrt {3} \sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+a x}}{\sqrt {3} \sqrt [3]{-1+a} \sqrt [3]{x}}\right )}{2 \sqrt [3]{-1+a} \sqrt [3]{x}}-\frac {\sqrt {3} \sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+a x}}{\sqrt {3} \sqrt [3]{1+a} \sqrt [3]{x}}\right )}{2 \sqrt [3]{1+a} \sqrt [3]{x}}+\frac {\sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \log (1-x)}{4 \sqrt [3]{1+a} \sqrt [3]{x}}-\frac {\sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \log (1+x)}{4 \sqrt [3]{-1+a} \sqrt [3]{x}}+\frac {3 \sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+a x}}{\sqrt [3]{-1+a}}\right )}{4 \sqrt [3]{-1+a} \sqrt [3]{x}}-\frac {3 \sqrt [3]{1+a x} \sqrt [3]{-\frac {x}{(1+a x) \left (1-x^2\right )^3}} \left (1-x^2\right ) \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+a x}}{\sqrt [3]{1+a}}\right )}{4 \sqrt [3]{1+a} \sqrt [3]{x}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 72, normalized size = 0.11 \begin {gather*} \frac {3}{2} \left (x^2-1\right ) \sqrt [3]{\frac {x}{\left (x^2-1\right )^3 (a x+1)}} \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(a-1) x}{a x+1}\right )-\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(a+1) x}{a x+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3),x]
[Out]
(3*(x/((1 + a*x)*(-1 + x^2)^3))^(1/3)*(-1 + x^2)*(Hypergeometric2F1[1/3, 1, 4/3, ((-1 + a)*x)/(1 + a*x)] - Hyp
ergeometric2F1[1/3, 1, 4/3, ((1 + a)*x)/(1 + a*x)]))/2
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IntegrateAlgebraic [A] time = 2.02, size = 669, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{1-a} \left (-2+2 x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1-a}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{1+a} \left (-2+2 x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1+a}}+\frac {\log \left (1+\left (-\sqrt [3]{1-a}+\sqrt [3]{1-a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right )}{2 \sqrt [3]{1-a}}+\frac {\log \left (-1+\left (-\sqrt [3]{1+a}+\sqrt [3]{1+a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right )}{2 \sqrt [3]{1+a}}-\frac {\log \left (1+\left (\sqrt [3]{1-a}-\sqrt [3]{1-a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left ((1-a)^{2/3}-2 (1-a)^{2/3} x^2+(1-a)^{2/3} x^4\right ) \left (\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right )^{2/3}\right )}{4 \sqrt [3]{1-a}}-\frac {\log \left (1+\left (-\sqrt [3]{1+a}+\sqrt [3]{1+a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left ((1+a)^{2/3}-2 (1+a)^{2/3} x^2+(1+a)^{2/3} x^4\right ) \left (\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right )^{2/3}\right )}{4 \sqrt [3]{1+a}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3),x]
[Out]
-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - ((1 - a)^(1/3)*(-2 + 2*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5
+ x^6 + a*x^7))^(1/3))/Sqrt[3]])/(1 - a)^(1/3) - (Sqrt[3]*ArcTan[1/Sqrt[3] + ((1 + a)^(1/3)*(-2 + 2*x^2)*(x/(-
1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3))/Sqrt[3]])/(2*(1 + a)^(1/3)) + Log[1 + (-(1
- a)^(1/3) + (1 - a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)]/(2*(1
- a)^(1/3)) + Log[-1 + (-(1 + a)^(1/3) + (1 + a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 +
x^6 + a*x^7))^(1/3)]/(2*(1 + a)^(1/3)) - Log[1 + ((1 - a)^(1/3) - (1 - a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3
*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3) + ((1 - a)^(2/3) - 2*(1 - a)^(2/3)*x^2 + (1 - a)^(2/3)*x^4)*(x/
(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(2/3)]/(4*(1 - a)^(1/3)) - Log[1 + (-(1 + a)^(1/
3) + (1 + a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3) + ((1 + a)^(2/3
) - 2*(1 + a)^(2/3)*x^2 + (1 + a)^(2/3)*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^
(2/3)]/(4*(1 + a)^(1/3))
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fricas [A] time = 0.60, size = 3267, normalized size = 4.88
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3),x, algorithm="fricas")
[Out]
[1/4*(sqrt(3)*(a^2 - 1)*sqrt((-a + 1)^(1/3)/(a - 1))*log(-((3*a - 2)*x - sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(-a
+ 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 - a)*x^5 + (a - 1)*
x^4 - 2*(a^2 - a)*x^3 - 2*(a - 1)*x^2 + (a^2 - a)*x + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x
+ 3*x^2 - 1))^(2/3) - (a*x + 1)*(-a + 1)^(1/3))*sqrt((-a + 1)^(1/3)/(a - 1)) + 3*(a*x^3 - a*x + x^2 - 1)*(-a
+ 1)^(1/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x + 1)) + sqrt(3)*(a^2
- 1)*sqrt(-1/(a + 1)^(2/3))*log(((3*a + 2)*x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*
x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 + a)*x^5 + (a + 1)*x^4 - 2*(a^2 + a)*x^3 - 2*(
a + 1)*x^2 + (a^2 + a)*x + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a*x
+ 1)*(a + 1)^(1/3))*sqrt(-1/(a + 1)^(2/3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a + 1)^(1/3)*(x/(a*x^7 - 3*a*x^5 + x^
6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x - 1)) - (a + 1)^(2/3)*(a - 1)*log((x^2 - 1)*(a + 1)^(2/3
)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 + a + 1
)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) + (a + 1)*(-a + 1)^(2
/3)*log((x^2 - 1)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a -
1)*x^4 - 2*(a - 1)*x^2 + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) - (-a +
1)^(1/3)) + 2*(a + 1)^(2/3)*(a - 1)*log(((a + 1)*x^2 - a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a
*x + 3*x^2 - 1))^(1/3) - (a + 1)^(2/3)) - 2*(a + 1)*(-a + 1)^(2/3)*log(((a - 1)*x^2 - a + 1)*(x/(a*x^7 - 3*a*x
^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (-a + 1)^(2/3)))/(a^2 - 1), 1/4*(2*sqrt(3)*(a^2 - 1)*sq
rt(-(-a + 1)^(1/3)/(a - 1))*arctan(1/3*sqrt(3)*(2*(x^2 - 1)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3
- 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (-a + 1)^(1/3))*sqrt(-(-a + 1)^(1/3)/(a - 1))) + sqrt(3)*(a^2 - 1)*sqrt(-
1/(a + 1)^(2/3))*log(((3*a + 2)*x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 +
3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 + a)*x^5 + (a + 1)*x^4 - 2*(a^2 + a)*x^3 - 2*(a + 1)*x^2
+ (a^2 + a)*x + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a*x + 1)*(a +
1)^(1/3))*sqrt(-1/(a + 1)^(2/3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a + 1)^(1/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3
- 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x - 1)) - (a + 1)^(2/3)*(a - 1)*log((x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7
- 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 + a + 1)*(x/(a*x^7
- 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) + (a + 1)*(-a + 1)^(2/3)*log((x^
2 - 1)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a - 1)*x^4 - 2
*(a - 1)*x^2 + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) - (-a + 1)^(1/3))
+ 2*(a + 1)^(2/3)*(a - 1)*log(((a + 1)*x^2 - a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2
- 1))^(1/3) - (a + 1)^(2/3)) - 2*(a + 1)*(-a + 1)^(2/3)*log(((a - 1)*x^2 - a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 +
3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (-a + 1)^(2/3)))/(a^2 - 1), 1/4*(sqrt(3)*(a^2 - 1)*sqrt((-a + 1)^(
1/3)/(a - 1))*log(-((3*a - 2)*x - sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 +
3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 - a)*x^5 + (a - 1)*x^4 - 2*(a^2 - a)*x^3 - 2*(a - 1)*x^2 +
(a^2 - a)*x + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) - (a*x + 1)*(-a +
1)^(1/3))*sqrt((-a + 1)^(1/3)/(a - 1)) + 3*(a*x^3 - a*x + x^2 - 1)*(-a + 1)^(1/3)*(x/(a*x^7 - 3*a*x^5 + x^6 +
3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x + 1)) - (a + 1)^(2/3)*(a - 1)*log((x^2 - 1)*(a + 1)^(2/3)*(x
/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 + a + 1)*(x
/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) + (a + 1)*(-a + 1)^(2/3)*
log((x^2 - 1)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a - 1)*
x^4 - 2*(a - 1)*x^2 + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) - (-a + 1)^
(1/3)) + 2*(a + 1)^(2/3)*(a - 1)*log(((a + 1)*x^2 - a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x +
3*x^2 - 1))^(1/3) - (a + 1)^(2/3)) - 2*(a + 1)*(-a + 1)^(2/3)*log(((a - 1)*x^2 - a + 1)*(x/(a*x^7 - 3*a*x^5 +
x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (-a + 1)^(2/3)) - 2*sqrt(3)*(a^2 - 1)*arctan(1/3*sqrt(3)*(2
*(x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + (a + 1)^(1/3)
)/(a + 1)^(1/3))/(a + 1)^(1/3))/(a^2 - 1), 1/4*(2*sqrt(3)*(a^2 - 1)*sqrt(-(-a + 1)^(1/3)/(a - 1))*arctan(1/3*s
qrt(3)*(2*(x^2 - 1)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (-a
+ 1)^(1/3))*sqrt(-(-a + 1)^(1/3)/(a - 1))) - (a + 1)^(2/3)*(a - 1)*log((x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*
a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 + a + 1)*(x/(a*x^7 - 3*
a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) + (a + 1)*(-a + 1)^(2/3)*log((x^2 - 1
)*(-a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a - 1)*x^4 - 2*(a -
1)*x^2 + a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) - (-a + 1)^(1/3)) + 2*(
a + 1)^(2/3)*(a - 1)*log(((a + 1)*x^2 - a - 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))
^(1/3) - (a + 1)^(2/3)) - 2*(a + 1)*(-a + 1)^(2/3)*log(((a - 1)*x^2 - a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x
^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - (-a + 1)^(2/3)) - 2*sqrt(3)*(a^2 - 1)*arctan(1/3*sqrt(3)*(2*(x^2 - 1)*(
a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + (a + 1)^(1/3))/(a + 1)^(1
/3))/(a + 1)^(1/3))/(a^2 - 1)]
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\frac {x}{a x^{7} - 3 \, a x^{5} + x^{6} + 3 \, a x^{3} - 3 \, x^{4} - a x + 3 \, x^{2} - 1}\right )^{\frac {1}{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3),x, algorithm="giac")
[Out]
integrate((x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3), x)
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \left (\frac {x}{a \,x^{7}-3 a \,x^{5}+x^{6}+3 a \,x^{3}-3 x^{4}-a x +3 x^{2}-1}\right )^{\frac {1}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3),x)
[Out]
int((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\frac {x}{a x^{7} - 3 \, a x^{5} + x^{6} + 3 \, a x^{3} - 3 \, x^{4} - a x + 3 \, x^{2} - 1}\right )^{\frac {1}{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3),x, algorithm="maxima")
[Out]
integrate((x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (-\frac {x}{-a\,x^7-x^6+3\,a\,x^5+3\,x^4-3\,a\,x^3-3\,x^2+a\,x+1}\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-x/(a*x - 3*a*x^3 + 3*a*x^5 - a*x^7 - 3*x^2 + 3*x^4 - x^6 + 1))^(1/3),x)
[Out]
int((-x/(a*x - 3*a*x^3 + 3*a*x^5 - a*x^7 - 3*x^2 + 3*x^4 - x^6 + 1))^(1/3), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{\frac {x}{a x^{7} - 3 a x^{5} + 3 a x^{3} - a x + x^{6} - 3 x^{4} + 3 x^{2} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x/(a*x**7-3*a*x**5+x**6+3*a*x**3-3*x**4-a*x+3*x**2-1))**(1/3),x)
[Out]
Integral((x/(a*x**7 - 3*a*x**5 + 3*a*x**3 - a*x + x**6 - 3*x**4 + 3*x**2 - 1))**(1/3), x)
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