3.24.22 \(\int \frac {1}{(1+x^3) \sqrt [3]{-x^2+x^3}} \, dx\)
Optimal. Leaf size=180 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right )}{\sqrt [3]{2} \sqrt {3}} \]
________________________________________________________________________________________
Rubi [B] time = 0.34, antiderivative size = 571, normalized size of antiderivative = 3.17,
number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used =
{2056, 6725, 91} \begin {gather*} -\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log (-x-1)}{6 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{-1} x-1\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (-(-1)^{2/3} x-1\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
-(((-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2^(2/3)*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(2^(1/3)*Sqrt[3]*(-x
^2 + x^3)^(1/3))) - ((-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*(1 - (-1)^(1/3))^(1
/3)*x^(1/3))])/(Sqrt[3]*(1 - (-1)^(1/3))^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3]
+ (2*(-1 + x)^(1/3))/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)*x^(1/3))])/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)*(-x^2 + x^3)^(
1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/2^(1/3) - x^(1/3)])/(2*2^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 +
x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/(1 - (-1)^(1/3))^(1/3) - x^(1/3)])/(2*(1 - (-1)^(1/3))^(1/3)*(-x^2 + x^3)
^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/(1 + (-1)^(2/3))^(1/3) - x^(1/3)])/(2*(1 + (-1)^(2/3))^(1
/3)*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*Log[-1 - x])/(6*2^(1/3)*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3
)*x^(2/3)*Log[-1 + (-1)^(1/3)*x])/(6*(1 - (-1)^(1/3))^(1/3)*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*Log[
-1 - (-1)^(2/3)*x])/(6*(1 + (-1)^(2/3))^(1/3)*(-x^2 + x^3)^(1/3))
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 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}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (1+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {1}{3 (-1-x) \sqrt [3]{-1+x} x^{2/3}}-\frac {1}{3 \sqrt [3]{-1+x} x^{2/3} \left (-1+\sqrt [3]{-1} x\right )}-\frac {1}{3 \sqrt [3]{-1+x} x^{2/3} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{6 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\sqrt [3]{-1} x\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1-(-1)^{2/3} x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 119, normalized size = 0.66 \begin {gather*} \frac {\left ((x-1) x^2\right )^{2/3} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x-1}{2 x}\right )+\left (1-i \sqrt {3}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {2 i (x-1)}{\left (-i+\sqrt {3}\right ) x}\right )+\left (1+i \sqrt {3}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 i (x-1)}{\left (i+\sqrt {3}\right ) x}\right )\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
(((-1 + x)*x^2)^(2/3)*(Hypergeometric2F1[2/3, 1, 5/3, (-1 + x)/(2*x)] + (1 - I*Sqrt[3])*Hypergeometric2F1[2/3,
1, 5/3, ((-2*I)*(-1 + x))/((-I + Sqrt[3])*x)] + (1 + I*Sqrt[3])*Hypergeometric2F1[2/3, 1, 5/3, ((2*I)*(-1 + x
))/((I + Sqrt[3])*x)]))/(4*x^2)
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 180, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) - Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)
]/(3*2^(1/3)) + Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(6*2^(1/3)) - RootSum[1
- #1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3
________________________________________________________________________________________
fricas [B] time = 0.60, size = 851, normalized size = 4.73
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3+1)/(x^3-x^2)^(1/3),x, algorithm="fricas")
[Out]
-1/6*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(-1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x - 2*sqrt(6)*(-1)^(1/3)*(x^3 - x^2)^(1/3
))/x) - 2/3*(sqrt(3)*cos(2/9*pi) - sin(2/9*pi))*arctan((8*(2*x*cos(2/9*pi)^3 - x*cos(2/9*pi))*sin(2/9*pi) + sq
rt(3)*x + 2*(2*sqrt(3)*x*cos(2/9*pi)^2 + 2*x*cos(2/9*pi)*sin(2/9*pi) - sqrt(3)*x)*sqrt((x^2 - (2*sqrt(3)*x*cos
(2/9*pi)*sin(2/9*pi) + 2*x*cos(2/9*pi)^2 - x)*(x^3 - x^2)^(1/3) + (x^3 - x^2)^(2/3))/x^2) - 2*(x^3 - x^2)^(1/3
)*(2*sqrt(3)*cos(2/9*pi)^2 + 2*cos(2/9*pi)*sin(2/9*pi) - sqrt(3)))/(16*x*cos(2/9*pi)^4 - 16*x*cos(2/9*pi)^2 +
3*x)) + 2/3*(sqrt(3)*cos(2/9*pi) + sin(2/9*pi))*arctan(-1/2*(2*x*cos(2/9*pi)^2 - x*sqrt((x^2 + 2*(x^3 - x^2)^(
1/3)*(2*x*cos(2/9*pi)^2 - x) + (x^3 - x^2)^(2/3))/x^2) - x + (x^3 - x^2)^(1/3))/(x*cos(2/9*pi)*sin(2/9*pi))) +
1/6*2^(2/3)*(-1)^(1/3)*log(-(2^(1/3)*(-1)^(2/3)*x - (x^3 - x^2)^(1/3))/x) - 1/12*2^(2/3)*(-1)^(1/3)*log(-(2^(
2/3)*(-1)^(1/3)*x^2 - 2^(1/3)*(-1)^(2/3)*(x^3 - x^2)^(1/3)*x - (x^3 - x^2)^(2/3))/x^2) - 1/6*(sqrt(3)*sin(2/9*
pi) + cos(2/9*pi))*log(64*(x^2 - (2*sqrt(3)*x*cos(2/9*pi)*sin(2/9*pi) + 2*x*cos(2/9*pi)^2 - x)*(x^3 - x^2)^(1/
3) + (x^3 - x^2)^(2/3))/x^2) + 1/3*cos(2/9*pi)*log(16*(x^2 + (2*sqrt(3)*x*cos(2/9*pi)*sin(2/9*pi) - 2*x*cos(2/
9*pi)^2 + x)*(x^3 - x^2)^(1/3) + (x^3 - x^2)^(2/3))/x^2) + 1/6*(sqrt(3)*sin(2/9*pi) - cos(2/9*pi))*log(64*(x^2
+ 2*(x^3 - x^2)^(1/3)*(2*x*cos(2/9*pi)^2 - x) + (x^3 - x^2)^(2/3))/x^2) - 4/3*arctan((8*(2*x*cos(2/9*pi)^3 -
x*cos(2/9*pi))*sin(2/9*pi) - sqrt(3)*x - 2*(2*sqrt(3)*x*cos(2/9*pi)^2 - 2*x*cos(2/9*pi)*sin(2/9*pi) - sqrt(3)*
x)*sqrt((x^2 + (2*sqrt(3)*x*cos(2/9*pi)*sin(2/9*pi) - 2*x*cos(2/9*pi)^2 + x)*(x^3 - x^2)^(1/3) + (x^3 - x^2)^(
2/3))/x^2) + 2*(x^3 - x^2)^(1/3)*(2*sqrt(3)*cos(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi) - sqrt(3)))/(16*x*cos(2/
9*pi)^4 - 16*x*cos(2/9*pi)^2 + 3*x))*sin(2/9*pi)
________________________________________________________________________________________
giac [B] time = 66.77, size = 967, normalized size = 5.37
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3+1)/(x^3-x^2)^(1/3),x, algorithm="giac")
[Out]
-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x + 1)^(1/3))) + 1/3*(sqrt(3)*cos(4/9*pi)^5 -
10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi) +
10*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqrt(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)
*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi
))) + 1/3*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^
4 - 5*cos(2/9*pi)^4*sin(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2/9*pi)^2 - sqr
t(3)*sin(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(2/9*pi) + 2*(-1/x + 1)^(1/3))
/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) - 1/3*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 +
5*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1/9*pi)^2*sin(1/9*pi)^3 + sin(1/9*p
i)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1
)*cos(1/9*pi) - 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) + 1/6*(5*sqrt(3)*cos(4/9*pi)^4*sin(4/
9*pi) - 10*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*pi)^3*sin(
4/9*pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos(4/9*pi)^2 - sin(4/9*pi)^2)*l
og((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3) + 1) + 1/6*(5*sqrt(3)*cos(2/9*pi
)^4*sin(2/9*pi) - 10*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*pi)^5 + cos(2/9*pi)^5 - 10*cos(2/9*
pi)^3*sin(2/9*pi)^2 + 5*cos(2/9*pi)*sin(2/9*pi)^4 + 2*sqrt(3)*cos(2/9*pi)*sin(2/9*pi) + cos(2/9*pi)^2 - sin(2/
9*pi)^2)*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3) + 1) + 1/6*(5*sqrt(3)*
cos(1/9*pi)^4*sin(1/9*pi) - 10*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + sqrt(3)*sin(1/9*pi)^5 - cos(1/9*pi)^5 + 1
0*cos(1/9*pi)^3*sin(1/9*pi)^2 - 5*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) + cos(1/9*pi)^
2 - sin(1/9*pi)^2)*log((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi))*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3) + 1) + 1/12*2
^(2/3)*log(2^(2/3) + 2^(1/3)*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3)) - 1/6*2^(2/3)*log(abs(-2^(1/3) + (-1/x + 1)^
(1/3)))
________________________________________________________________________________________
maple [B] time = 18.88, size = 3979, normalized size = 22.11
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(3979\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^3+1)/(x^3-x^2)^(1/3),x,method=_RETURNVERBOSE)
[Out]
-1/6*ln((6*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^3*x^2+27*RootOf(RootOf(_Z^3+4)^2
+3*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^2*x^2-12*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*Roo
tOf(_Z^3+4)^3*x-54*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^2*x-72*(x^3-x^2)^(2/3)
*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)+18*(x^3-x^2)^(1/3)*RootOf(_Z^3+4)^2*x+14
4*(x^3-x^2)^(1/3)*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*x-52*RootOf(_Z^3+4)*x^2-2
34*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x^2+20*RootOf(_Z^3+4)*x+90*RootOf(RootOf(_Z^3+4)^2+3*_Z
*RootOf(_Z^3+4)+9*_Z^2)*x+36*(x^3-x^2)^(2/3))/x/(1+x))*RootOf(_Z^3+4)-1/2*ln((6*RootOf(RootOf(_Z^3+4)^2+3*_Z*R
ootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^3*x^2+27*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^
3+4)^2*x^2-12*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^3*x-54*RootOf(RootOf(_Z^3+4)^
2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^2*x-72*(x^3-x^2)^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2
+3*_Z*RootOf(_Z^3+4)+9*_Z^2)+18*(x^3-x^2)^(1/3)*RootOf(_Z^3+4)^2*x+144*(x^3-x^2)^(1/3)*RootOf(RootOf(_Z^3+4)^2
+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*x-52*RootOf(_Z^3+4)*x^2-234*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z
^3+4)+9*_Z^2)*x^2+20*RootOf(_Z^3+4)*x+90*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x+36*(x^3-x^2)^(2
/3))/x/(1+x))*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)+1/2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3
+4)+9*_Z^2)*ln((3*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^3*x^2+27*RootOf(RootOf(_Z
^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^2*x^2-6*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^
2)*RootOf(_Z^3+4)^3*x-54*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^2*x+72*(x^3-x^2)
^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)-30*(x^3-x^2)^(1/3)*RootOf(_Z^3+4)^
2*x-144*(x^3-x^2)^(1/3)*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)*x+22*RootOf(_Z^3+4)
*x^2+198*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x^2-2*RootOf(_Z^3+4)*x-18*RootOf(RootOf(_Z^3+4)^2
+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x-60*(x^3-x^2)^(2/3))/x/(1+x))+1/6*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_
Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*ln((9*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf
(_Z^3+4)+9*_Z^2))^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*(x^3-x^2)^(1/3)*x+24*
RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*RootOf(RootOf(_Z^3+4)^2+3*
_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*x^2+36*(x^3-x^2)^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_
Z*RootOf(_Z^3+4)+9*_Z^2)-12*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)
)*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*x-12*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*Roo
tOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*(x^3-x^2)^(1/3)*x-16*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf
(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*x^2+8*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*
_Z*RootOf(_Z^3+4)+9*_Z^2))*x)/(3*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x-6*Root
Of(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)-8*x+4)/x)+1/8*ln((3*RootOf(_Z^3+6*RootOf(_Z^3
+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)
^2*RootOf(_Z^3+4)^4*x^2-6*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*
RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^4*x+6*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*Root
Of(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(
_Z^3+4)^2*(x^3-x^2)^(1/3)*x+4*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^
2))*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*x^2+24*(x^3-x^2)^(2/3)*RootOf(_Z^3+4)
^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)+4*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^
2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*x-8*RootOf
(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*(x^3-x^2)^(1/3)*x)/(3*RootOf(_
Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x-6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z
*RootOf(_Z^3+4)+9*_Z^2)-8*x+4)/x)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(
_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))-1/6*ln((3*RootOf(_Z^3+6*RootOf(_Z
^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^
2)^2*RootOf(_Z^3+4)^4*x^2-6*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)
)*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)^2*RootOf(_Z^3+4)^4*x+6*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*Ro
otOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootO
f(_Z^3+4)^2*(x^3-x^2)^(1/3)*x+4*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_
Z^2))*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*x^2+24*(x^3-x^2)^(2/3)*RootOf(_Z^3+
4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)+4*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4
)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*x-8*Root
Of(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*(x^3-x^2)^(1/3)*x)/(3*RootOf
(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x-6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*
_Z*RootOf(_Z^3+4)+9*_Z^2)-8*x+4)/x)*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)
+9*_Z^2))-1/16*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*RootOf(_Z
^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*ln(-(3*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootO
f(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^
2*x^2-6*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*RootOf(RootOf(_Z
^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*x-8*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^
2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*x^2+24*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^
3+4)+9*_Z^2))*(x^3-x^2)^(1/3)*x+4*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9
*_Z^2))^2*x-48*(x^3-x^2)^(2/3))/(3*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x-6*Ro
otOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)+4*x+4)/x)+1/12*RootOf(_Z^3+6*RootOf(_Z^3+4)
^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*ln(-(6*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(
_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*
x^2-9*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*RootOf(RootOf(_Z^3+4
)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4)^2*(x^3-x^2)^(1/3)*x-3*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf(Roo
tOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*RootOf(_Z^3+4
)^2*x+18*(x^3-x^2)^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)-4*RootOf(_Z^3+6*
RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*x^2+12*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*
RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))*(x^3-x^2)^(1/3)*x+2*RootOf(_Z^3+6*RootOf(_Z^3+4)^2*RootOf
(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2))^2*x-24*(x^3-x^2)^(2/3))/(3*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+
4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)*x-6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+3*_Z*RootOf(_Z^3+4)+9*_Z^2)+4*x+
4)/x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3+1)/(x^3-x^2)^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((x^3 - x^2)^(1/3)*(x^3 + 1)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x^3+1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((x^3 + 1)*(x^3 - x^2)^(1/3)),x)
[Out]
int(1/((x^3 + 1)*(x^3 - x^2)^(1/3)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x^{2} \left (x - 1\right )} \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**3+1)/(x**3-x**2)**(1/3),x)
[Out]
Integral(1/((x**2*(x - 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)
________________________________________________________________________________________