3.22.45 \(\int \frac {-1+x+x^3}{(1-x+x^3) \sqrt [3]{-x^2+x^3}} \, dx\) [2145]
Optimal. Leaf size=157 \[ \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )+2 \text {RootSum}\left [-1+\text {$\#$1}^3-2 \text {$\#$1}^6+\text {$\#$1}^9\& ,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+3 \text {$\#$1}^3}\& \right ] \]
[Out]
Unintegrable
________________________________________________________________________________________
Rubi [F]
time = 0.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-1+x+x^3}{\left (1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-1 + x + x^3)/((1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
(Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[(1 + (2*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(-x^2 + x^3)^(1/3) - (3*(-1
+ x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - x^(1/3)])/(2*(-x^2 + x^3)^(1/3)) + (6*(-1 + x)^(1/3)*x^(2/3)*Defer[Sub
st][Defer[Int][(-1 + x^3)^(2/3)/(1 - x^3 + x^9), x], x, x^(1/3)])/(-x^2 + x^3)^(1/3)
Rubi steps
\begin {align*} \int \frac {-1+x+x^3}{\left (1-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {-1+x+x^3}{\sqrt [3]{-1+x} x^{2/3} \left (1-x+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-1+x^3+x^9}{\sqrt [3]{-1+x^3} \left (1-x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{-1+x^3}}-\frac {2 \left (1-x^3\right )}{\sqrt [3]{-1+x^3} \left (1-x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {\left (6 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1-x^3}{\sqrt [3]{-1+x^3} \left (1-x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^3}}+\frac {\left (6 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {\left (-1+x^3\right )^{2/3}}{1-x^3+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 177, normalized size = 1.13 \begin {gather*} \frac {\sqrt [3]{-1+x} x^{2/3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )+4 \text {RootSum}\left [-1+\text {$\#$1}^3-2 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right ) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^2}{-1+3 \text {$\#$1}^3}\&\right ]\right )}{2 \sqrt [3]{(-1+x) x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-1 + x + x^3)/((1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
((-1 + x)^(1/3)*x^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(2*(-1 + x)^(1/3) + x^(1/3))] - 2*Log[(-1 + x)^(1/
3) - x^(1/3)] + Log[(-1 + x)^(2/3) + (-1 + x)^(1/3)*x^(1/3) + x^(2/3)] + 4*RootSum[-1 + #1^3 - 2*#1^6 + #1^9 &
, (-(Log[x^(1/3)]*#1^2) + Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^2)/(-1 + 3*#1^3) & ]))/(2*((-1 + x)*x^2)^(1/3))
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 111.77, size = 222275, normalized size = 1415.76
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(222275\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^3+x-1)/(x^3-x+1)/(x^3-x^2)^(1/3),x,method=_RETURNVERBOSE)
[Out]
result too large to display
________________________________________________________________________________________
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^3+x-1)/(x^3-x+1)/(x^3-x^2)^(1/3),x, algorithm="maxima")
[Out]
integrate((x^3 + x - 1)/((x^3 - x^2)^(1/3)*(x^3 - x + 1)), x)
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 1.28, size = 6525, normalized size = 41.56 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3+x-1)/(x^3-x+1)/(x^3-x^2)^(1/3),x, algorithm="fricas")
[Out]
1/529*1058^(2/3)*sqrt(3)*(sqrt(23)*sqrt(-1472/3*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)
/(129*sqrt(69) + 1265)^(1/3) - 1)^2 - 2944/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/
3)/(129*sqrt(69) + 1265)^(1/3) + 18880/3) - 184/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 71944/3*(4/529)^
(2/3)/(129*sqrt(69) + 1265)^(1/3) - 368/3)^(1/3)*arctan(1/7463892676608*(sqrt(43)*sqrt(2)*(2024*1058^(2/3)*sqr
t(3)*x*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1)^2 - 110
4*1058^(2/3)*sqrt(3)*x*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1
/3) - 1) - 81144*1058^(2/3)*sqrt(3)*x + 3*(11*1058^(2/3)*sqrt(23)*sqrt(3)*x*((4/529)^(1/3)*(129*sqrt(69) + 126
5)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1) + 39*1058^(2/3)*sqrt(23)*sqrt(3)*x)*sqrt(-1472/3
*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1)^2 - 2944/3*(4
/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) + 18880/3))*(sqr
t(23)*sqrt(-1472/3*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3)
- 1)^2 - 2944/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3
) + 18880/3) - 184/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 71944/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(
1/3) - 368/3)^(1/3)*sqrt(-((14536*1058^(1/3)*(x^3 - x^2)^(1/3)*x*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) +
391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1)^2 + 30912*1058^(1/3)*(x^3 - x^2)^(1/3)*x*((4/529)^(1/3)*(12
9*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1) - 77832*1058^(1/3)*(x^3 - x^2)^(
1/3)*x + 3*(79*1058^(1/3)*sqrt(23)*(x^3 - x^2)^(1/3)*x*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529
)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1) + 69*1058^(1/3)*sqrt(23)*(x^3 - x^2)^(1/3)*x)*sqrt(-1472/3*((4/529)^(
1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1)^2 - 2944/3*(4/529)^(1/3)
*(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) + 18880/3))*(sqrt(23)*sqrt(
-1472/3*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1)^2 - 29
44/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) + 18880/3
) - 184/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 71944/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 368/
3)^(2/3) + 12*(920*1058^(2/3)*x^2*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69)
+ 1265)^(1/3) - 1)^2 + 1656*1058^(2/3)*x^2*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(12
9*sqrt(69) + 1265)^(1/3) - 1) - 8832*1058^(2/3)*x^2 + 3*(5*1058^(2/3)*sqrt(23)*x^2*((4/529)^(1/3)*(129*sqrt(69
) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1) + 6*1058^(2/3)*sqrt(23)*x^2)*sqrt(-1472/3
*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1)^2 - 2944/3*(4
/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) + 18880/3))*(sqr
t(23)*sqrt(-1472/3*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3)
- 1)^2 - 2944/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3
) + 18880/3) - 184/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 71944/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(
1/3) - 368/3)^(1/3) - 104818176*(x^3 - x^2)^(2/3))/x^2) + 2487964225536*sqrt(3)*x - 94944*(2024*1058^(2/3)*sqr
t(3)*(x^3 - x^2)^(1/3)*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1
/3) - 1)^2 - 1104*1058^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3)*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529
)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1) - 81144*1058^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3) + 3*(11*1058^(2/3)*sqrt(
23)*sqrt(3)*(x^3 - x^2)^(1/3)*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1
265)^(1/3) - 1) + 39*1058^(2/3)*sqrt(23)*sqrt(3)*(x^3 - x^2)^(1/3))*sqrt(-1472/3*((4/529)^(1/3)*(129*sqrt(69)
+ 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1)^2 - 2944/3*(4/529)^(1/3)*(129*sqrt(69) + 12
65)^(1/3) - 1151104/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) + 18880/3))*(sqrt(23)*sqrt(-1472/3*((4/529)^(1
/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 1)^2 - 2944/3*(4/529)^(1/3)*
(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) + 18880/3) - 184/3*(4/529)^(
1/3)*(129*sqrt(69) + 1265)^(1/3) - 71944/3*(4/529)^(2/3)/(129*sqrt(69) + 1265)^(1/3) - 368/3)^(1/3))/x) - 1/52
9*1058^(2/3)*sqrt(3)*(-sqrt(23)*sqrt(-1472/3*((4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) + 391*(4/529)^(2/3)/(1
29*sqrt(69) + 1265)^(1/3) - 1)^2 - 2944/3*(4/529)^(1/3)*(129*sqrt(69) + 1265)^(1/3) - 1151104/3*(4/529)^(2/3)/
(129*sqrt(69) + 1265)^(1/3) + 18880/3) - 184/3*...
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**3+x-1)/(x**3-x+1)/(x**3-x**2)**(1/3),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3+x-1)/(x^3-x+1)/(x^3-x^2)^(1/3),x, algorithm="giac")
[Out]
integrate((x^3 + x - 1)/((x^3 - x^2)^(1/3)*(x^3 - x + 1)), x)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3+x-1}{{\left (x^3-x^2\right )}^{1/3}\,\left (x^3-x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x + x^3 - 1)/((x^3 - x^2)^(1/3)*(x^3 - x + 1)),x)
[Out]
int((x + x^3 - 1)/((x^3 - x^2)^(1/3)*(x^3 - x + 1)), x)
________________________________________________________________________________________