3.22.44 \(\int \frac {-1+x+x^3}{(1-x+x^3) \sqrt [3]{-x^2+x^3}} \, dx\)

Optimal. Leaf size=157 \[ 2 \text {RootSum}\left [\text {$\#$1}^9-2 \text {$\#$1}^6+\text {$\#$1}^3-1\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)}{3 \text {$\#$1}^3-1}\& \right ]-\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right ) \]

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Rubi [F]  time = 0.58, 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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [F]  time = 0.47, size = 0, normalized size = 0.00 \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]

Integrate[(-1 + x + x^3)/((1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]

[Out]

Integrate[(-1 + x + x^3)/((1 - x + x^3)*(-x^2 + x^3)^(1/3)), x]

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IntegrateAlgebraic [A]  time = 0.43, size = 157, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x + x^3)/((1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^
3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 + 2*RootSum[-1 + #1^3 - 2*#1^6 + #1^9 & , (-(Log[x]*#1^2) + Log[(-x^2 + x^3)^
(1/3) - x*#1]*#1^2)/(-1 + 3*#1^3) & ]

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fricas [F(-1)]  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, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} + x - 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x + 1\right )}}\,{d x} \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)

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maple [B]  time = 196.66, size = 189720, normalized size = 1208.41

method	result	size
trager	\(\text {Expression too large to display}\)	\(189720\)

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} + x - 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x + 1\right )}}\,{d x} \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)

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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)

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sympy [F(-1)]  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

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