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

Optimal. Leaf size=53 \[ -\frac {3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )+\log (x+1) \]

[Out]

ln(1+x)-3/2*ln(2+x-(x^3+2)^(1/3))+arctan(1/3*(1+2*(2+x)/(x^3+2)^(1/3))*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2151} \[ -\frac {3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )+\log (x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[3]*ArcTan[(1 + (2*(2 + x))/(2 + x^3)^(1/3))/Sqrt[3]] + Log[1 + x] - (3*Log[2 + x - (2 + x^3)^(1/3)])/2

Rule 2151

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*f*ArcTan
[(1 + (2*Rt[b, 3]*(2*c + d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Rt[b, 3]*d), x] + (Simp[(f*Log[c + d*x])/(Rt[
b, 3]*d), x] - Simp[(3*f*Log[Rt[b, 3]*(2*c + d*x) - d*(a + b*x^3)^(1/3)])/(2*Rt[b, 3]*d), x]) /; FreeQ[{a, b,
c, d, e, f}, x] && EqQ[d*e + c*f, 0] && EqQ[2*b*c^3 - a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx &=\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )+\log (1+x)-\frac {3}{2} \log \left (2+x-\sqrt [3]{2+x^3}\right )\\ \end {align*}

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(1+x)/(x^3+2)^(1/3),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(1+x)/(x^3+2)^(1/3),x, algorithm="giac")

[Out]

integrate((x - 1)/((x^3 + 2)^(1/3)*(x + 1)), x)

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maple [C]  time = 3.00, size = 818, normalized size = 15.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-1)/(x+1)/(x^3+2)^(1/3),x)

[Out]

RootOf(_Z^2-_Z+1)*ln(-(1239*RootOf(_Z^2-_Z+1)^2*x^3-2478*RootOf(_Z^2-_Z+1)^2*x^2+4504*RootOf(_Z^2-_Z+1)*(x^3+2
)^(2/3)*x+4504*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x^2+3265*RootOf(_Z^2-_Z+1)*x^3-4956*RootOf(_Z^2-_Z+1)^2*x+9008*
RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)+18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x+10816*RootOf(_Z^2-_Z+1)*x^2+335*(x^3+2
)^(2/3)*x+335*(x^3+2)^(1/3)*x^2+1574*x^3+18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)+21632*RootOf(_Z^2-_Z+1)*x+670*(
x^3+2)^(2/3)+1340*(x^3+2)^(1/3)*x+7870*x^2+17346*RootOf(_Z^2-_Z+1)+1340*(x^3+2)^(1/3)+15740*x+11018)/(x+1)^2)-
ln(-(1239*RootOf(_Z^2-_Z+1)^2*x^3-2478*RootOf(_Z^2-_Z+1)^2*x^2-4504*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)*x-4504*Roo
tOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x^2-5743*RootOf(_Z^2-_Z+1)*x^3-4956*RootOf(_Z^2-_Z+1)^2*x-9008*RootOf(_Z^2-_Z+1)*
(x^3+2)^(2/3)-18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x-5860*RootOf(_Z^2-_Z+1)*x^2+4839*(x^3+2)^(2/3)*x+4839*(x^
3+2)^(1/3)*x^2+6078*x^3-18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)-11720*RootOf(_Z^2-_Z+1)*x+9678*(x^3+2)^(2/3)+193
56*(x^3+2)^(1/3)*x+16208*x^2-17346*RootOf(_Z^2-_Z+1)+19356*(x^3+2)^(1/3)+32416*x+28364)/(x+1)^2)*RootOf(_Z^2-_
Z+1)+ln(-(1239*RootOf(_Z^2-_Z+1)^2*x^3-2478*RootOf(_Z^2-_Z+1)^2*x^2-4504*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)*x-450
4*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x^2-5743*RootOf(_Z^2-_Z+1)*x^3-4956*RootOf(_Z^2-_Z+1)^2*x-9008*RootOf(_Z^2-_
Z+1)*(x^3+2)^(2/3)-18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x-5860*RootOf(_Z^2-_Z+1)*x^2+4839*(x^3+2)^(2/3)*x+483
9*(x^3+2)^(1/3)*x^2+6078*x^3-18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)-11720*RootOf(_Z^2-_Z+1)*x+9678*(x^3+2)^(2/3
)+19356*(x^3+2)^(1/3)*x+16208*x^2-17346*RootOf(_Z^2-_Z+1)+19356*(x^3+2)^(1/3)+32416*x+28364)/(x+1)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(1+x)/(x^3+2)^(1/3),x, algorithm="maxima")

[Out]

integrate((x - 1)/((x^3 + 2)^(1/3)*(x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x-1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/((x^3 + 2)^(1/3)*(x + 1)),x)

[Out]

int((x - 1)/((x^3 + 2)^(1/3)*(x + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x - 1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(1+x)/(x**3+2)**(1/3),x)

[Out]

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

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