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

Optimal. Leaf size=92 \[ -\log \left (\sqrt [3]{x^3+2}-x-2\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^3+2}}{\sqrt [3]{x^3+2}+2 x+4}\right )+\frac {1}{2} \log \left (\left (x^3+2\right )^{2/3}+(x+2) \sqrt [3]{x^3+2}+x^2+4 x+4\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 53, normalized size of antiderivative = 0.58, 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} \begin {gather*} -\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) \end {gather*}

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 = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.84, size = 92, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{2+x^3}}{4+2 x+\sqrt [3]{2+x^3}}\right )-\log \left (-2-x+\sqrt [3]{2+x^3}\right )+\frac {1}{2} \log \left (4+4 x+x^2+(2+x) \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3]*ArcTan[(Sqrt[3]*(2 + x^3)^(1/3))/(4 + 2*x + (2 + x^3)^(1/3))]) - Log[-2 - x + (2 + x^3)^(1/3)] + Log
[4 + 4*x + x^2 + (2 + x)*(2 + x^3)^(1/3) + (2 + x^3)^(2/3)]/2

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

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 \begin {gather*} \int \frac {x - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \end {gather*}

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 = 3.56, size = 816, normalized size = 8.87

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {23 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+80 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +80 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-46 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+57 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+160 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}}-x \left (x^{3}+2\right )^{\frac {2}{3}}+320 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x -\left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-92 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +208 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+22 x^{3}-2 \left (x^{3}+2\right )^{\frac {2}{3}}+320 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}-4 x \left (x^{3}+2\right )^{\frac {1}{3}}+416 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +110 x^{2}-4 \left (x^{3}+2\right )^{\frac {1}{3}}+322 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+220 x +154}{\left (1+x \right )^{2}}\right )-\ln \left (\frac {-1239 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+4504 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +4504 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}+2478 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+5743 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+9008 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}}-4839 x \left (x^{3}+2\right )^{\frac {2}{3}}+18016 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x -4839 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}+4956 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +5860 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-6078 x^{3}-9678 \left (x^{3}+2\right )^{\frac {2}{3}}+18016 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}-19356 x \left (x^{3}+2\right )^{\frac {1}{3}}+11720 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -16208 x^{2}-19356 \left (x^{3}+2\right )^{\frac {1}{3}}+17346 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-32416 x -28364}{\left (1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {-1239 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+4504 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +4504 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}+2478 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+5743 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+9008 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}}-4839 x \left (x^{3}+2\right )^{\frac {2}{3}}+18016 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x -4839 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}+4956 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +5860 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-6078 x^{3}-9678 \left (x^{3}+2\right )^{\frac {2}{3}}+18016 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}-19356 x \left (x^{3}+2\right )^{\frac {1}{3}}+11720 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -16208 x^{2}-19356 \left (x^{3}+2\right )^{\frac {1}{3}}+17346 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-32416 x -28364}{\left (1+x \right )^{2}}\right )\)	\(816\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(_Z^2-_Z+1)*ln(-(23*RootOf(_Z^2-_Z+1)^2*x^3+80*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)*x+80*RootOf(_Z^2-_Z+1)*(x
^3+2)^(1/3)*x^2-46*RootOf(_Z^2-_Z+1)^2*x^2+57*RootOf(_Z^2-_Z+1)*x^3+160*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)-x*(x^3
+2)^(2/3)+320*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x-(x^3+2)^(1/3)*x^2-92*RootOf(_Z^2-_Z+1)^2*x+208*RootOf(_Z^2-_Z+
1)*x^2+22*x^3-2*(x^3+2)^(2/3)+320*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)-4*x*(x^3+2)^(1/3)+416*RootOf(_Z^2-_Z+1)*x+11
0*x^2-4*(x^3+2)^(1/3)+322*RootOf(_Z^2-_Z+1)+220*x+154)/(1+x)^2)-ln((-1239*RootOf(_Z^2-_Z+1)^2*x^3+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+2478*RootOf(_Z^2-_Z+1)^2*x^2+5743*RootOf(_
Z^2-_Z+1)*x^3+9008*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)-4839*x*(x^3+2)^(2/3)+18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*
x-4839*(x^3+2)^(1/3)*x^2+4956*RootOf(_Z^2-_Z+1)^2*x+5860*RootOf(_Z^2-_Z+1)*x^2-6078*x^3-9678*(x^3+2)^(2/3)+180
16*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)-19356*x*(x^3+2)^(1/3)+11720*RootOf(_Z^2-_Z+1)*x-16208*x^2-19356*(x^3+2)^(1/
3)+17346*RootOf(_Z^2-_Z+1)-32416*x-28364)/(1+x)^2)*RootOf(_Z^2-_Z+1)+ln((-1239*RootOf(_Z^2-_Z+1)^2*x^3+4504*Ro
otOf(_Z^2-_Z+1)*(x^3+2)^(2/3)*x+4504*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x^2+2478*RootOf(_Z^2-_Z+1)^2*x^2+5743*Roo
tOf(_Z^2-_Z+1)*x^3+9008*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)-4839*x*(x^3+2)^(2/3)+18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(
1/3)*x-4839*(x^3+2)^(1/3)*x^2+4956*RootOf(_Z^2-_Z+1)^2*x+5860*RootOf(_Z^2-_Z+1)*x^2-6078*x^3-9678*(x^3+2)^(2/3
)+18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)-19356*x*(x^3+2)^(1/3)+11720*RootOf(_Z^2-_Z+1)*x-16208*x^2-19356*(x^3+2
)^(1/3)+17346*RootOf(_Z^2-_Z+1)-32416*x-28364)/(1+x)^2)

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

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 = -0.01 \begin {gather*} \int \frac {x-1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \end {gather*}

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

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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