3.1.92 \(\int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx\) [92]

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

[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 = 0.04, 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 = {2176} \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 2176

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 [A]
time = 0.57, size = 92, normalized size = 1.74 \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]

Integrate[(-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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.30, size = 544, normalized size = 10.26

method	result	size
trager	\(-\ln \left (-\frac {787 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+9008 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x -9678 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-1574 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-904 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+18016 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {2}{3}}+1340 x \left (x^{3}+2\right )^{\frac {2}{3}}-38712 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x +18016 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-3148 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x +23844 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-16208 x^{3}+2680 \left (x^{3}+2\right )^{\frac {2}{3}}-38712 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}}+72064 \left (x^{3}+2\right )^{\frac {1}{3}} x +47688 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -81040 x^{2}+72064 \left (x^{3}+2\right )^{\frac {1}{3}}+22036 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-162080 x -113456}{\left (1+x \right )^{2}}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {1013 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+4504 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +335 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-2026 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-6865 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+9008 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {2}{3}}-9678 x \left (x^{3}+2\right )^{\frac {2}{3}}+1340 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x +9008 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-4052 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -14634 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+4722 x^{3}-19356 \left (x^{3}+2\right )^{\frac {2}{3}}+1340 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2\right )^{\frac {1}{3}}+36032 \left (x^{3}+2\right )^{\frac {1}{3}} x -29268 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +12592 x^{2}+36032 \left (x^{3}+2\right )^{\frac {1}{3}}-28364 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+25184 x +22036}{\left (1+x \right )^{2}}\right )}{2}\)	\(544\)

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]

-ln(-(787*RootOf(_Z^2-2*_Z+4)^2*x^3+9008*RootOf(_Z^2-2*_Z+4)*(x^3+2)^(2/3)*x-9678*RootOf(_Z^2-2*_Z+4)*(x^3+2)^
(1/3)*x^2-1574*RootOf(_Z^2-2*_Z+4)^2*x^2-904*RootOf(_Z^2-2*_Z+4)*x^3+18016*RootOf(_Z^2-2*_Z+4)*(x^3+2)^(2/3)+1
340*x*(x^3+2)^(2/3)-38712*RootOf(_Z^2-2*_Z+4)*(x^3+2)^(1/3)*x+18016*(x^3+2)^(1/3)*x^2-3148*RootOf(_Z^2-2*_Z+4)
^2*x+23844*RootOf(_Z^2-2*_Z+4)*x^2-16208*x^3+2680*(x^3+2)^(2/3)-38712*RootOf(_Z^2-2*_Z+4)*(x^3+2)^(1/3)+72064*
(x^3+2)^(1/3)*x+47688*RootOf(_Z^2-2*_Z+4)*x-81040*x^2+72064*(x^3+2)^(1/3)+22036*RootOf(_Z^2-2*_Z+4)-162080*x-1
13456)/(1+x)^2)+1/2*RootOf(_Z^2-2*_Z+4)*ln((1013*RootOf(_Z^2-2*_Z+4)^2*x^3+4504*RootOf(_Z^2-2*_Z+4)*(x^3+2)^(2
/3)*x+335*RootOf(_Z^2-2*_Z+4)*(x^3+2)^(1/3)*x^2-2026*RootOf(_Z^2-2*_Z+4)^2*x^2-6865*RootOf(_Z^2-2*_Z+4)*x^3+90
08*RootOf(_Z^2-2*_Z+4)*(x^3+2)^(2/3)-9678*x*(x^3+2)^(2/3)+1340*RootOf(_Z^2-2*_Z+4)*(x^3+2)^(1/3)*x+9008*(x^3+2
)^(1/3)*x^2-4052*RootOf(_Z^2-2*_Z+4)^2*x-14634*RootOf(_Z^2-2*_Z+4)*x^2+4722*x^3-19356*(x^3+2)^(2/3)+1340*RootO
f(_Z^2-2*_Z+4)*(x^3+2)^(1/3)+36032*(x^3+2)^(1/3)*x-29268*RootOf(_Z^2-2*_Z+4)*x+12592*x^2+36032*(x^3+2)^(1/3)-2
8364*RootOf(_Z^2-2*_Z+4)+25184*x+22036)/(1+x)^2)

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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((-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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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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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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Giac [F] N/A
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((-1+x)/(1+x)/(x^3+2)^(1/3),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \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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