\(\int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx\) [1264]

Optimal result

Integrand size = 18, antiderivative size = 92 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{2+x^3}}{4+2 x+\sqrt [3]{2+x^3}}\right )-\log \left (-2-x+\sqrt [3]{2+x^3}\right )+\frac {1}{2} \log \left (4+4 x+x^2+(2+x) \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right ) \]

[Out]

-3^(1/2)*arctan(3^(1/2)*(x^3+2)^(1/3)/(4+2*x+(x^3+2)^(1/3)))-ln(-2-x+(x^3+2)^(1/3))+1/2*ln(4+4*x+x^2+(2+x)*(x^
3+2)^(1/3)+(x^3+2)^(2/3))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2176} \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\log (x+1) \]

[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*} \text {integral}& = \sqrt {3} \arctan \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*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{2+x^3}}{4+2 x+\sqrt [3]{2+x^3}}\right )-\log \left (-2-x+\sqrt [3]{2+x^3}\right )+\frac {1}{2} \log \left (4+4 x+x^2+(2+x) \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right ) \]

[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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.85 (sec) , antiderivative size = 543, normalized size of antiderivative = 5.90

method	result	size
trager	\(-\ln \left (-\frac {787 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+4504 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x -4839 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-1574 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-452 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+9008 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}}+335 x \left (x^{3}+2\right )^{\frac {2}{3}}-19356 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x +4504 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-3148 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +11922 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-4052 x^{3}+670 \left (x^{3}+2\right )^{\frac {2}{3}}-19356 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}+18016 x \left (x^{3}+2\right )^{\frac {1}{3}}+23844 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -20260 x^{2}+18016 \left (x^{3}+2\right )^{\frac {1}{3}}+11018 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-40520 x -28364}{\left (1+x \right )^{2}}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2026 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+4504 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +335 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-4052 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-6865 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+9008 \operatorname {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}}+1340 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x +4504 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-8104 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -14634 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2361 x^{3}-9678 \left (x^{3}+2\right )^{\frac {2}{3}}+1340 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}}+18016 x \left (x^{3}+2\right )^{\frac {1}{3}}-29268 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +6296 x^{2}+18016 \left (x^{3}+2\right )^{\frac {1}{3}}-28364 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+12592 x +11018}{\left (1+x \right )^{2}}\right )\)	\(543\)

[In]

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

[Out]

-ln(-(787*RootOf(_Z^2-_Z+1)^2*x^3+4504*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)*x-4839*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*
x^2-1574*RootOf(_Z^2-_Z+1)^2*x^2-452*RootOf(_Z^2-_Z+1)*x^3+9008*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)+335*x*(x^3+2)^
(2/3)-19356*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x+4504*(x^3+2)^(1/3)*x^2-3148*RootOf(_Z^2-_Z+1)^2*x+11922*RootOf(_
Z^2-_Z+1)*x^2-4052*x^3+670*(x^3+2)^(2/3)-19356*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)+18016*x*(x^3+2)^(1/3)+23844*Roo
tOf(_Z^2-_Z+1)*x-20260*x^2+18016*(x^3+2)^(1/3)+11018*RootOf(_Z^2-_Z+1)-40520*x-28364)/(1+x)^2)+RootOf(_Z^2-_Z+
1)*ln((2026*RootOf(_Z^2-_Z+1)^2*x^3+4504*RootOf(_Z^2-_Z+1)*(x^3+2)^(2/3)*x+335*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)
*x^2-4052*RootOf(_Z^2-_Z+1)^2*x^2-6865*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)+1340*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x+4504*(x^3+2)^(1/3)*x^2-8104*RootOf(_Z^2-_Z+1)^2*x-14634*RootOf
(_Z^2-_Z+1)*x^2+2361*x^3-9678*(x^3+2)^(2/3)+1340*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)+18016*x*(x^3+2)^(1/3)-29268*R
ootOf(_Z^2-_Z+1)*x+6296*x^2+18016*(x^3+2)^(1/3)-28364*RootOf(_Z^2-_Z+1)+12592*x+11018)/(1+x)^2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int \frac {x-1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \]

[In]

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

[Out]

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