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

Optimal result

Integrand size = 18, antiderivative size = 53 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {1+\frac {2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )+\log (1+x)-\frac {3}{2} \log \left (2+x-\sqrt [3]{2+x^3}\right ) \]

[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)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, 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 = 0.76 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.74 \[ \int \frac {-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{2+x^3}}{4+2 x+\sqrt [3]{2+x^3}}\right )-\log \left (-2-x+\sqrt [3]{2+x^3}\right )+\frac {1}{2} \log \left (4+4 x+x^2+(2+x) \sqrt [3]{2+x^3}+\left (2+x^3\right )^{2/3}\right ) \]

[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 = 2.09 (sec) , antiderivative size = 816, normalized size of antiderivative = 15.40

method	result	size
trager	\(\text {Expression too large to display}\)	\(816\)

[In]

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

[Out]

RootOf(_Z^2-_Z+1)*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+3265*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)+18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)*x+335*(x^3+2)^(1/3)*x^2-4956*RootOf(_Z^2-_Z+1)^
2*x+10816*RootOf(_Z^2-_Z+1)*x^2+1574*x^3+670*(x^3+2)^(2/3)+18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)+1340*x*(x^3+2
)^(1/3)+21632*RootOf(_Z^2-_Z+1)*x+7870*x^2+1340*(x^3+2)^(1/3)+17346*RootOf(_Z^2-_Z+1)+15740*x+11018)/(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)+18016*RootOf(_Z^2-_Z+1)*(x^3+2)^(1/3)-19356*x*(x^3+2)^(1/3)+11720*R
ootOf(_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*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*Ro
otOf(_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)+11
720*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)

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)