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

Optimal. Leaf size=482 \[ \sqrt [3]{1-x^3}-\frac {1}{3} \sqrt [3]{2} \log \left (x^3+1\right )+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2^{2/3}}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

(-x^3+1)^(1/3)-1/3*2^(1/3)*ln(x^3+1)+1/6*ln(2^(2/3)+(-1+x)/(-x^3+1)^(1/3))*2^(1/3)-1/6*ln(1+2^(2/3)*(1-x)^2/(-
x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/3*2^(1/3)*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))-1/12*ln(2*2^
(1/3)+(1-x)^2/(-x^3+1)^(2/3)+2^(2/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/2*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(1/3)-1/2*
ln(-x-(-x^3+1)^(1/3))+1/2*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(1/3)+1/3*2^(1/3)*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^
3+1)^(1/3))*3^(1/2))*3^(1/2)+1/6*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)-1/3*arct
an(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+1/3*2^(1/3)*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*
3^(1/2)-1/3*2^(1/3)*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)

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Rubi [F]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - x^3)^(1/3)/(1 + x),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx &=\int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - x^3)^(1/3)/(1 + x),x]

[Out]

Integrate[(1 - x^3)^(1/3)/(1 + x), 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((-x^3+1)^(1/3)/(1+x),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 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 19.96, size = 2972, normalized size = 6.17 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^3-1)/(-x^3+1)^(2/3)+(1/2*RootOf(_Z^3-2)*ln((2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^
3-2)^2*x^3+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^3+2*RootOf(RootOf(_Z^3-2)^2+_Z
*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x^2+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^
3*x^2+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x+4*RootOf(RootOf(_Z^3-2)^2+_Z*Root
Of(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x+5*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_
Z^3-2)+_Z^2)-2*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2-7*RootOf
(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^4+8*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x^2-14*RootOf(_Z^3-2)*x^4
-2*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x-9*RootOf(RootOf(_Z^3-2
)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^3+8*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x-18*RootOf(_Z^3-2)*x^3-2*(x^6-2*x^3+1)
^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)-16*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^
3-2)+_Z^2)*x^2+8*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3-2)^2-32*RootOf(_Z^3-2)*x^2-9*RootOf(RootOf(_Z^3-2)^2+_Z*RootO
f(_Z^3-2)+_Z^2)*x-18*RootOf(_Z^3-2)*x+2*(x^6-2*x^3+1)^(2/3)-7*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-
14*RootOf(_Z^3-2))/(x+1)^2/(x^2+x+1))-1/2*ln((2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-
2)^2*x^3+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^3+2*RootOf(RootOf(_Z^3-2)^2+_Z*R
ootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x^2+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*
x^2+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf
(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x-5*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^
3-2)+_Z^2)+8*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+7*RootOf(R
ootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^4-2*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x^2+14*RootOf(_Z^3-2)*x^4+8
*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x+13*RootOf(RootOf(_Z^3-2)
^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^3-2*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x+26*RootOf(_Z^3-2)*x^3+8*(x^6-2*x^3+1)^
(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)+20*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3
-2)+_Z^2)*x^2-2*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3-2)^2+40*RootOf(_Z^3-2)*x^2+13*RootOf(RootOf(_Z^3-2)^2+_Z*RootO
f(_Z^3-2)+_Z^2)*x+26*RootOf(_Z^3-2)*x-8*(x^6-2*x^3+1)^(2/3)+7*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)+
14*RootOf(_Z^3-2))/(x+1)^2/(x^2+x+1))*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-1/2*ln((2*RootOf(RootOf(
_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x^3+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*Roo
tOf(_Z^3-2)^3*x^3+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x^2+4*RootOf(RootOf(_Z^
3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^2+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf
(_Z^3-2)^2*x+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x-5*(x^6-2*x^3+1)^(2/3)*RootOf
(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)+8*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*Ro
otOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*x^2+7*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^4-2*(x^6-2*x^3+1)^(1
/3)*RootOf(_Z^3-2)^2*x^2+14*RootOf(_Z^3-2)*x^4+8*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)
+_Z^2)*RootOf(_Z^3-2)*x+13*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^3-2*(x^6-2*x^3+1)^(1/3)*RootOf(_Z
^3-2)^2*x+26*RootOf(_Z^3-2)*x^3+8*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_
Z^3-2)+20*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^2-2*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3-2)^2+40*RootOf
(_Z^3-2)*x^2+13*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x+26*RootOf(_Z^3-2)*x-8*(x^6-2*x^3+1)^(2/3)+7*
RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)+14*RootOf(_Z^3-2))/(x+1)^2/(x^2+x+1))*RootOf(_Z^3-2)-1/3*ln((R
ootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^6-RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2
+_Z*RootOf(_Z^3-2)+_Z^2)^2*x^3+8*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^6-6*(x^6-2
*x^3+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^2-10*RootOf(RootOf(_Z^3-2)^2+
_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^3+16*x^6-12*(x^6-2*x^3+1)^(1/3)*x^4+2*RootOf(_Z^3-2)^2*RootOf(RootO
f(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-24*x^3+12*(x^6-2*x^3+1)^(1/3)*x+8)/(x-1)/(x^2+x+1))+1/3*ln((RootOf(RootOf(
_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^6-RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z
^3-2)+_Z^2)^2*x^3+2*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^6-6*(x^6-2*x^3+1)^(1/3)
*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^4+6*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3-2)^2*R
ootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^2-8*x^6+6*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*Root
Of(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x+12*(x^6-2*x^3+1)^(2/3)*x^2-2*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*R
ootOf(_Z^3-2)+_Z^2)+16*x^3-8)/(x-1)/(x^2+x+1))+1/6*ln((RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootO
f(_Z^3-2)^4*x^6-RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*x^3+2*RootOf(_Z^3-2)^2*Root
Of(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^6-6*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2
)+_Z^2)*RootOf(_Z^3-2)^2*x^4+6*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+
_Z^2)*x^2-8*x^6+6*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x+12*(x
^6-2*x^3+1)^(2/3)*x^2-2*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)+16*x^3-8)/(x-1)/(x^2+
x+1))*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2))/(-x^3+1)^(2/3)*((x^3-1)^2)^(1/3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^3)^(1/3)/(x + 1),x)

[Out]

int((1 - x^3)^(1/3)/(x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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