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

Optimal. Leaf size=123 \[ \frac{\log \left (1-\frac{\sqrt [3]{3} (x+1)}{\sqrt [3]{(x+1)^3+2}}\right )}{3 \sqrt [3]{3}}-\frac{\log \left (\frac{3^{2/3} (x+1)^2}{\left ((x+1)^3+2\right )^{2/3}}+\frac{\sqrt [3]{3} (x+1)}{\sqrt [3]{(x+1)^3+2}}+1\right )}{6 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{2 (x+1)}{\sqrt [6]{3} \sqrt [3]{(x+1)^3+2}}+\frac{1}{\sqrt{3}}\right )}{3^{5/6}} \]

[Out]

-(ArcTan[1/Sqrt[3] + (2*(1 + x))/(3^(1/6)*(2 + (1 + x)^3)^(1/3))]/3^(5/6)) + Log
[1 - (3^(1/3)*(1 + x))/(2 + (1 + x)^3)^(1/3)]/(3*3^(1/3)) - Log[1 + (3^(2/3)*(1
+ x)^2)/(2 + (1 + x)^3)^(2/3) + (3^(1/3)*(1 + x))/(2 + (1 + x)^3)^(1/3)]/(6*3^(1
/3))

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Rubi [A]  time = 0.261326, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29 \[ \frac{\log \left (1-\frac{\sqrt [3]{3} (x+1)}{\sqrt [3]{(x+1)^3+2}}\right )}{3 \sqrt [3]{3}}-\frac{\log \left (\frac{3^{2/3} (x+1)^2}{\left ((x+1)^3+2\right )^{2/3}}+\frac{\sqrt [3]{3} (x+1)}{\sqrt [3]{(x+1)^3+2}}+1\right )}{6 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{2 (x+1)}{\sqrt [6]{3} \sqrt [3]{(x+1)^3+2}}+\frac{1}{\sqrt{3}}\right )}{3^{5/6}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(3 + 3*x + x^2)*(3 + 3*x + 3*x^2 + x^3)^(1/3)),x]

[Out]

-(ArcTan[1/Sqrt[3] + (2*(1 + x))/(3^(1/6)*(2 + (1 + x)^3)^(1/3))]/3^(5/6)) + Log
[1 - (3^(1/3)*(1 + x))/(2 + (1 + x)^3)^(1/3)]/(3*3^(1/3)) - Log[1 + (3^(2/3)*(1
+ x)^2)/(2 + (1 + x)^3)^(2/3) + (3^(1/3)*(1 + x))/(2 + (1 + x)^3)^(1/3)]/(6*3^(1
/3))

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Rubi in Sympy [A]  time = 16.1275, size = 116, normalized size = 0.94 \[ \frac{3^{\frac{2}{3}} \log{\left (- \frac{\sqrt [3]{3} \left (x + 1\right )}{\sqrt [3]{\left (x + 1\right )^{3} + 2}} + 1 \right )}}{9} - \frac{3^{\frac{2}{3}} \log{\left (\frac{3^{\frac{2}{3}} \left (x + 1\right )^{2}}{\left (\left (x + 1\right )^{3} + 2\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{3} \left (x + 1\right )}{\sqrt [3]{\left (x + 1\right )^{3} + 2}} + 1 \right )}}{18} - \frac{\sqrt [6]{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{3} \left (x + 1\right )}{3 \sqrt [3]{\left (x + 1\right )^{3} + 2}} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(x**2+3*x+3)/(x**3+3*x**2+3*x+3)**(1/3),x)

[Out]

3**(2/3)*log(-3**(1/3)*(x + 1)/((x + 1)**3 + 2)**(1/3) + 1)/9 - 3**(2/3)*log(3**
(2/3)*(x + 1)**2/((x + 1)**3 + 2)**(2/3) + 3**(1/3)*(x + 1)/((x + 1)**3 + 2)**(1
/3) + 1)/18 - 3**(1/6)*atan(sqrt(3)*(2*3**(1/3)*(x + 1)/(3*((x + 1)**3 + 2)**(1/
3)) + 1/3))/3

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Mathematica [A]  time = 0.0800684, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (3+3 x+x^2\right ) \sqrt [3]{3+3 x+3 x^2+x^3}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/(x*(3 + 3*x + x^2)*(3 + 3*x + 3*x^2 + x^3)^(1/3)),x]

[Out]

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

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Maple [F]  time = 0.217, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ({x}^{2}+3\,x+3 \right ) }{\frac{1}{\sqrt [3]{{x}^{3}+3\,{x}^{2}+3\,x+3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^2+3*x+3)/(x^3+3*x^2+3*x+3)^(1/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{1}{3}}{\left (x^{2} + 3 \, x + 3\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 3*x^2 + 3*x + 3)^(1/3)*(x^2 + 3*x + 3)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 7.25573, size = 489, normalized size = 3.98 \[ \frac{1}{54} \cdot 3^{\frac{1}{6}}{\left (2 \, \sqrt{3} \log \left (\frac{3 \cdot 3^{\frac{2}{3}}{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{2}{3}}{\left (x + 1\right )} + 2 \cdot 3^{\frac{1}{3}}{\left (x^{3} + 3 \, x^{2} + 3 \, x\right )} - 9 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{1}{3}}{\left (x^{2} + 2 \, x + 1\right )}}{x^{3} + 3 \, x^{2} + 3 \, x}\right ) - \sqrt{3} \log \left (\frac{3^{\frac{2}{3}}{\left (31 \, x^{6} + 186 \, x^{5} + 465 \, x^{4} + 666 \, x^{3} + 603 \, x^{2} + 324 \, x + 81\right )} + 9 \cdot 3^{\frac{1}{3}}{\left (5 \, x^{5} + 25 \, x^{4} + 50 \, x^{3} + 54 \, x^{2} + 33 \, x + 9\right )}{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{1}{3}} + 9 \,{\left (7 \, x^{4} + 28 \, x^{3} + 42 \, x^{2} + 30 \, x + 9\right )}{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{2}{3}}}{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 18 \, x^{3} + 9 \, x^{2}}\right ) + 6 \, \arctan \left (-\frac{2 \cdot 3^{\frac{5}{6}}{\left (x^{3} + 3 \, x^{2} + 3 \, x\right )} - 9 \, \sqrt{3}{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{1}{3}}{\left (x^{2} + 2 \, x + 1\right )} - 18 \cdot 3^{\frac{1}{6}}{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{2}{3}}{\left (x + 1\right )}}{3 \,{\left (2 \cdot 3^{\frac{1}{3}}{\left (x^{3} + 3 \, x^{2} + 3 \, x\right )} + 9 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{1}{3}}{\left (x^{2} + 2 \, x + 1\right )}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 3*x^2 + 3*x + 3)^(1/3)*(x^2 + 3*x + 3)*x),x, algorithm="fricas")

[Out]

1/54*3^(1/6)*(2*sqrt(3)*log((3*3^(2/3)*(x^3 + 3*x^2 + 3*x + 3)^(2/3)*(x + 1) + 2
*3^(1/3)*(x^3 + 3*x^2 + 3*x) - 9*(x^3 + 3*x^2 + 3*x + 3)^(1/3)*(x^2 + 2*x + 1))/
(x^3 + 3*x^2 + 3*x)) - sqrt(3)*log((3^(2/3)*(31*x^6 + 186*x^5 + 465*x^4 + 666*x^
3 + 603*x^2 + 324*x + 81) + 9*3^(1/3)*(5*x^5 + 25*x^4 + 50*x^3 + 54*x^2 + 33*x +
9)*(x^3 + 3*x^2 + 3*x + 3)^(1/3) + 9*(7*x^4 + 28*x^3 + 42*x^2 + 30*x + 9)*(x^3
+ 3*x^2 + 3*x + 3)^(2/3))/(x^6 + 6*x^5 + 15*x^4 + 18*x^3 + 9*x^2)) + 6*arctan(-1
/3*(2*3^(5/6)*(x^3 + 3*x^2 + 3*x) - 9*sqrt(3)*(x^3 + 3*x^2 + 3*x + 3)^(1/3)*(x^2
+ 2*x + 1) - 18*3^(1/6)*(x^3 + 3*x^2 + 3*x + 3)^(2/3)*(x + 1))/(2*3^(1/3)*(x^3
+ 3*x^2 + 3*x) + 9*(x^3 + 3*x^2 + 3*x + 3)^(1/3)*(x^2 + 2*x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**2+3*x+3)/(x**3+3*x**2+3*x+3)**(1/3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac{1}{3}}{\left (x^{2} + 3 \, x + 3\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 3*x^2 + 3*x + 3)^(1/3)*(x^2 + 3*x + 3)*x),x, algorithm="giac")

[Out]

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