3.182 \(\int \frac{1}{2+x^3+x^6} \, dx\)

Optimal. Leaf size=381 \[ \frac{i \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt{7}\right )} x+\left (1-i \sqrt{7}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{7} \left (1-i \sqrt{7}\right )^{2/3}}-\frac{i \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt{7}\right )} x+\left (1+i \sqrt{7}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{7} \left (1+i \sqrt{7}\right )^{2/3}}-\frac{i \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{7}}\right )}{3 \sqrt{7} \left (\frac{1}{2} \left (1-i \sqrt{7}\right )\right )^{2/3}}+\frac{i \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{7}}\right )}{3 \sqrt{7} \left (\frac{1}{2} \left (1+i \sqrt{7}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21} \left (\frac{1}{2} \left (1-i \sqrt{7}\right )\right )^{2/3}}-\frac{i \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21} \left (\frac{1}{2} \left (1+i \sqrt{7}\right )\right )^{2/3}} \]

[Out]

(I*ArcTan[(1 - (2*x)/((1 - I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/(Sqrt[21]*((1 - I*Sqrt
[7])/2)^(2/3)) - (I*ArcTan[(1 - (2*x)/((1 + I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/(Sqrt
[21]*((1 + I*Sqrt[7])/2)^(2/3)) - ((I/3)*Log[(1 - I*Sqrt[7])^(1/3) + 2^(1/3)*x])
/(Sqrt[7]*((1 - I*Sqrt[7])/2)^(2/3)) + ((I/3)*Log[(1 + I*Sqrt[7])^(1/3) + 2^(1/3
)*x])/(Sqrt[7]*((1 + I*Sqrt[7])/2)^(2/3)) + ((I/3)*Log[(1 - I*Sqrt[7])^(2/3) - (
2*(1 - I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^2])/(2^(1/3)*Sqrt[7]*(1 - I*Sqrt[7])^(2/3
)) - ((I/3)*Log[(1 + I*Sqrt[7])^(2/3) - (2*(1 + I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^
2])/(2^(1/3)*Sqrt[7]*(1 + I*Sqrt[7])^(2/3))

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Rubi [A]  time = 0.888122, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7 \[ \frac{i \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt{7}\right )} x+\left (1-i \sqrt{7}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{7} \left (1-i \sqrt{7}\right )^{2/3}}-\frac{i \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt{7}\right )} x+\left (1+i \sqrt{7}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{7} \left (1+i \sqrt{7}\right )^{2/3}}-\frac{i \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{7}}\right )}{3 \sqrt{7} \left (\frac{1}{2} \left (1-i \sqrt{7}\right )\right )^{2/3}}+\frac{i \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{7}}\right )}{3 \sqrt{7} \left (\frac{1}{2} \left (1+i \sqrt{7}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21} \left (\frac{1}{2} \left (1-i \sqrt{7}\right )\right )^{2/3}}-\frac{i \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21} \left (\frac{1}{2} \left (1+i \sqrt{7}\right )\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + x^3 + x^6)^(-1),x]

[Out]

(I*ArcTan[(1 - (2*x)/((1 - I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/(Sqrt[21]*((1 - I*Sqrt
[7])/2)^(2/3)) - (I*ArcTan[(1 - (2*x)/((1 + I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/(Sqrt
[21]*((1 + I*Sqrt[7])/2)^(2/3)) - ((I/3)*Log[(1 - I*Sqrt[7])^(1/3) + 2^(1/3)*x])
/(Sqrt[7]*((1 - I*Sqrt[7])/2)^(2/3)) + ((I/3)*Log[(1 + I*Sqrt[7])^(1/3) + 2^(1/3
)*x])/(Sqrt[7]*((1 + I*Sqrt[7])/2)^(2/3)) + ((I/3)*Log[(1 - I*Sqrt[7])^(2/3) - (
2*(1 - I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^2])/(2^(1/3)*Sqrt[7]*(1 - I*Sqrt[7])^(2/3
)) - ((I/3)*Log[(1 + I*Sqrt[7])^(2/3) - (2*(1 + I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^
2])/(2^(1/3)*Sqrt[7]*(1 + I*Sqrt[7])^(2/3))

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Rubi in Sympy [A]  time = 96.7762, size = 345, normalized size = 0.91 \[ - \frac{2^{\frac{2}{3}} \sqrt{7} i \log{\left (\sqrt [3]{2} x + \sqrt [3]{1 - \sqrt{7} i} \right )}}{21 \left (1 - \sqrt{7} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} \sqrt{7} i \log{\left (\sqrt [3]{2} x + \sqrt [3]{1 + \sqrt{7} i} \right )}}{21 \left (1 + \sqrt{7} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} \sqrt{7} i \log{\left (x^{2} - \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{7} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{7} i\right )^{\frac{2}{3}}}{2} \right )}}{42 \left (1 - \sqrt{7} i\right )^{\frac{2}{3}}} - \frac{2^{\frac{2}{3}} \sqrt{7} i \log{\left (x^{2} - \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{7} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{7} i\right )^{\frac{2}{3}}}{2} \right )}}{42 \left (1 + \sqrt{7} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} \sqrt{21} i \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{7} i}} + \frac{1}{3}\right ) \right )}}{21 \left (1 - \sqrt{7} i\right )^{\frac{2}{3}}} - \frac{2^{\frac{2}{3}} \sqrt{21} i \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{7} i}} + \frac{1}{3}\right ) \right )}}{21 \left (1 + \sqrt{7} i\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**6+x**3+2),x)

[Out]

-2**(2/3)*sqrt(7)*I*log(2**(1/3)*x + (1 - sqrt(7)*I)**(1/3))/(21*(1 - sqrt(7)*I)
**(2/3)) + 2**(2/3)*sqrt(7)*I*log(2**(1/3)*x + (1 + sqrt(7)*I)**(1/3))/(21*(1 +
sqrt(7)*I)**(2/3)) + 2**(2/3)*sqrt(7)*I*log(x**2 - 2**(2/3)*x*(1 - sqrt(7)*I)**(
1/3)/2 + 2**(1/3)*(1 - sqrt(7)*I)**(2/3)/2)/(42*(1 - sqrt(7)*I)**(2/3)) - 2**(2/
3)*sqrt(7)*I*log(x**2 - 2**(2/3)*x*(1 + sqrt(7)*I)**(1/3)/2 + 2**(1/3)*(1 + sqrt
(7)*I)**(2/3)/2)/(42*(1 + sqrt(7)*I)**(2/3)) + 2**(2/3)*sqrt(21)*I*atan(sqrt(3)*
(-2*2**(1/3)*x/(3*(1 - sqrt(7)*I)**(1/3)) + 1/3))/(21*(1 - sqrt(7)*I)**(2/3)) -
2**(2/3)*sqrt(21)*I*atan(sqrt(3)*(-2*2**(1/3)*x/(3*(1 + sqrt(7)*I)**(1/3)) + 1/3
))/(21*(1 + sqrt(7)*I)**(2/3))

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Mathematica [C]  time = 0.0146363, size = 38, normalized size = 0.1 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6+\text{$\#$1}^3+2\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5+\text{$\#$1}^2}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x^3 + x^6)^(-1),x]

[Out]

RootSum[2 + #1^3 + #1^6 & , Log[x - #1]/(#1^2 + 2*#1^5) & ]/3

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Maple [C]  time = 0.008, size = 33, normalized size = 0.1 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+{{\it \_Z}}^{3}+2 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}+{{\it \_R}}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^6+x^3+2),x)

[Out]

1/3*sum(1/(2*_R^5+_R^2)*ln(x-_R),_R=RootOf(_Z^6+_Z^3+2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{6} + x^{3} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 + x^3 + 2),x, algorithm="maxima")

[Out]

integrate(1/(x^6 + x^3 + 2), x)

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Fricas [A]  time = 0.277735, size = 1477, normalized size = 3.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 + x^3 + 2),x, algorithm="fricas")

[Out]

1/8232*1372^(5/6)*(4*(sqrt(3)*cos(2/3*arctan(3/(sqrt(7) - 4))) - sin(2/3*arctan(
3/(sqrt(7) - 4))))*arctan(7*((sqrt(7)*sqrt(3) - 1)*cos(2/3*arctan(3/(sqrt(7) - 4
))) - (sqrt(7) + sqrt(3))*sin(2/3*arctan(3/(sqrt(7) - 4))))/(7*(sqrt(7) + sqrt(3
))*cos(2/3*arctan(3/(sqrt(7) - 4))) + 7*(sqrt(7)*sqrt(3) - 1)*sin(2/3*arctan(3/(
sqrt(7) - 4))) - 2*1372^(1/6)*sqrt(7)*sqrt(1/14)*sqrt(4^(2/3)*(14*4^(1/3)*x^2 -
1372^(1/6)*(sqrt(7)*sqrt(3)*x + 7*x)*cos(2/3*arctan(3/(sqrt(7) - 4))) + 1372^(1/
6)*(sqrt(7)*x - 7*sqrt(3)*x)*sin(2/3*arctan(3/(sqrt(7) - 4))) + 28*cos(2/3*arcta
n(3/(sqrt(7) - 4)))^2 + 28*sin(2/3*arctan(3/(sqrt(7) - 4)))^2)) - 4*1372^(1/6)*s
qrt(7)*x)) - 4*(sqrt(3)*cos(2/3*arctan(3/(sqrt(7) - 4))) + sin(2/3*arctan(3/(sqr
t(7) - 4))))*arctan(-7*((sqrt(7)*sqrt(3) + 1)*cos(2/3*arctan(3/(sqrt(7) - 4))) +
(sqrt(7) - sqrt(3))*sin(2/3*arctan(3/(sqrt(7) - 4))))/(7*(sqrt(7) - sqrt(3))*co
s(2/3*arctan(3/(sqrt(7) - 4))) - 7*(sqrt(7)*sqrt(3) + 1)*sin(2/3*arctan(3/(sqrt(
7) - 4))) - 2*1372^(1/6)*sqrt(7)*sqrt(1/14)*sqrt(4^(2/3)*(14*4^(1/3)*x^2 + 1372^
(1/6)*(sqrt(7)*sqrt(3)*x - 7*x)*cos(2/3*arctan(3/(sqrt(7) - 4))) + 1372^(1/6)*(s
qrt(7)*x + 7*sqrt(3)*x)*sin(2/3*arctan(3/(sqrt(7) - 4))) + 28*cos(2/3*arctan(3/(
sqrt(7) - 4)))^2 + 28*sin(2/3*arctan(3/(sqrt(7) - 4)))^2)) - 4*1372^(1/6)*sqrt(7
)*x)) + 2*cos(2/3*arctan(3/(sqrt(7) - 4)))*log(-1372^(1/6)*sqrt(7)*x*sin(2/3*arc
tan(3/(sqrt(7) - 4))) + 7*4^(1/3)*x^2 + 7*1372^(1/6)*x*cos(2/3*arctan(3/(sqrt(7)
- 4))) + 14*cos(2/3*arctan(3/(sqrt(7) - 4)))^2 + 14*sin(2/3*arctan(3/(sqrt(7) -
4)))^2) - (sqrt(3)*sin(2/3*arctan(3/(sqrt(7) - 4))) + cos(2/3*arctan(3/(sqrt(7)
- 4))))*log(14*4^(1/3)*x^2 - 1372^(1/6)*(sqrt(7)*sqrt(3)*x + 7*x)*cos(2/3*arcta
n(3/(sqrt(7) - 4))) + 1372^(1/6)*(sqrt(7)*x - 7*sqrt(3)*x)*sin(2/3*arctan(3/(sqr
t(7) - 4))) + 28*cos(2/3*arctan(3/(sqrt(7) - 4)))^2 + 28*sin(2/3*arctan(3/(sqrt(
7) - 4)))^2) + (sqrt(3)*sin(2/3*arctan(3/(sqrt(7) - 4))) - cos(2/3*arctan(3/(sqr
t(7) - 4))))*log(14*4^(1/3)*x^2 + 1372^(1/6)*(sqrt(7)*sqrt(3)*x - 7*x)*cos(2/3*a
rctan(3/(sqrt(7) - 4))) + 1372^(1/6)*(sqrt(7)*x + 7*sqrt(3)*x)*sin(2/3*arctan(3/
(sqrt(7) - 4))) + 28*cos(2/3*arctan(3/(sqrt(7) - 4)))^2 + 28*sin(2/3*arctan(3/(s
qrt(7) - 4)))^2) - 8*arctan(7*(sqrt(7)*sin(2/3*arctan(3/(sqrt(7) - 4))) + cos(2/
3*arctan(3/(sqrt(7) - 4))))/(1372^(1/6)*sqrt(7)*sqrt(1/7)*sqrt(-4^(2/3)*(1372^(1
/6)*sqrt(7)*x*sin(2/3*arctan(3/(sqrt(7) - 4))) - 7*4^(1/3)*x^2 - 7*1372^(1/6)*x*
cos(2/3*arctan(3/(sqrt(7) - 4))) - 14*cos(2/3*arctan(3/(sqrt(7) - 4)))^2 - 14*si
n(2/3*arctan(3/(sqrt(7) - 4)))^2)) + 2*1372^(1/6)*sqrt(7)*x + 7*sqrt(7)*cos(2/3*
arctan(3/(sqrt(7) - 4))) - 7*sin(2/3*arctan(3/(sqrt(7) - 4)))))*sin(2/3*arctan(3
/(sqrt(7) - 4))))

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Sympy [A]  time = 0.364926, size = 24, normalized size = 0.06 \[ \operatorname{RootSum}{\left (1000188 t^{6} + 1323 t^{3} + 1, \left ( t \mapsto t \log{\left (- 5292 t^{4} + 7 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**6+x**3+2),x)

[Out]

RootSum(1000188*_t**6 + 1323*_t**3 + 1, Lambda(_t, _t*log(-5292*_t**4 + 7*_t + x
)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{6} + x^{3} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 + x^3 + 2),x, algorithm="giac")

[Out]

integrate(1/(x^6 + x^3 + 2), x)