3.125 \(\int \frac{1}{x^5 \left (a^3+x^3\right )} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\log (a+x)}{3 a^7}-\frac{\tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3} a^7}+\frac{1}{a^6 x}-\frac{1}{4 a^3 x^4}+\frac{\log \left (a^2-a x+x^2\right )}{6 a^7} \]

[Out]

-1/(4*a^3*x^4) + 1/(a^6*x) - ArcTan[(a - 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a^7) - Log[a
+ x]/(3*a^7) + Log[a^2 - a*x + x^2]/(6*a^7)

_______________________________________________________________________________________

Rubi [A]  time = 0.0952836, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\log (a+x)}{3 a^7}-\frac{\tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3} a^7}+\frac{1}{a^6 x}-\frac{1}{4 a^3 x^4}+\frac{\log \left (a^2-a x+x^2\right )}{6 a^7} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a^3 + x^3)),x]

[Out]

-1/(4*a^3*x^4) + 1/(a^6*x) - ArcTan[(a - 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a^7) - Log[a
+ x]/(3*a^7) + Log[a^2 - a*x + x^2]/(6*a^7)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.53334, size = 70, normalized size = 0.96 \[ - \frac{1}{4 a^{3} x^{4}} + \frac{1}{a^{6} x} - \frac{\log{\left (a + x \right )}}{3 a^{7}} + \frac{\log{\left (a^{2} - a x + x^{2} \right )}}{6 a^{7}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} - \frac{2 x}{3}\right )}{a} \right )}}{3 a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(a**3+x**3),x)

[Out]

-1/(4*a**3*x**4) + 1/(a**6*x) - log(a + x)/(3*a**7) + log(a**2 - a*x + x**2)/(6*
a**7) - sqrt(3)*atan(sqrt(3)*(a/3 - 2*x/3)/a)/(3*a**7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0200911, size = 74, normalized size = 1.01 \[ -\frac{\log (a+x)}{3 a^7}+\frac{\tan ^{-1}\left (\frac{2 x-a}{\sqrt{3} a}\right )}{\sqrt{3} a^7}+\frac{1}{a^6 x}-\frac{1}{4 a^3 x^4}+\frac{\log \left (a^2-a x+x^2\right )}{6 a^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a^3 + x^3)),x]

[Out]

-1/(4*a^3*x^4) + 1/(a^6*x) + ArcTan[(-a + 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a^7) - Log[
a + x]/(3*a^7) + Log[a^2 - a*x + x^2]/(6*a^7)

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 67, normalized size = 0.9 \[ -{\frac{\ln \left ( a+x \right ) }{3\,{a}^{7}}}+{\frac{\ln \left ({a}^{2}-ax+{x}^{2} \right ) }{6\,{a}^{7}}}+{\frac{\sqrt{3}}{3\,{a}^{7}}\arctan \left ({\frac{ \left ( 2\,x-a \right ) \sqrt{3}}{3\,a}} \right ) }-{\frac{1}{4\,{a}^{3}{x}^{4}}}+{\frac{1}{{a}^{6}x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(a^3+x^3),x)

[Out]

-1/3*ln(a+x)/a^7+1/6*ln(a^2-a*x+x^2)/a^7+1/3/a^7*3^(1/2)*arctan(1/3*(2*x-a)*3^(1
/2)/a)-1/4/a^3/x^4+1/a^6/x

_______________________________________________________________________________________

Maxima [A]  time = 1.52047, size = 89, normalized size = 1.22 \[ \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{7}} + \frac{\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{7}} - \frac{\log \left (a + x\right )}{3 \, a^{7}} - \frac{a^{3} - 4 \, x^{3}}{4 \, a^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^3 + x^3)*x^5),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*x)/a)/a^7 + 1/6*log(a^2 - a*x + x^2)/a^7
- 1/3*log(a + x)/a^7 - 1/4*(a^3 - 4*x^3)/(a^6*x^4)

_______________________________________________________________________________________

Fricas [A]  time = 0.211786, size = 105, normalized size = 1.44 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} x^{4} \log \left (a^{2} - a x + x^{2}\right ) - 4 \, \sqrt{3} x^{4} \log \left (a + x\right ) + 12 \, x^{4} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (a^{4} - 4 \, a x^{3}\right )}\right )}}{36 \, a^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^3 + x^3)*x^5),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(2*sqrt(3)*x^4*log(a^2 - a*x + x^2) - 4*sqrt(3)*x^4*log(a + x) + 12
*x^4*arctan(-1/3*sqrt(3)*(a - 2*x)/a) - 3*sqrt(3)*(a^4 - 4*a*x^3))/(a^7*x^4)

_______________________________________________________________________________________

Sympy [A]  time = 0.754398, size = 90, normalized size = 1.23 \[ \frac{- a^{3} + 4 x^{3}}{4 a^{6} x^{4}} + \frac{- \frac{\log{\left (a + x \right )}}{3} + \left (\frac{1}{6} - \frac{\sqrt{3} i}{6}\right ) \log{\left (9 a \left (\frac{1}{6} - \frac{\sqrt{3} i}{6}\right )^{2} + x \right )} + \left (\frac{1}{6} + \frac{\sqrt{3} i}{6}\right ) \log{\left (9 a \left (\frac{1}{6} + \frac{\sqrt{3} i}{6}\right )^{2} + x \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(a**3+x**3),x)

[Out]

(-a**3 + 4*x**3)/(4*a**6*x**4) + (-log(a + x)/3 + (1/6 - sqrt(3)*I/6)*log(9*a*(1
/6 - sqrt(3)*I/6)**2 + x) + (1/6 + sqrt(3)*I/6)*log(9*a*(1/6 + sqrt(3)*I/6)**2 +
x))/a**7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.202021, size = 90, normalized size = 1.23 \[ \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{7}} + \frac{{\rm ln}\left (a^{2} - a x + x^{2}\right )}{6 \, a^{7}} - \frac{{\rm ln}\left ({\left | a + x \right |}\right )}{3 \, a^{7}} - \frac{a^{3} - 4 \, x^{3}}{4 \, a^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^3 + x^3)*x^5),x, algorithm="giac")

[Out]

1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*x)/a)/a^7 + 1/6*ln(a^2 - a*x + x^2)/a^7 -
1/3*ln(abs(a + x))/a^7 - 1/4*(a^3 - 4*x^3)/(a^6*x^4)