3.48 \(\int \frac{a+a x+c x^2}{1-x^3} \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{3} (a-c) \log \left (x^2+x+1\right )-\frac{1}{3} (2 a+c) \log (1-x) \]

[Out]

-((2*a + c)*Log[1 - x])/3 + ((a - c)*Log[1 + x + x^2])/3

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Rubi [A]  time = 0.060001, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{1}{3} (a-c) \log \left (x^2+x+1\right )-\frac{1}{3} (2 a+c) \log (1-x) \]

Antiderivative was successfully verified.

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

[Out]

-((2*a + c)*Log[1 - x])/3 + ((a - c)*Log[1 + x + x^2])/3

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Rubi in Sympy [A]  time = 11.8263, size = 27, normalized size = 0.84 \[ \left (\frac{a}{3} - \frac{c}{3}\right ) \log{\left (x^{2} + x + 1 \right )} - \left (\frac{2 a}{3} + \frac{c}{3}\right ) \log{\left (- x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a/3 - c/3)*log(x**2 + x + 1) - (2*a/3 + c/3)*log(-x + 1)

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Mathematica [A]  time = 0.0234506, size = 31, normalized size = 0.97 \[ \frac{1}{3} \left ((a-c) \log \left (x^2+x+1\right )-(2 a+c) \log (1-x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((2*a + c)*Log[1 - x]) + (a - c)*Log[1 + x + x^2])/3

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Maple [A]  time = 0.009, size = 36, normalized size = 1.1 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) a}{3}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) c}{3}}-{\frac{\ln \left ( -1+x \right ) c}{3}}-{\frac{2\,\ln \left ( -1+x \right ) a}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a*x+a)/(-x^3+1),x)

[Out]

1/3*ln(x^2+x+1)*a-1/3*ln(x^2+x+1)*c-1/3*ln(-1+x)*c-2/3*ln(-1+x)*a

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Maxima [A]  time = 1.57875, size = 35, normalized size = 1.09 \[ \frac{1}{3} \,{\left (a - c\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{3} \,{\left (2 \, a + c\right )} \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a - c)*log(x^2 + x + 1) - 1/3*(2*a + c)*log(x - 1)

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Fricas [A]  time = 0.21926, size = 35, normalized size = 1.09 \[ \frac{1}{3} \,{\left (a - c\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{3} \,{\left (2 \, a + c\right )} \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a - c)*log(x^2 + x + 1) - 1/3*(2*a + c)*log(x - 1)

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Sympy [A]  time = 0.496307, size = 24, normalized size = 0.75 \[ \frac{\left (a - c\right ) \log{\left (x^{2} + x + 1 \right )}}{3} - \frac{\left (2 a + c\right ) \log{\left (x - 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a - c)*log(x**2 + x + 1)/3 - (2*a + c)*log(x - 1)/3

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GIAC/XCAS [A]  time = 0.219611, size = 36, normalized size = 1.12 \[ \frac{1}{3} \,{\left (a - c\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{3} \,{\left (2 \, a + c\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a - c)*ln(x^2 + x + 1) - 1/3*(2*a + c)*ln(abs(x - 1))