3.166 \(\int \frac{1}{-x^3+x^4} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2 x^2}+\frac{1}{x}+\log (1-x)-\log (x) \]

[Out]

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

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Rubi [A]  time = 0.0185005, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2 x^2}+\frac{1}{x}+\log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(-x^3 + x^4)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.50932, size = 17, normalized size = 0.81 \[ - \log{\left (x \right )} + \log{\left (- x + 1 \right )} + \frac{1}{x} + \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4-x**3),x)

[Out]

-log(x) + log(-x + 1) + 1/x + 1/(2*x**2)

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Mathematica [A]  time = 0.00311023, size = 21, normalized size = 1. \[ \frac{1}{2 x^2}+\frac{1}{x}+\log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(-x^3 + x^4)^(-1),x]

[Out]

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

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Maple [A]  time = 0.01, size = 18, normalized size = 0.9 \[{\frac{1}{2\,{x}^{2}}}+{x}^{-1}-\ln \left ( x \right ) +\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4-x^3),x)

[Out]

1/2/x^2+1/x-ln(x)+ln(-1+x)

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Maxima [A]  time = 1.36104, size = 26, normalized size = 1.24 \[ \frac{2 \, x + 1}{2 \, x^{2}} + \log \left (x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 - x^3),x, algorithm="maxima")

[Out]

1/2*(2*x + 1)/x^2 + log(x - 1) - log(x)

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Fricas [A]  time = 0.203377, size = 35, normalized size = 1.67 \[ \frac{2 \, x^{2} \log \left (x - 1\right ) - 2 \, x^{2} \log \left (x\right ) + 2 \, x + 1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 - x^3),x, algorithm="fricas")

[Out]

1/2*(2*x^2*log(x - 1) - 2*x^2*log(x) + 2*x + 1)/x^2

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Sympy [A]  time = 0.104677, size = 17, normalized size = 0.81 \[ - \log{\left (x \right )} + \log{\left (x - 1 \right )} + \frac{2 x + 1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4-x**3),x)

[Out]

-log(x) + log(x - 1) + (2*x + 1)/(2*x**2)

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GIAC/XCAS [A]  time = 0.221906, size = 28, normalized size = 1.33 \[ \frac{2 \, x + 1}{2 \, x^{2}} +{\rm ln}\left ({\left | x - 1 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 - x^3),x, algorithm="giac")

[Out]

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