3.100 \(\int \frac{1}{(-4+x) (-3+x) (-2+x) (-1+x)} \, dx\)

Optimal. Leaf size=41 \[ -\frac{1}{6} \log (1-x)+\frac{1}{2} \log (2-x)-\frac{1}{2} \log (3-x)+\frac{1}{6} \log (4-x) \]

[Out]

-Log[1 - x]/6 + Log[2 - x]/2 - Log[3 - x]/2 + Log[4 - x]/6

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Rubi [A]  time = 0.0204782, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {180} \[ -\frac{1}{6} \log (1-x)+\frac{1}{2} \log (2-x)-\frac{1}{2} \log (3-x)+\frac{1}{6} \log (4-x) \]

Antiderivative was successfully verified.

[In]

Int[1/((-4 + x)*(-3 + x)*(-2 + x)*(-1 + x)),x]

[Out]

-Log[1 - x]/6 + Log[2 - x]/2 - Log[3 - x]/2 + Log[4 - x]/6

Rule 180

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

Rubi steps

\begin{align*} \int \frac{1}{(-4+x) (-3+x) (-2+x) (-1+x)} \, dx &=\int \left (\frac{1}{6 (-4+x)}-\frac{1}{2 (-3+x)}+\frac{1}{2 (-2+x)}-\frac{1}{6 (-1+x)}\right ) \, dx\\ &=-\frac{1}{6} \log (1-x)+\frac{1}{2} \log (2-x)-\frac{1}{2} \log (3-x)+\frac{1}{6} \log (4-x)\\ \end{align*}

Mathematica [A]  time = 0.0073356, size = 41, normalized size = 1. \[ -\frac{1}{6} \log (1-x)+\frac{1}{2} \log (2-x)-\frac{1}{2} \log (3-x)+\frac{1}{6} \log (4-x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((-4 + x)*(-3 + x)*(-2 + x)*(-1 + x)),x]

[Out]

-Log[1 - x]/6 + Log[2 - x]/2 - Log[3 - x]/2 + Log[4 - x]/6

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Maple [A]  time = 0.008, size = 26, normalized size = 0.6 \begin{align*} -{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{\ln \left ( -3+x \right ) }{2}}+{\frac{\ln \left ( -2+x \right ) }{2}}+{\frac{\ln \left ( x-4 \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x-4)/(-3+x)/(-2+x)/(-1+x),x)

[Out]

-1/6*ln(-1+x)-1/2*ln(-3+x)+1/2*ln(-2+x)+1/6*ln(x-4)

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Maxima [A]  time = 0.923479, size = 34, normalized size = 0.83 \begin{align*} -\frac{1}{6} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x - 2\right ) - \frac{1}{2} \, \log \left (x - 3\right ) + \frac{1}{6} \, \log \left (x - 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4+x)/(-3+x)/(-2+x)/(-1+x),x, algorithm="maxima")

[Out]

-1/6*log(x - 1) + 1/2*log(x - 2) - 1/2*log(x - 3) + 1/6*log(x - 4)

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Fricas [A]  time = 2.20441, size = 92, normalized size = 2.24 \begin{align*} -\frac{1}{6} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x - 2\right ) - \frac{1}{2} \, \log \left (x - 3\right ) + \frac{1}{6} \, \log \left (x - 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4+x)/(-3+x)/(-2+x)/(-1+x),x, algorithm="fricas")

[Out]

-1/6*log(x - 1) + 1/2*log(x - 2) - 1/2*log(x - 3) + 1/6*log(x - 4)

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Sympy [A]  time = 0.154118, size = 26, normalized size = 0.63 \begin{align*} \frac{\log{\left (x - 4 \right )}}{6} - \frac{\log{\left (x - 3 \right )}}{2} + \frac{\log{\left (x - 2 \right )}}{2} - \frac{\log{\left (x - 1 \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4+x)/(-3+x)/(-2+x)/(-1+x),x)

[Out]

log(x - 4)/6 - log(x - 3)/2 + log(x - 2)/2 - log(x - 1)/6

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Giac [A]  time = 1.04729, size = 39, normalized size = 0.95 \begin{align*} -\frac{1}{6} \, \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x - 2 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x - 3 \right |}\right ) + \frac{1}{6} \, \log \left ({\left | x - 4 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4+x)/(-3+x)/(-2+x)/(-1+x),x, algorithm="giac")

[Out]

-1/6*log(abs(x - 1)) + 1/2*log(abs(x - 2)) - 1/2*log(abs(x - 3)) + 1/6*log(abs(x - 4))