\(\int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx\) [265]

Optimal result

Integrand size = 20, antiderivative size = 39 \[ \int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx=\frac {1}{8} \log (1-x)-\frac {1}{5} \log (2-x)+\frac {1}{12} \log (3-x)-\frac {1}{120} \log (3+x) \]

[Out]

1/8*ln(1-x)-1/5*ln(2-x)+1/12*ln(3-x)-1/120*ln(3+x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2083} \[ \int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx=\frac {1}{8} \log (1-x)-\frac {1}{5} \log (2-x)+\frac {1}{12} \log (3-x)-\frac {1}{120} \log (x+3) \]

[In]

Int[(-18 + 27*x - 7*x^2 - 3*x^3 + x^4)^(-1),x]

[Out]

Log[1 - x]/8 - Log[2 - x]/5 + Log[3 - x]/12 - Log[3 + x]/120

Rule 2083

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{12 (-3+x)}-\frac {1}{5 (-2+x)}+\frac {1}{8 (-1+x)}-\frac {1}{120 (3+x)}\right ) \, dx \\ & = \frac {1}{8} \log (1-x)-\frac {1}{5} \log (2-x)+\frac {1}{12} \log (3-x)-\frac {1}{120} \log (3+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx=\frac {1}{8} \log (1-x)-\frac {1}{5} \log (2-x)+\frac {1}{12} \log (3-x)-\frac {1}{120} \log (3+x) \]

[In]

Integrate[(-18 + 27*x - 7*x^2 - 3*x^3 + x^4)^(-1),x]

[Out]

Log[1 - x]/8 - Log[2 - x]/5 + Log[3 - x]/12 - Log[3 + x]/120

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.67

method	result	size
default	\(\frac {\ln \left (-3+x \right )}{12}-\frac {\ln \left (3+x \right )}{120}+\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x -2\right )}{5}\)	\(26\)
norman	\(\frac {\ln \left (-3+x \right )}{12}-\frac {\ln \left (3+x \right )}{120}+\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x -2\right )}{5}\)	\(26\)
risch	\(\frac {\ln \left (-3+x \right )}{12}-\frac {\ln \left (3+x \right )}{120}+\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x -2\right )}{5}\)	\(26\)
parallelrisch	\(\frac {\ln \left (-3+x \right )}{12}-\frac {\ln \left (3+x \right )}{120}+\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x -2\right )}{5}\)	\(26\)

[In]

int(1/(x^4-3*x^3-7*x^2+27*x-18),x,method=_RETURNVERBOSE)

[Out]

1/12*ln(-3+x)-1/120*ln(3+x)+1/8*ln(x-1)-1/5*ln(x-2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64 \[ \int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx=-\frac {1}{120} \, \log \left (x + 3\right ) + \frac {1}{8} \, \log \left (x - 1\right ) - \frac {1}{5} \, \log \left (x - 2\right ) + \frac {1}{12} \, \log \left (x - 3\right ) \]

[In]

integrate(1/(x^4-3*x^3-7*x^2+27*x-18),x, algorithm="fricas")

[Out]

-1/120*log(x + 3) + 1/8*log(x - 1) - 1/5*log(x - 2) + 1/12*log(x - 3)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.67 \[ \int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx=\frac {\log {\left (x - 3 \right )}}{12} - \frac {\log {\left (x - 2 \right )}}{5} + \frac {\log {\left (x - 1 \right )}}{8} - \frac {\log {\left (x + 3 \right )}}{120} \]

[In]

integrate(1/(x**4-3*x**3-7*x**2+27*x-18),x)

[Out]

log(x - 3)/12 - log(x - 2)/5 + log(x - 1)/8 - log(x + 3)/120

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64 \[ \int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx=-\frac {1}{120} \, \log \left (x + 3\right ) + \frac {1}{8} \, \log \left (x - 1\right ) - \frac {1}{5} \, \log \left (x - 2\right ) + \frac {1}{12} \, \log \left (x - 3\right ) \]

[In]

integrate(1/(x^4-3*x^3-7*x^2+27*x-18),x, algorithm="maxima")

[Out]

-1/120*log(x + 3) + 1/8*log(x - 1) - 1/5*log(x - 2) + 1/12*log(x - 3)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx=-\frac {1}{120} \, \log \left ({\left | x + 3 \right |}\right ) + \frac {1}{8} \, \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{5} \, \log \left ({\left | x - 2 \right |}\right ) + \frac {1}{12} \, \log \left ({\left | x - 3 \right |}\right ) \]

[In]

integrate(1/(x^4-3*x^3-7*x^2+27*x-18),x, algorithm="giac")

[Out]

-1/120*log(abs(x + 3)) + 1/8*log(abs(x - 1)) - 1/5*log(abs(x - 2)) + 1/12*log(abs(x - 3))

Mupad [B] (verification not implemented)

Time = 9.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64 \[ \int \frac {1}{-18+27 x-7 x^2-3 x^3+x^4} \, dx=\frac {\ln \left (x-1\right )}{8}-\frac {\ln \left (x-2\right )}{5}+\frac {\ln \left (x-3\right )}{12}-\frac {\ln \left (x+3\right )}{120} \]

[In]

int(-1/(7*x^2 - 27*x + 3*x^3 - x^4 + 18),x)

[Out]

log(x - 1)/8 - log(x - 2)/5 + log(x - 3)/12 - log(x + 3)/120