\(\int \frac {87+3 x-87 x^2-3 x^3+(-87-6 x-87 x^2) \log (x)}{(1-2 x^2+x^4) \log ^2(x)} \, dx\) [10313]

Optimal result

Integrand size = 45, antiderivative size = 18 \[ \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {3 (29+x)}{\left (-\frac {1}{x}+x\right ) \log (x)} \]

[Out]

3/ln(x)/(x-1/x)*(x+29)

Rubi [F]

\[ \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx \]

[In]

Int[(87 + 3*x - 87*x^2 - 3*x^3 + (-87 - 6*x - 87*x^2)*Log[x])/((1 - 2*x^2 + x^4)*Log[x]^2),x]

[Out]

-3*Defer[Int][(29 + x)/((-1 + x^2)*Log[x]^2), x] - 3*Defer[Int][(29 + 2*x + 29*x^2)/((-1 + x^2)^2*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (-1+x^2\right )^2 \log ^2(x)} \, dx \\ & = \int \left (-\frac {3 (29+x)}{\left (-1+x^2\right ) \log ^2(x)}-\frac {3 \left (29+2 x+29 x^2\right )}{\left (-1+x^2\right )^2 \log (x)}\right ) \, dx \\ & = -\left (3 \int \frac {29+x}{\left (-1+x^2\right ) \log ^2(x)} \, dx\right )-3 \int \frac {29+2 x+29 x^2}{\left (-1+x^2\right )^2 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {3 x (29+x)}{\left (-1+x^2\right ) \log (x)} \]

[In]

Integrate[(87 + 3*x - 87*x^2 - 3*x^3 + (-87 - 6*x - 87*x^2)*Log[x])/((1 - 2*x^2 + x^4)*Log[x]^2),x]

[Out]

(3*x*(29 + x))/((-1 + x^2)*Log[x])

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {3 x \left (x +29\right )}{\left (x^{2}-1\right ) \ln \left (x \right )}\)	\(18\)
norman	\(\frac {3 x^{2}+87 x}{\left (x^{2}-1\right ) \ln \left (x \right )}\)	\(22\)
parallelrisch	\(\frac {3 x^{2}+87 x}{\left (x^{2}-1\right ) \ln \left (x \right )}\)	\(22\)

[In]

int(((-87*x^2-6*x-87)*ln(x)-3*x^3-87*x^2+3*x+87)/(x^4-2*x^2+1)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

3*x*(x+29)/(x^2-1)/ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (x^{2} + 29 \, x\right )}}{{\left (x^{2} - 1\right )} \log \left (x\right )} \]

[In]

integrate(((-87*x^2-6*x-87)*log(x)-3*x^3-87*x^2+3*x+87)/(x^4-2*x^2+1)/log(x)^2,x, algorithm="fricas")

[Out]

3*(x^2 + 29*x)/((x^2 - 1)*log(x))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {3 x^{2} + 87 x}{\left (x^{2} - 1\right ) \log {\left (x \right )}} \]

[In]

integrate(((-87*x**2-6*x-87)*ln(x)-3*x**3-87*x**2+3*x+87)/(x**4-2*x**2+1)/ln(x)**2,x)

[Out]

(3*x**2 + 87*x)/((x**2 - 1)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (x^{2} + 29 \, x\right )}}{{\left (x^{2} - 1\right )} \log \left (x\right )} \]

[In]

integrate(((-87*x^2-6*x-87)*log(x)-3*x^3-87*x^2+3*x+87)/(x^4-2*x^2+1)/log(x)^2,x, algorithm="maxima")

[Out]

3*(x^2 + 29*x)/((x^2 - 1)*log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (x^{2} + 29 \, x\right )}}{x^{2} \log \left (x\right ) - \log \left (x\right )} \]

[In]

integrate(((-87*x^2-6*x-87)*log(x)-3*x^3-87*x^2+3*x+87)/(x^4-2*x^2+1)/log(x)^2,x, algorithm="giac")

[Out]

3*(x^2 + 29*x)/(x^2*log(x) - log(x))

Mupad [B] (verification not implemented)

Time = 17.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {87+3 x-87 x^2-3 x^3+\left (-87-6 x-87 x^2\right ) \log (x)}{\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {3\,x\,\left (x+29\right )}{\ln \left (x\right )\,\left (x^2-1\right )} \]

[In]

int(-(log(x)*(6*x + 87*x^2 + 87) - 3*x + 87*x^2 + 3*x^3 - 87)/(log(x)^2*(x^4 - 2*x^2 + 1)),x)

[Out]

(3*x*(x + 29))/(log(x)*(x^2 - 1))