\(\int \frac {3 x+\log (2)+(-x-\log (2)) \log (x^3+2 x^2 \log (2)+x \log ^2(2))}{(x^2+x \log (2)) \log (x^3+2 x^2 \log (2)+x \log ^2(2))} \, dx\) [3694]

Optimal result

Integrand size = 64, antiderivative size = 19 \[ \int \frac {3 x+\log (2)+(-x-\log (2)) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )}{\left (x^2+x \log (2)\right ) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )} \, dx=\log \left (\frac {3 \log \left (x (-x-\log (2))^2\right )}{x}\right ) \]

[Out]

ln(3*ln(x*(-x-ln(2))^2)/x)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {3 x+\log (2)+(-x-\log (2)) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )}{\left (x^2+x \log (2)\right ) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )} \, dx=-\log (x)+\log \left (\log \left (x (x+\log (2))^2\right )\right ) \]

[In]

Integrate[(3*x + Log[2] + (-x - Log[2])*Log[x^3 + 2*x^2*Log[2] + x*Log[2]^2])/((x^2 + x*Log[2])*Log[x^3 + 2*x^
2*Log[2] + x*Log[2]^2]),x]

[Out]

-Log[x] + Log[Log[x*(x + Log[2])^2]]

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32

method	result	size
default	\(\ln \left (\ln \left (x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+x^{3}\right )\right )-\ln \left (x \right )\)	\(25\)
norman	\(\ln \left (\ln \left (x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+x^{3}\right )\right )-\ln \left (x \right )\)	\(25\)
risch	\(\ln \left (\ln \left (x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+x^{3}\right )\right )-\ln \left (x \right )\)	\(25\)
parallelrisch	\(\ln \left (\ln \left (x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+x^{3}\right )\right )-\ln \left (x \right )\)	\(25\)
parts	\(\ln \left (\ln \left (x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+x^{3}\right )\right )-\ln \left (x \right )\)	\(25\)

[In]

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

[Out]

ln(ln(x*ln(2)^2+2*x^2*ln(2)+x^3))-ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {3 x+\log (2)+(-x-\log (2)) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )}{\left (x^2+x \log (2)\right ) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )} \, dx=-\log \left (x\right ) + \log \left (\log \left (x^{3} + 2 \, x^{2} \log \left (2\right ) + x \log \left (2\right )^{2}\right )\right ) \]

[In]

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

[Out]

-log(x) + log(log(x^3 + 2*x^2*log(2) + x*log(2)^2))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {3 x+\log (2)+(-x-\log (2)) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )}{\left (x^2+x \log (2)\right ) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )} \, dx=- \log {\left (x \right )} + \log {\left (\log {\left (x^{3} + 2 x^{2} \log {\left (2 \right )} + x \log {\left (2 \right )}^{2} \right )} \right )} \]

[In]

integrate(((-x-ln(2))*ln(x*ln(2)**2+2*x**2*ln(2)+x**3)+ln(2)+3*x)/(x*ln(2)+x**2)/ln(x*ln(2)**2+2*x**2*ln(2)+x*
*3),x)

[Out]

-log(x) + log(log(x**3 + 2*x**2*log(2) + x*log(2)**2))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {3 x+\log (2)+(-x-\log (2)) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )}{\left (x^2+x \log (2)\right ) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )} \, dx=-\log \left (x\right ) + \log \left (\log \left (x + \log \left (2\right )\right ) + \frac {1}{2} \, \log \left (x\right )\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {3 x+\log (2)+(-x-\log (2)) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )}{\left (x^2+x \log (2)\right ) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )} \, dx=-\log \left (x\right ) + \log \left (\log \left (x^{3} + 2 \, x^{2} \log \left (2\right ) + x \log \left (2\right )^{2}\right )\right ) \]

[In]

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

[Out]

-log(x) + log(log(x^3 + 2*x^2*log(2) + x*log(2)^2))

Mupad [B] (verification not implemented)

Time = 8.75 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {3 x+\log (2)+(-x-\log (2)) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )}{\left (x^2+x \log (2)\right ) \log \left (x^3+2 x^2 \log (2)+x \log ^2(2)\right )} \, dx=\ln \left (\ln \left (x^3+2\,\ln \left (2\right )\,x^2+{\ln \left (2\right )}^2\,x\right )\right )-\ln \left (x\right ) \]

[In]

int((3*x + log(2) - log(x*log(2)^2 + 2*x^2*log(2) + x^3)*(x + log(2)))/(log(x*log(2)^2 + 2*x^2*log(2) + x^3)*(
x*log(2) + x^2)),x)

[Out]

log(log(x*log(2)^2 + 2*x^2*log(2) + x^3)) - log(x)