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

Optimal. Leaf size=19 \[ \log \left (\frac {3 \log \left (x (-x-\log (2))^2\right )}{x}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.53, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.51, size = 15, normalized size = 0.79 \begin {gather*} -\log (x)+\log \left (\log \left (x (x+\log (2))^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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]]

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 24, normalized size = 1.26 \begin {gather*} -\log \relax (x) + \log \left (\log \left (x^{3} + 2 \, x^{2} \log \relax (2) + x \log \relax (2)^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [A]  time = 0.20, size = 24, normalized size = 1.26 \begin {gather*} -\log \relax (x) + \log \left (\log \left (x^{3} + 2 \, x^{2} \log \relax (2) + x \log \relax (2)^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maple [A]  time = 0.09, size = 25, normalized size = 1.32

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 16, normalized size = 0.84 \begin {gather*} -\log \relax (x) + \log \left (\log \left (x + \log \relax (2)\right ) + \frac {1}{2} \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

mupad [B]  time = 2.64, size = 24, normalized size = 1.26 \begin {gather*} \ln \left (\ln \left (x^3+2\,\ln \relax (2)\,x^2+{\ln \relax (2)}^2\,x\right )\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 24, normalized size = 1.26 \begin {gather*} - \log {\relax (x )} + \log {\left (\log {\left (x^{3} + 2 x^{2} \log {\relax (2 )} + x \log {\relax (2 )}^{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________