3.1.30 \(\int \frac {1+2 x+(1+x+2 x^2+2 x^3) \log ((x+x^2) \log (4))}{(x+x^2) \log ((x+x^2) \log (4))} \, dx\)

Optimal. Leaf size=19 \[ \log \left (\frac {1}{40} e^{x^2} x \log (x (1+x) \log (4))\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.63, 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 {1+2 x+\left (1+x+2 x^2+2 x^3\right ) \log \left (\left (x+x^2\right ) \log (4)\right )}{\left (x+x^2\right ) \log \left (\left (x+x^2\right ) \log (4)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x^2 + Log[x] + Defer[Int][(1 + 2*x)/(x*(1 + x)*Log[x*(1 + x)*Log[4]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+2 x+\left (1+x+2 x^2+2 x^3\right ) \log \left (\left (x+x^2\right ) \log (4)\right )}{x (1+x) \log \left (\left (x+x^2\right ) \log (4)\right )} \, dx\\ &=\int \frac {1+2 x+\left (1+x+2 x^2+2 x^3\right ) \log \left (\left (x+x^2\right ) \log (4)\right )}{x (1+x) \log (x (1+x) \log (4))} \, dx\\ &=\int \left (\frac {1+2 x^2}{x}+\frac {1+2 x}{x (1+x) \log (x (1+x) \log (4))}\right ) \, dx\\ &=\int \frac {1+2 x^2}{x} \, dx+\int \frac {1+2 x}{x (1+x) \log (x (1+x) \log (4))} \, dx\\ &=\int \left (\frac {1}{x}+2 x\right ) \, dx+\int \frac {1+2 x}{x (1+x) \log (x (1+x) \log (4))} \, dx\\ &=x^2+\log (x)+\int \frac {1+2 x}{x (1+x) \log (x (1+x) \log (4))} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 15, normalized size = 0.79 \begin {gather*} x^2+\log (x)+\log (\log (x (1+x) \log (4))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2 + Log[x] + Log[Log[x*(1 + x)*Log[4]]]

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 17, normalized size = 0.89 \begin {gather*} x^{2} + \log \relax (x) + \log \left (\log \left (2 \, {\left (x^{2} + x\right )} \log \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.44, size = 22, normalized size = 1.16 \begin {gather*} x^{2} + \log \relax (x) + \log \left (\log \relax (2) + \log \left (x^{2} \log \relax (2) + x \log \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 18, normalized size = 0.95

method	result	size
norman	\(x^{2}+\ln \relax (x )+\ln \left (\ln \left (2 \left (x^{2}+x \right ) \ln \relax (2)\right )\right )\)	\(18\)
risch	\(x^{2}+\ln \relax (x )+\ln \left (\ln \left (2 \left (x^{2}+x \right ) \ln \relax (2)\right )\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.67, size = 19, normalized size = 1.00 \begin {gather*} x^{2} + \log \relax (x) + \log \left (\log \relax (2) + \log \left (x + 1\right ) + \log \relax (x) + \log \left (\log \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + log(x) + log(log(2) + log(x + 1) + log(x) + log(log(2)))

________________________________________________________________________________________

mupad [B]  time = 0.90, size = 17, normalized size = 0.89 \begin {gather*} \ln \left (\ln \left (2\,\ln \relax (2)\,\left (x^2+x\right )\right )\right )+\ln \relax (x)+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + log(2*log(2)*(x + x^2))*(x + 2*x^2 + 2*x^3 + 1) + 1)/(log(2*log(2)*(x + x^2))*(x + x^2)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 20, normalized size = 1.05 \begin {gather*} x^{2} + \log {\relax (x )} + \log {\left (\log {\left (\left (2 x^{2} + 2 x\right ) \log {\relax (2 )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+2*x**2+x+1)*ln(2*(x**2+x)*ln(2))+2*x+1)/(x**2+x)/ln(2*(x**2+x)*ln(2)),x)

[Out]

x**2 + log(x) + log(log((2*x**2 + 2*x)*log(2)))

________________________________________________________________________________________