3.26.2 \(\int \frac {4+9 x+(1+3 x) \log (x)+(3+\log (x)) \log (3+\log (x))}{3+\log (x)} \, dx\)

Optimal. Leaf size=14 \[ x \left (1+\frac {3 x}{2}+\log (3+\log (x))\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.12, 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 {4+9 x+(1+3 x) \log (x)+(3+\log (x)) \log (3+\log (x))}{3+\log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + 9*x + (1 + 3*x)*Log[x] + (3 + Log[x])*Log[3 + Log[x]])/(3 + Log[x]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4+9 x+\log (x)+3 x \log (x)}{3+\log (x)}+\log (3+\log (x))\right ) \, dx\\ &=\int \frac {4+9 x+\log (x)+3 x \log (x)}{3+\log (x)} \, dx+\int \log (3+\log (x)) \, dx\\ &=\int \left (1+3 x+\frac {1}{3+\log (x)}\right ) \, dx+\int \log (3+\log (x)) \, dx\\ &=x+\frac {3 x^2}{2}+\int \frac {1}{3+\log (x)} \, dx+\int \log (3+\log (x)) \, dx\\ &=x+\frac {3 x^2}{2}+\int \log (3+\log (x)) \, dx+\operatorname {Subst}\left (\int \frac {e^x}{3+x} \, dx,x,\log (x)\right )\\ &=x+\frac {3 x^2}{2}+\frac {\text {Ei}(3+\log (x))}{e^3}+\int \log (3+\log (x)) \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 16, normalized size = 1.14 \begin {gather*} x+\frac {3 x^2}{2}+x \log (3+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 9*x + (1 + 3*x)*Log[x] + (3 + Log[x])*Log[3 + Log[x]])/(3 + Log[x]),x]

[Out]

x + (3*x^2)/2 + x*Log[3 + Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 14, normalized size = 1.00 \begin {gather*} \frac {3}{2} \, x^{2} + x \log \left (\log \relax (x) + 3\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+log(x))*log(3+log(x))+(3*x+1)*log(x)+9*x+4)/(3+log(x)),x, algorithm="fricas")

[Out]

3/2*x^2 + x*log(log(x) + 3) + x

________________________________________________________________________________________

giac [A]  time = 0.16, size = 14, normalized size = 1.00 \begin {gather*} \frac {3}{2} \, x^{2} + x \log \left (\log \relax (x) + 3\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+log(x))*log(3+log(x))+(3*x+1)*log(x)+9*x+4)/(3+log(x)),x, algorithm="giac")

[Out]

3/2*x^2 + x*log(log(x) + 3) + x

________________________________________________________________________________________

maple [A]  time = 0.03, size = 15, normalized size = 1.07

method	result	size
norman	\(x +x \ln \left (3+\ln \relax (x )\right )+\frac {3 x^{2}}{2}\)	\(15\)
risch	\(x +x \ln \left (3+\ln \relax (x )\right )+\frac {3 x^{2}}{2}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3+ln(x))*ln(3+ln(x))+(3*x+1)*ln(x)+9*x+4)/(3+ln(x)),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -e^{\left (-3\right )} E_{1}\left (-\log \relax (x) - 3\right ) \log \relax (x) - 3 \, e^{\left (-6\right )} E_{1}\left (-2 \, \log \relax (x) - 6\right ) \log \relax (x) + e^{\left (-3\right )} E_{2}\left (-\log \relax (x) - 3\right ) + \frac {3}{2} \, e^{\left (-6\right )} E_{2}\left (-2 \, \log \relax (x) - 6\right ) - 4 \, e^{\left (-3\right )} E_{1}\left (-\log \relax (x) - 3\right ) - 9 \, e^{\left (-6\right )} E_{1}\left (-2 \, \log \relax (x) - 6\right ) + x \log \left (\log \relax (x) + 3\right ) - \int \frac {1}{\log \relax (x) + 3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+log(x))*log(3+log(x))+(3*x+1)*log(x)+9*x+4)/(3+log(x)),x, algorithm="maxima")

[Out]

-e^(-3)*exp_integral_e(1, -log(x) - 3)*log(x) - 3*e^(-6)*exp_integral_e(1, -2*log(x) - 6)*log(x) + e^(-3)*exp_
integral_e(2, -log(x) - 3) + 3/2*e^(-6)*exp_integral_e(2, -2*log(x) - 6) - 4*e^(-3)*exp_integral_e(1, -log(x)
- 3) - 9*e^(-6)*exp_integral_e(1, -2*log(x) - 6) + x*log(log(x) + 3) - integrate(1/(log(x) + 3), x)

________________________________________________________________________________________

mupad [B]  time = 1.58, size = 15, normalized size = 1.07 \begin {gather*} \frac {x\,\left (3\,x+2\,\ln \left (\ln \relax (x)+3\right )+2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x + log(log(x) + 3)*(log(x) + 3) + log(x)*(3*x + 1) + 4)/(log(x) + 3),x)

[Out]

(x*(3*x + 2*log(log(x) + 3) + 2))/2

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 15, normalized size = 1.07 \begin {gather*} \frac {3 x^{2}}{2} + x \log {\left (\log {\relax (x )} + 3 \right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+ln(x))*ln(3+ln(x))+(3*x+1)*ln(x)+9*x+4)/(3+ln(x)),x)

[Out]

3*x**2/2 + x*log(log(x) + 3) + x

________________________________________________________________________________________