Optimal. Leaf size=26 \[ \log \left (\frac {1}{3} x \left (\left (4+e^2\right )^2-\frac {3}{\log (4-x+\log (x))}\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.52, antiderivative size = 34, normalized size of antiderivative = 1.31, number of steps
used = 5, number of rules used = 3, integrand size = 161, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6820, 6874,
6816} \begin {gather*} \log (x)-\log (\log (-x+\log (x)+4))+\log \left (3-\left (4+e^2\right )^2 \log (-x+\log (x)+4)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6816
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 (-1+x)-3 (-4+x-\log (x)) \log (4-x+\log (x))+\left (4+e^2\right )^2 (-4+x-\log (x)) \log ^2(4-x+\log (x))}{x (4-x+\log (x)) \log (4-x+\log (x)) \left (3-\left (4+e^2\right )^2 \log (4-x+\log (x))\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {1-x}{x (-4+x-\log (x)) \log (4-x+\log (x))}+\frac {\left (4+e^2\right )^2 (-1+x)}{x (4-x+\log (x)) \left (3-16 \left (1+\frac {1}{16} e^2 \left (8+e^2\right )\right ) \log (4-x+\log (x))\right )}\right ) \, dx\\ &=\log (x)+\left (4+e^2\right )^2 \int \frac {-1+x}{x (4-x+\log (x)) \left (3-16 \left (1+\frac {1}{16} e^2 \left (8+e^2\right )\right ) \log (4-x+\log (x))\right )} \, dx+\int \frac {1-x}{x (-4+x-\log (x)) \log (4-x+\log (x))} \, dx\\ &=\log (x)-\log (\log (4-x+\log (x)))+\log \left (3-\left (4+e^2\right )^2 \log (4-x+\log (x))\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(26)=52\).
time = 0.06, size = 53, normalized size = 2.04 \begin {gather*} \log (x)-\log (\log (4-x+\log (x)))+\log \left (3-16 \log (4-x+\log (x))-8 e^2 \log (4-x+\log (x))-e^4 \log (4-x+\log (x))\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs.
\(2(23)=46\).
time = 0.71, size = 53, normalized size = 2.04
method | result | size |
risch | \(\ln \left (x \right )-\ln \left (\ln \left (\ln \left (x \right )-x +4\right )\right )+\ln \left (\ln \left (\ln \left (x \right )-x +4\right )-\frac {3}{{\mathrm e}^{4}+8 \,{\mathrm e}^{2}+16}\right )\) | \(37\) |
default | \(\ln \left (x \right )-\ln \left (\ln \left (\ln \left (x \right )-x +4\right )\right )+\ln \left ({\mathrm e}^{4} \ln \left (\ln \left (x \right )-x +4\right )+8 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )-x +4\right )+16 \ln \left (\ln \left (x \right )-x +4\right )-3\right )\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.34, size = 45, normalized size = 1.73 \begin {gather*} \log \left (x\right ) + \log \left (\frac {{\left (e^{4} + 8 \, e^{2} + 16\right )} \log \left (-x + \log \left (x\right ) + 4\right ) - 3}{e^{4} + 8 \, e^{2} + 16}\right ) - \log \left (\log \left (-x + \log \left (x\right ) + 4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.52, size = 34, normalized size = 1.31 \begin {gather*} \log \left ({\left (e^{4} + 8 \, e^{2} + 16\right )} \log \left (-x + \log \left (x\right ) + 4\right ) - 3\right ) + \log \left (x\right ) - \log \left (\log \left (-x + \log \left (x\right ) + 4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (23) = 46\).
time = 0.49, size = 51, normalized size = 1.96 \begin {gather*} \log \left (-e^{4} \log \left (-x + \log \left (x\right ) + 4\right ) - 8 \, e^{2} \log \left (-x + \log \left (x\right ) + 4\right ) - 16 \, \log \left (-x + \log \left (x\right ) + 4\right ) + 3\right ) + \log \left (x\right ) - \log \left (\log \left (-x + \log \left (x\right ) + 4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\left (16\,x+{\mathrm {e}}^4\,\left (x-4\right )-\ln \left (x\right )\,\left (8\,{\mathrm {e}}^2+{\mathrm {e}}^4+16\right )+{\mathrm {e}}^2\,\left (8\,x-32\right )-64\right )\,{\ln \left (\ln \left (x\right )-x+4\right )}^2+\left (3\,\ln \left (x\right )-3\,x+12\right )\,\ln \left (\ln \left (x\right )-x+4\right )+3\,x-3}{{\ln \left (\ln \left (x\right )-x+4\right )}^2\,\left (64\,x+{\mathrm {e}}^4\,\left (4\,x-x^2\right )+{\mathrm {e}}^2\,\left (32\,x-8\,x^2\right )-16\,x^2+\ln \left (x\right )\,\left (16\,x+8\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4\right )\right )-\ln \left (\ln \left (x\right )-x+4\right )\,\left (12\,x+3\,x\,\ln \left (x\right )-3\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________