Optimal. Leaf size=14 \[ \frac {(2+x) \log ^2(x)}{96 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(14)=28\).
time = 0.08, antiderivative size = 61, normalized size of antiderivative = 4.36, number of steps
used = 13, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {12, 14, 37,
2372, 45, 2395, 2342, 2341} \begin {gather*} \frac {\log ^2(x)}{48 x^2}-\frac {(x+2)^2 \log (x)}{192 x^2}+\frac {\log (x)}{48 x^2}+\frac {\log ^2(x)}{96 x}+\frac {\log (x)}{48 x}+\frac {\log (x)}{192} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 37
Rule 45
Rule 2341
Rule 2342
Rule 2372
Rule 2395
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{96} \int \frac {(4+2 x) \log (x)+(-4-x) \log ^2(x)}{x^3} \, dx\\ &=\frac {1}{96} \int \left (\frac {2 (2+x) \log (x)}{x^3}-\frac {(4+x) \log ^2(x)}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{96} \int \frac {(4+x) \log ^2(x)}{x^3} \, dx\right )+\frac {1}{48} \int \frac {(2+x) \log (x)}{x^3} \, dx\\ &=-\frac {(2+x)^2 \log (x)}{192 x^2}-\frac {1}{96} \int \left (\frac {4 \log ^2(x)}{x^3}+\frac {\log ^2(x)}{x^2}\right ) \, dx-\frac {1}{48} \int -\frac {(2+x)^2}{4 x^3} \, dx\\ &=-\frac {(2+x)^2 \log (x)}{192 x^2}+\frac {1}{192} \int \frac {(2+x)^2}{x^3} \, dx-\frac {1}{96} \int \frac {\log ^2(x)}{x^2} \, dx-\frac {1}{24} \int \frac {\log ^2(x)}{x^3} \, dx\\ &=-\frac {(2+x)^2 \log (x)}{192 x^2}+\frac {\log ^2(x)}{48 x^2}+\frac {\log ^2(x)}{96 x}+\frac {1}{192} \int \left (\frac {4}{x^3}+\frac {4}{x^2}+\frac {1}{x}\right ) \, dx-\frac {1}{48} \int \frac {\log (x)}{x^2} \, dx-\frac {1}{24} \int \frac {\log (x)}{x^3} \, dx\\ &=\frac {\log (x)}{192}+\frac {\log (x)}{48 x^2}+\frac {\log (x)}{48 x}-\frac {(2+x)^2 \log (x)}{192 x^2}+\frac {\log ^2(x)}{48 x^2}+\frac {\log ^2(x)}{96 x}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 14, normalized size = 1.00 \begin {gather*} \frac {(2+x) \log ^2(x)}{96 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 20, normalized size = 1.43
method | result | size |
risch | \(\frac {\ln \left (x \right )^{2} \left (2+x \right )}{96 x^{2}}\) | \(13\) |
norman | \(\frac {\frac {\ln \left (x \right )^{2}}{48}+\frac {x \ln \left (x \right )^{2}}{96}}{x^{2}}\) | \(19\) |
default | \(\frac {\ln \left (x \right )^{2}}{96 x}+\frac {\ln \left (x \right )^{2}}{48 x^{2}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (12) = 24\).
time = 0.27, size = 57, normalized size = 4.07 \begin {gather*} \frac {\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2}{96 \, x} - \frac {\log \left (x\right )}{48 \, x} + \frac {2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1}{96 \, x^{2}} - \frac {1}{48 \, x} - \frac {\log \left (x\right )}{48 \, x^{2}} - \frac {1}{96 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (x + 2\right )} \log \left (x\right )^{2}}{96 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 12, normalized size = 0.86 \begin {gather*} \frac {\left (x + 2\right ) \log {\left (x \right )}^{2}}{96 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (x + 2\right )} \log \left (x\right )^{2}}{96 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.13, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\ln \left (x\right )}^2\,\left (x+2\right )}{96\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________