Optimal. Leaf size=29 \[ x-\log (x)+\frac {1}{4} x \left (5+x+x^2-\frac {5 \left (3+\log \left (x^2\right )\right )}{\log (x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.42, antiderivative size = 46, normalized size of antiderivative = 1.59, number of steps
used = 16, number of rules used = 8, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {12, 6874,
6820, 2334, 2335, 2407, 2408, 6631} \begin {gather*} \frac {x^3}{4}+\frac {x^2}{4}-\frac {5 x \log \left (x^2\right )}{4 \log (x)}+\frac {9 x}{4}-\frac {15 x}{4 \log (x)}-\log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2334
Rule 2335
Rule 2407
Rule 2408
Rule 6631
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {15 x-25 x \log (x)+\left (-4+9 x+2 x^2+3 x^3\right ) \log ^2(x)+(5 x-5 x \log (x)) \log \left (x^2\right )}{x \log ^2(x)} \, dx\\ &=\frac {1}{4} \int \left (\frac {15 x-25 x \log (x)-4 \log ^2(x)+9 x \log ^2(x)+2 x^2 \log ^2(x)+3 x^3 \log ^2(x)}{x \log ^2(x)}-\frac {5 (-1+\log (x)) \log \left (x^2\right )}{\log ^2(x)}\right ) \, dx\\ &=\frac {1}{4} \int \frac {15 x-25 x \log (x)-4 \log ^2(x)+9 x \log ^2(x)+2 x^2 \log ^2(x)+3 x^3 \log ^2(x)}{x \log ^2(x)} \, dx-\frac {5}{4} \int \frac {(-1+\log (x)) \log \left (x^2\right )}{\log ^2(x)} \, dx\\ &=\frac {1}{4} \int \left (9-\frac {4}{x}+2 x+3 x^2+\frac {15}{\log ^2(x)}-\frac {25}{\log (x)}\right ) \, dx-\frac {5}{4} \int \left (-\frac {\log \left (x^2\right )}{\log ^2(x)}+\frac {\log \left (x^2\right )}{\log (x)}\right ) \, dx\\ &=\frac {9 x}{4}+\frac {x^2}{4}+\frac {x^3}{4}-\log (x)+\frac {5}{4} \int \frac {\log \left (x^2\right )}{\log ^2(x)} \, dx-\frac {5}{4} \int \frac {\log \left (x^2\right )}{\log (x)} \, dx+\frac {15}{4} \int \frac {1}{\log ^2(x)} \, dx-\frac {25}{4} \int \frac {1}{\log (x)} \, dx\\ &=\frac {9 x}{4}+\frac {x^2}{4}+\frac {x^3}{4}-\frac {15 x}{4 \log (x)}-\log (x)-\frac {5 x \log \left (x^2\right )}{4 \log (x)}-\frac {25 \text {li}(x)}{4}+\frac {5}{2} \int \frac {\text {li}(x)}{x} \, dx-\frac {5}{2} \int \left (-\frac {1}{\log (x)}+\frac {\text {li}(x)}{x}\right ) \, dx+\frac {15}{4} \int \frac {1}{\log (x)} \, dx\\ &=-\frac {x}{4}+\frac {x^2}{4}+\frac {x^3}{4}-\frac {15 x}{4 \log (x)}-\log (x)-\frac {5 x \log \left (x^2\right )}{4 \log (x)}-\frac {5 \text {li}(x)}{2}+\frac {5}{2} \log (x) \text {li}(x)+\frac {5}{2} \int \frac {1}{\log (x)} \, dx-\frac {5}{2} \int \frac {\text {li}(x)}{x} \, dx\\ &=\frac {9 x}{4}+\frac {x^2}{4}+\frac {x^3}{4}-\frac {15 x}{4 \log (x)}-\log (x)-\frac {5 x \log \left (x^2\right )}{4 \log (x)}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{4} \left (x \left (9+x+x^2\right )-4 \log (x)-\frac {5 x \left (3+\log \left (x^2\right )\right )}{\log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.60, size = 42, normalized size = 1.45
method | result | size |
default | \(-\frac {x}{4}-\frac {5 \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right ) x}{4 \ln \left (x \right )}-\frac {15 x}{4 \ln \left (x \right )}+\frac {x^{3}}{4}+\frac {x^{2}}{4}-\ln \left (x \right )\) | \(42\) |
risch | \(\frac {x^{3}}{4}+\frac {x^{2}}{4}-\frac {x}{4}-\ln \left (x \right )+\frac {5 i x \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+6 i\right )}{8 \ln \left (x \right )}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.32, size = 69, normalized size = 2.38 \begin {gather*} \frac {1}{4} \, x^{3} + \frac {1}{4} \, x^{2} - \frac {5}{4} \, {\rm Ei}\left (\log \left (x\right )\right ) \log \left (x^{2}\right ) + \frac {5}{4} \, \Gamma \left (-1, -\log \left (x\right )\right ) \log \left (x^{2}\right ) + \frac {5}{2} \, {\rm Ei}\left (\log \left (x\right )\right ) \log \left (x\right ) - \frac {5}{2} \, \Gamma \left (-1, -\log \left (x\right )\right ) \log \left (x\right ) - \frac {1}{4} \, x - \frac {15}{4} \, {\rm Ei}\left (\log \left (x\right )\right ) + \frac {15}{4} \, \Gamma \left (-1, -\log \left (x\right )\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 29, normalized size = 1.00 \begin {gather*} \frac {{\left (x^{3} + x^{2} - x\right )} \log \left (x\right ) - 4 \, \log \left (x\right )^{2} - 15 \, x}{4 \, \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 24, normalized size = 0.83 \begin {gather*} \frac {x^{3}}{4} + \frac {x^{2}}{4} - \frac {x}{4} - \frac {15 x}{4 \log {\left (x \right )}} - \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{4} \, x^{3} + \frac {1}{4} \, x^{2} - \frac {1}{4} \, x - \frac {15 \, x}{4 \, \log \left (x\right )} - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.31, size = 35, normalized size = 1.21 \begin {gather*} \frac {9\,x}{4}-\ln \left (x\right )+\frac {x^2}{4}+\frac {x^3}{4}-\frac {\frac {15\,x}{4}+\frac {5\,x\,\ln \left (x^2\right )}{4}}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________