Optimal. Leaf size=15 \[ \frac {\left (-5+e^x\right ) x}{\log \left (4 x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 25, normalized size of antiderivative = 1.67, number of steps
used = 10, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6874, 2407,
2334, 2337, 2209, 2326} \begin {gather*} \frac {e^x x}{\log \left (4 x^2\right )}-\frac {5 x}{\log \left (4 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2209
Rule 2326
Rule 2334
Rule 2337
Rule 2407
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {5 \left (-2+\log \left (4 x^2\right )\right )}{\log ^2\left (4 x^2\right )}+\frac {e^x \left (-2+\log \left (4 x^2\right )+x \log \left (4 x^2\right )\right )}{\log ^2\left (4 x^2\right )}\right ) \, dx\\ &=-\left (5 \int \frac {-2+\log \left (4 x^2\right )}{\log ^2\left (4 x^2\right )} \, dx\right )+\int \frac {e^x \left (-2+\log \left (4 x^2\right )+x \log \left (4 x^2\right )\right )}{\log ^2\left (4 x^2\right )} \, dx\\ &=\frac {e^x x}{\log \left (4 x^2\right )}-5 \int \left (-\frac {2}{\log ^2\left (4 x^2\right )}+\frac {1}{\log \left (4 x^2\right )}\right ) \, dx\\ &=\frac {e^x x}{\log \left (4 x^2\right )}-5 \int \frac {1}{\log \left (4 x^2\right )} \, dx+10 \int \frac {1}{\log ^2\left (4 x^2\right )} \, dx\\ &=-\frac {5 x}{\log \left (4 x^2\right )}+\frac {e^x x}{\log \left (4 x^2\right )}+5 \int \frac {1}{\log \left (4 x^2\right )} \, dx-\frac {(5 x) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (4 x^2\right )\right )}{4 \sqrt {x^2}}\\ &=-\frac {5 x \text {Ei}\left (\frac {1}{2} \log \left (4 x^2\right )\right )}{4 \sqrt {x^2}}-\frac {5 x}{\log \left (4 x^2\right )}+\frac {e^x x}{\log \left (4 x^2\right )}+\frac {(5 x) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (4 x^2\right )\right )}{4 \sqrt {x^2}}\\ &=-\frac {5 x}{\log \left (4 x^2\right )}+\frac {e^x x}{\log \left (4 x^2\right )}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 15, normalized size = 1.00 \begin {gather*} \frac {\left (-5+e^x\right ) x}{\log \left (4 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.66, size = 29, normalized size = 1.93
method | result | size |
norman | \(\frac {{\mathrm e}^{x} x -5 x}{\ln \left (4 x^{2}\right )}\) | \(22\) |
default | \(-\frac {5 x}{\ln \left (4 x^{2}\right )}+\frac {x \,{\mathrm e}^{x}}{\ln \left (4 x^{2}\right )}\) | \(29\) |
risch | \(\frac {2 i x \left ({\mathrm e}^{x}-5\right )}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (2\right )+4 i \ln \left (x \right )}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 17, normalized size = 1.13 \begin {gather*} \frac {x e^{x} - 5 \, x}{2 \, {\left (\log \left (2\right ) + \log \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 17, normalized size = 1.13 \begin {gather*} \frac {x e^{x} - 5 \, x}{\log \left (4 \, x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 20, normalized size = 1.33 \begin {gather*} \frac {x e^{x}}{\log {\left (4 x^{2} \right )}} - \frac {5 x}{\log {\left (4 x^{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 17, normalized size = 1.13 \begin {gather*} \frac {x e^{x} - 5 \, x}{\log \left (4 \, x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.08, size = 14, normalized size = 0.93 \begin {gather*} \frac {x\,\left ({\mathrm {e}}^x-5\right )}{\ln \left (4\,x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________