Optimal. Leaf size=24 \[ \frac{2 \text{Ei}(2 \log (c x))}{c^2}-\frac{x^2}{\log (c x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0263859, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2306, 2309, 2178} \[ \frac{2 \text{Ei}(2 \log (c x))}{c^2}-\frac{x^2}{\log (c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2306
Rule 2309
Rule 2178
Rubi steps
\begin{align*} \int \frac{x}{\log ^2(c x)} \, dx &=-\frac{x^2}{\log (c x)}+2 \int \frac{x}{\log (c x)} \, dx\\ &=-\frac{x^2}{\log (c x)}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x}}{x} \, dx,x,\log (c x)\right )}{c^2}\\ &=\frac{2 \text{Ei}(2 \log (c x))}{c^2}-\frac{x^2}{\log (c x)}\\ \end{align*}
Mathematica [A] time = 0.013267, size = 24, normalized size = 1. \[ \frac{2 \text{Ei}(2 \log (c x))}{c^2}-\frac{x^2}{\log (c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 26, normalized size = 1.1 \begin{align*} -{\frac{{x}^{2}}{\ln \left ( cx \right ) }}-2\,{\frac{{\it Ei} \left ( 1,-2\,\ln \left ( cx \right ) \right ) }{{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.25397, size = 18, normalized size = 0.75 \begin{align*} \frac{2 \, \Gamma \left (-1, -2 \, \log \left (c x\right )\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.887326, size = 84, normalized size = 3.5 \begin{align*} -\frac{c^{2} x^{2} - 2 \, \log \left (c x\right ) \logintegral \left (c^{2} x^{2}\right )}{c^{2} \log \left (c x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{x^{2}}{\log{\left (c x \right )}} + 2 \int \frac{x}{\log{\left (c x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14104, size = 32, normalized size = 1.33 \begin{align*} -\frac{x^{2}}{\log \left (c x\right )} + \frac{2 \,{\rm Ei}\left (2 \, \log \left (c x\right )\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]