Optimal. Leaf size=24 \[ -2 c^2 \text{Ei}(-2 \log (c x))-\frac{1}{x^2 \log (c x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0370607, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2306, 2309, 2178} \[ -2 c^2 \text{Ei}(-2 \log (c x))-\frac{1}{x^2 \log (c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2306
Rule 2309
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{x^3 \log ^2(c x)} \, dx &=-\frac{1}{x^2 \log (c x)}-2 \int \frac{1}{x^3 \log (c x)} \, dx\\ &=-\frac{1}{x^2 \log (c x)}-\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{x} \, dx,x,\log (c x)\right )\\ &=-2 c^2 \text{Ei}(-2 \log (c x))-\frac{1}{x^2 \log (c x)}\\ \end{align*}
Mathematica [A] time = 0.0145643, size = 24, normalized size = 1. \[ -2 c^2 \text{Ei}(-2 \log (c x))-\frac{1}{x^2 \log (c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.035, size = 26, normalized size = 1.1 \begin{align*} -{\frac{1}{{x}^{2}\ln \left ( cx \right ) }}+2\,{c}^{2}{\it Ei} \left ( 1,2\,\ln \left ( cx \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.2607, size = 18, normalized size = 0.75 \begin{align*} -2 \, c^{2} \Gamma \left (-1, 2 \, \log \left (c x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.686952, size = 92, normalized size = 3.83 \begin{align*} -\frac{2 \, c^{2} x^{2} \log \left (c x\right ) \logintegral \left (\frac{1}{c^{2} x^{2}}\right ) + 1}{x^{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*} - 2 \int \frac{1}{x^{3} \log{\left (c x \right )}}\, dx - \frac{1}{x^{2} \log{\left (c x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \log \left (c x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]