Optimal. Leaf size=30 \[ \frac{1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac{\cot ^{-1}(a x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0178592, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4853, 266, 36, 29, 31} \[ \frac{1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac{\cot ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4853
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a x)}{x^2} \, dx &=-\frac{\cot ^{-1}(a x)}{x}-a \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)}{x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{\cot ^{-1}(a x)}{x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{\cot ^{-1}(a x)}{x}-a \log (x)+\frac{1}{2} a \log \left (1+a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0022753, size = 30, normalized size = 1. \[ \frac{1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac{\cot ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 31, normalized size = 1. \begin{align*} -{\frac{{\rm arccot} \left (ax\right )}{x}}+{\frac{a\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2}}-a\ln \left ( ax \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.966614, size = 41, normalized size = 1.37 \begin{align*} \frac{1}{2} \, a{\left (\log \left (a^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} - \frac{\operatorname{arccot}\left (a x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.91349, size = 82, normalized size = 2.73 \begin{align*} \frac{a x \log \left (a^{2} x^{2} + 1\right ) - 2 \, a x \log \left (x\right ) - 2 \, \operatorname{arccot}\left (a x\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.34515, size = 24, normalized size = 0.8 \begin{align*} - a \log{\left (x \right )} + \frac{a \log{\left (a^{2} x^{2} + 1 \right )}}{2} - \frac{\operatorname{acot}{\left (a x \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09297, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{2} \, a{\left (\log \left (a^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} - \frac{\arctan \left (\frac{1}{a x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]