Optimal. Leaf size=34 \[ \frac{1}{4} a \log \left (a^2 x^4+1\right )-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0173108, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5034, 266, 36, 29, 31} \[ \frac{1}{4} a \log \left (a^2 x^4+1\right )-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5034
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}\left (a x^2\right )}{x^3} \, dx &=-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \int \frac{1}{x \left (1+a^2 x^4\right )} \, dx\\ &=-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^4\right )\\ &=-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^4\right )+\frac{1}{4} a^3 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^4\right )\\ &=-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x)+\frac{1}{4} a \log \left (1+a^2 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0058354, size = 34, normalized size = 1. \[ \frac{1}{4} a \log \left (a^2 x^4+1\right )-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 31, normalized size = 0.9 \begin{align*} -{\frac{{\rm arccot} \left (a{x}^{2}\right )}{2\,{x}^{2}}}-a\ln \left ( x \right ) +{\frac{a\ln \left ({a}^{2}{x}^{4}+1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.965668, size = 43, normalized size = 1.26 \begin{align*} \frac{1}{4} \, a{\left (\log \left (a^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} - \frac{\operatorname{arccot}\left (a x^{2}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.29115, size = 93, normalized size = 2.74 \begin{align*} \frac{a x^{2} \log \left (a^{2} x^{4} + 1\right ) - 4 \, a x^{2} \log \left (x\right ) - 2 \, \operatorname{arccot}\left (a x^{2}\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.05433, size = 29, normalized size = 0.85 \begin{align*} - a \log{\left (x \right )} + \frac{a \log{\left (a^{2} x^{4} + 1 \right )}}{4} - \frac{\operatorname{acot}{\left (a x^{2} \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14325, size = 46, normalized size = 1.35 \begin{align*} \frac{1}{4} \, a{\left (\log \left (a^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} - \frac{\arctan \left (\frac{1}{a x^{2}}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]