Optimal. Leaf size=46 \[ -\frac{1}{6} a^3 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^3 \log (x)+\frac{a}{6 x^2}-\frac{\cot ^{-1}(a x)}{3 x^3} \]
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Rubi [A] time = 0.0271508, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4853, 266, 44} \[ -\frac{1}{6} a^3 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^3 \log (x)+\frac{a}{6 x^2}-\frac{\cot ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 4853
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a x)}{x^4} \, dx &=-\frac{\cot ^{-1}(a x)}{3 x^3}-\frac{1}{3} a \int \frac{1}{x^3 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)}{3 x^3}-\frac{1}{6} a \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{\cot ^{-1}(a x)}{3 x^3}-\frac{1}{6} a \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{a^2}{x}+\frac{a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{6 x^2}-\frac{\cot ^{-1}(a x)}{3 x^3}+\frac{1}{3} a^3 \log (x)-\frac{1}{6} a^3 \log \left (1+a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0113659, size = 44, normalized size = 0.96 \[ -\frac{1}{6} a \left (a^2 \log \left (a^2 x^2+1\right )-2 a^2 \log (x)-\frac{1}{x^2}\right )-\frac{\cot ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 41, normalized size = 0.9 \begin{align*} -{\frac{{\rm arccot} \left (ax\right )}{3\,{x}^{3}}}-{\frac{{a}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{6}}+{\frac{a}{6\,{x}^{2}}}+{\frac{{a}^{3}\ln \left ( ax \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980529, size = 57, normalized size = 1.24 \begin{align*} -\frac{1}{6} \,{\left (a^{2} \log \left (a^{2} x^{2} + 1\right ) - a^{2} \log \left (x^{2}\right ) - \frac{1}{x^{2}}\right )} a - \frac{\operatorname{arccot}\left (a x\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99549, size = 105, normalized size = 2.28 \begin{align*} -\frac{a^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{3} x^{3} \log \left (x\right ) - a x + 2 \, \operatorname{arccot}\left (a x\right )}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.869719, size = 39, normalized size = 0.85 \begin{align*} \frac{a^{3} \log{\left (x \right )}}{3} - \frac{a^{3} \log{\left (a^{2} x^{2} + 1 \right )}}{6} + \frac{a}{6 x^{2}} - \frac{\operatorname{acot}{\left (a x \right )}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12538, size = 73, normalized size = 1.59 \begin{align*} -\frac{1}{6} \,{\left (a^{2} \log \left (a^{2} x^{2} + 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac{a^{2} x^{2} - 1}{x^{2}}\right )} a - \frac{\arctan \left (\frac{1}{a x}\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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