Optimal. Leaf size=30 \[ \frac {1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac {\cot ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, 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.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4853
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*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.00 \[ \frac {1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac {\cot ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 31, normalized size = 1.03 \[ \frac {a x \log \left (a^{2} x^{2} + 1\right ) - 2 \, a x \log \relax (x) - 2 \, \operatorname {arccot}\left (a x\right )}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 32, normalized size = 1.07 \[ -\frac {1}{2} \, a {\left (\frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x} - \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 1.03 \[ -\frac {\mathrm {arccot}\left (a x \right )}{x}-a \ln \left (a x \right )+\frac {a \ln \left (a^{2} x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 30, normalized size = 1.00 \[ \frac {1}{2} \, a {\left (\log \left (a^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} - \frac {\operatorname {arccot}\left (a x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 28, normalized size = 0.93 \[ \frac {a\,\left (\ln \left (a^2\,x^2+1\right )-2\,\ln \relax (x)\right )}{2}-\frac {\mathrm {acot}\left (a\,x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 24, normalized size = 0.80 \[ - a \log {\relax (x )} + \frac {a \log {\left (a^{2} x^{2} + 1 \right )}}{2} - \frac {\operatorname {acot}{\left (a x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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