Optimal. Leaf size=31 \[ \frac {1}{2} a^2 \tan ^{-1}(a x)-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {a}{2 x} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4853, 325, 203} \[ \frac {1}{2} a^2 \tan ^{-1}(a x)-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {a}{2 x} \]
Antiderivative was successfully verified.
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Rule 203
Rule 325
Rule 4853
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)}{x^3} \, dx &=-\frac {\cot ^{-1}(a x)}{2 x^2}-\frac {1}{2} a \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=\frac {a}{2 x}-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^3 \int \frac {1}{1+a^2 x^2} \, dx\\ &=\frac {a}{2 x}-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \tan ^{-1}(a x)\\ \end {align*}
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Mathematica [C] time = 0.00, size = 36, normalized size = 1.16 \[ \frac {a \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-a^2 x^2\right )}{2 x}-\frac {\cot ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 24, normalized size = 0.77 \[ \frac {a x - {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 40, normalized size = 1.29 \[ \frac {1}{2} \, {\left (a {\left (\frac {1}{a x} - \arctan \left (\frac {1}{a x}\right )\right )} - \frac {\arctan \left (\frac {1}{a x}\right )}{a x^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 0.84 \[ \frac {a}{2 x}-\frac {\mathrm {arccot}\left (a x \right )}{2 x^{2}}+\frac {a^{2} \arctan \left (a x \right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 23, normalized size = 0.74 \[ \frac {1}{2} \, {\left (a \arctan \left (a x\right ) + \frac {1}{x}\right )} a - \frac {\operatorname {arccot}\left (a x\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 44, normalized size = 1.42 \[ \left \{\begin {array}{cl} -\frac {\pi }{4\,x^2} & \text {\ if\ \ }a=0\\ \frac {a^3\,\mathrm {atan}\left (a\,x\right )+\frac {a^2}{x}}{2\,a}-\frac {\mathrm {acot}\left (a\,x\right )}{2\,x^2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 24, normalized size = 0.77 \[ - \frac {a^{2} \operatorname {acot}{\left (a x \right )}}{2} + \frac {a}{2 x} - \frac {\operatorname {acot}{\left (a x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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