Optimal. Leaf size=31 \[ \frac {a}{2 x}-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \text {ArcTan}(a x) \]
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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 = {4947, 331, 209}
\begin {gather*} \frac {1}{2} a^2 \text {ArcTan}(a x)-\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {a}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 331
Rule 4947
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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.00, size = 36, normalized size = 1.16 \begin {gather*} -\frac {\cot ^{-1}(a x)}{2 x^2}+\frac {a \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-a^2 x^2\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 32, normalized size = 1.03
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\mathrm {arccot}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}\right )\) | \(32\) |
default | \(a^{2} \left (-\frac {\mathrm {arccot}\left (a x \right )}{2 a^{2} x^{2}}+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}\right )\) | \(32\) |
risch | \(-\frac {i \ln \left (i a x +1\right )}{4 x^{2}}-\frac {i a^{2} \ln \left (-a x +i\right ) x^{2}-i a^{2} \ln \left (-a x -i\right ) x^{2}-i \ln \left (-i a x +1\right )-2 a x +\pi }{4 x^{2}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 23, normalized size = 0.74 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.57, size = 24, normalized size = 0.77 \begin {gather*} \frac {a x - {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 24, normalized size = 0.77 \begin {gather*} - \frac {a^{2} \operatorname {acot}{\left (a x \right )}}{2} + \frac {a}{2 x} - \frac {\operatorname {acot}{\left (a x \right )}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 40, normalized size = 1.29 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \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 . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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