Optimal. Leaf size=46 \[ \frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (a^2 x^2+1\right )-\frac {\cot ^{-1}(a x)}{3 x^3}+\frac {a}{6 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 44
Rule 266
Rule 4853
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.62, size = 43, normalized size = 0.93 \[ -\frac {a^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{3} x^{3} \log \relax (x) - a x + 2 \, \operatorname {arccot}\left (a x\right )}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 44, normalized size = 0.96 \[ \frac {1}{6} \, {\left (a^{2} {\left (\frac {1}{a^{2} x^{2}} - \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} - \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.89 \[ -\frac {\mathrm {arccot}\left (a x \right )}{3 x^{3}}+\frac {a}{6 x^{2}}+\frac {a^{3} \ln \left (a x \right )}{3}-\frac {a^{3} \ln \left (a^{2} x^{2}+1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 42, normalized size = 0.91 \[ -\frac {1}{6} \, {\left (a^{2} \log \left (a^{2} x^{2} + 1\right ) - a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} a - \frac {\operatorname {arccot}\left (a x\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 58, normalized size = 1.26 \[ \left \{\begin {array}{cl} -\frac {\pi }{6\,x^3} & \text {\ if\ \ }a=0\\ \frac {a^4\,\ln \relax (x)-\frac {a^4\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {a^2}{2\,x^2}}{3\,a}-\frac {\mathrm {acot}\left (a\,x\right )}{3\,x^3} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 39, normalized size = 0.85 \[ \frac {a^{3} \log {\relax (x )}}{3} - \frac {a^{3} \log {\left (a^{2} x^{2} + 1 \right )}}{6} + \frac {a}{6 x^{2}} - \frac {\operatorname {acot}{\left (a x \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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