Optimal. Leaf size=34 \[ \frac {1}{4} a \log \left (a^2 x^4+1\right )-\frac {\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5034, 266, 36, 29, 31} \[ \frac {1}{4} a \log \left (a^2 x^4+1\right )-\frac {\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 5034
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}\left (a x^2\right )}{x^3} \, dx &=-\frac {\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \int \frac {1}{x \left (1+a^2 x^4\right )} \, dx\\ &=-\frac {\cot ^{-1}\left (a x^2\right )}{2 x^2}-\frac {1}{4} a \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^4\right )\\ &=-\frac {\cot ^{-1}\left (a x^2\right )}{2 x^2}-\frac {1}{4} a \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )+\frac {1}{4} a^3 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^4\right )\\ &=-\frac {\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x)+\frac {1}{4} a \log \left (1+a^2 x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ \frac {1}{4} a \log \left (a^2 x^4+1\right )-\frac {\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 37, normalized size = 1.09 \[ \frac {a x^{2} \log \left (a^{2} x^{4} + 1\right ) - 4 \, a x^{2} \log \relax (x) - 2 \, \operatorname {arccot}\left (a x^{2}\right )}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 34, normalized size = 1.00 \[ \frac {1}{4} \, a {\left (\log \left (a^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} - \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 0.91 \[ -\frac {\mathrm {arccot}\left (a \,x^{2}\right )}{2 x^{2}}-a \ln \relax (x )+\frac {a \ln \left (a^{2} x^{4}+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 32, normalized size = 0.94 \[ \frac {1}{4} \, a {\left (\log \left (a^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} - \frac {\operatorname {arccot}\left (a x^{2}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 31, normalized size = 0.91 \[ \frac {a\,\ln \left (-a^2\,x^4-1\right )}{4}-\frac {\mathrm {acot}\left (a\,x^2\right )}{2\,x^2}-a\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 29, normalized size = 0.85 \[ - a \log {\relax (x )} + \frac {a \log {\left (a^{2} x^{4} + 1 \right )}}{4} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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