Optimal. Leaf size=35 \[ \frac {1}{4} a^2 \tan ^{-1}\left (a x^2\right )+\frac {a}{4 x^2}-\frac {\cot ^{-1}\left (a x^2\right )}{4 x^4} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5034, 275, 325, 203} \[ \frac {1}{4} a^2 \tan ^{-1}\left (a x^2\right )+\frac {a}{4 x^2}-\frac {\cot ^{-1}\left (a x^2\right )}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 203
Rule 275
Rule 325
Rule 5034
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}\left (a x^2\right )}{x^5} \, dx &=-\frac {\cot ^{-1}\left (a x^2\right )}{4 x^4}-\frac {1}{2} a \int \frac {1}{x^3 \left (1+a^2 x^4\right )} \, dx\\ &=-\frac {\cot ^{-1}\left (a x^2\right )}{4 x^4}-\frac {1}{4} a \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx,x,x^2\right )\\ &=\frac {a}{4 x^2}-\frac {\cot ^{-1}\left (a x^2\right )}{4 x^4}+\frac {1}{4} a^3 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x^2} \, dx,x,x^2\right )\\ &=\frac {a}{4 x^2}-\frac {\cot ^{-1}\left (a x^2\right )}{4 x^4}+\frac {1}{4} a^2 \tan ^{-1}\left (a x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 1.09 \[ \frac {a \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-a^2 x^4\right )}{4 x^2}-\frac {\cot ^{-1}\left (a x^2\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 28, normalized size = 0.80 \[ \frac {a x^{2} - {\left (a^{2} x^{4} + 1\right )} \operatorname {arccot}\left (a x^{2}\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 29, normalized size = 0.83 \[ \frac {1}{4} \, {\left (a \arctan \left (a x^{2}\right ) + \frac {1}{x^{2}}\right )} a - \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.86 \[ \frac {a}{4 x^{2}}-\frac {\mathrm {arccot}\left (a \,x^{2}\right )}{4 x^{4}}+\frac {a^{2} \arctan \left (a \,x^{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 27, normalized size = 0.77 \[ \frac {1}{4} \, {\left (a \arctan \left (a x^{2}\right ) + \frac {1}{x^{2}}\right )} a - \frac {\operatorname {arccot}\left (a x^{2}\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 32, normalized size = 0.91 \[ \frac {a\,x^2-\mathrm {acot}\left (a\,x^2\right )+a^2\,x^4\,\mathrm {atan}\left (a\,x^2\right )}{4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.05, size = 29, normalized size = 0.83 \[ - \frac {a^{2} \operatorname {acot}{\left (a x^{2} \right )}}{4} + \frac {a}{4 x^{2}} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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