Optimal. Leaf size=37 \[ -\frac {\tan ^{-1}\left (a x^2\right )}{4 a^2}+\frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 37, 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, 321, 203} \[ -\frac {\tan ^{-1}\left (a x^2\right )}{4 a^2}+\frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 275
Rule 321
Rule 5034
Rubi steps
\begin {align*} \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )+\frac {1}{2} a \int \frac {x^5}{1+a^2 x^4} \, dx\\ &=\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )+\frac {1}{4} a \operatorname {Subst}\left (\int \frac {x^2}{1+a^2 x^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{1+a^2 x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )-\frac {\tan ^{-1}\left (a x^2\right )}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ -\frac {\tan ^{-1}\left (a x^2\right )}{4 a^2}+\frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 27, normalized size = 0.73 \[ \frac {a x^{2} + {\left (a^{2} x^{4} + 1\right )} \operatorname {arccot}\left (a x^{2}\right )}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 38, normalized size = 1.03 \[ \frac {1}{4} \, {\left (\frac {x^{4} \arctan \left (\frac {1}{a x^{2}}\right )}{a} + \frac {x^{2}}{a^{2}} + \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{a^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 32, normalized size = 0.86 \[ \frac {x^{2}}{4 a}+\frac {x^{4} \mathrm {arccot}\left (a \,x^{2}\right )}{4}-\frac {\arctan \left (a \,x^{2}\right )}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 34, normalized size = 0.92 \[ \frac {1}{4} \, x^{4} \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{4} \, a {\left (\frac {x^{2}}{a^{2}} - \frac {\arctan \left (a x^{2}\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 31, normalized size = 0.84 \[ \frac {x^4\,\mathrm {acot}\left (a\,x^2\right )}{4}-\frac {\mathrm {atan}\left (a\,x^2\right )}{4\,a^2}+\frac {x^2}{4\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 36, normalized size = 0.97 \[ \begin {cases} \frac {x^{4} \operatorname {acot}{\left (a x^{2} \right )}}{4} + \frac {x^{2}}{4 a} + \frac {\operatorname {acot}{\left (a x^{2} \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi x^{4}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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