Optimal. Leaf size=31 \[ \frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right )+\frac {\log \left (1+a^2 x^4\right )}{4 a} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4947, 266}
\begin {gather*} \frac {\log \left (a^2 x^4+1\right )}{4 a}+\frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4947
Rubi steps
\begin {align*} \int x \cot ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right )+a \int \frac {x^3}{1+a^2 x^4} \, dx\\ &=\frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right )+\frac {\log \left (1+a^2 x^4\right )}{4 a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right )+\frac {\log \left (1+a^2 x^4\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 30, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\mathrm {arccot}\left (a \,x^{2}\right ) a \,x^{2}+\frac {\ln \left (a^{2} x^{4}+1\right )}{2}}{2 a}\) | \(30\) |
default | \(\frac {\mathrm {arccot}\left (a \,x^{2}\right ) a \,x^{2}+\frac {\ln \left (a^{2} x^{4}+1\right )}{2}}{2 a}\) | \(30\) |
risch | \(\frac {i x^{2} \ln \left (i a \,x^{2}+1\right )}{4}-\frac {i x^{2} \ln \left (-i a \,x^{2}+1\right )}{4}+\frac {\pi \,x^{2}}{4}+\frac {\ln \left (-a^{2} x^{4}-1\right )}{4 a}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 28, normalized size = 0.90 \begin {gather*} \frac {2 \, a x^{2} \operatorname {arccot}\left (a x^{2}\right ) + \log \left (a^{2} x^{4} + 1\right )}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.42, size = 28, normalized size = 0.90 \begin {gather*} \frac {2 \, a x^{2} \operatorname {arccot}\left (a x^{2}\right ) + \log \left (a^{2} x^{4} + 1\right )}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 31, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {acot}{\left (a x^{2} \right )}}{2} + \frac {\log {\left (a^{2} x^{4} + 1 \right )}}{4 a} & \text {for}\: a \neq 0 \\\frac {\pi x^{2}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 47, normalized size = 1.52 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, x^{2} \arctan \left (\frac {1}{a x^{2}}\right )}{a} + \frac {\log \left (\frac {1}{a^{2} x^{4}} + 1\right )}{a^{2}} - \frac {\log \left (\frac {1}{a^{2} x^{4}}\right )}{a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 27, normalized size = 0.87 \begin {gather*} \frac {x^2\,\mathrm {acot}\left (a\,x^2\right )}{2}+\frac {\ln \left (a^2\,x^4+1\right )}{4\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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