Optimal. Leaf size=27 \[ \frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^3}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {218, 212, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {x}{a}\right )}{2 a^3}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rubi steps
\begin {align*} \int \frac {1}{a^4-x^4} \, dx &=\frac {\int \frac {1}{a^2-x^2} \, dx}{2 a^2}+\frac {\int \frac {1}{a^2+x^2} \, dx}{2 a^2}\\ &=\frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^3}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 38, normalized size = 1.41 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^3}-\frac {\log (a-x)}{4 a^3}+\frac {\log (a+x)}{4 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 33, normalized size = 1.22
method | result | size |
default | \(\frac {\arctan \left (\frac {x}{a}\right )}{2 a^{3}}+\frac {\ln \left (a +x \right )}{4 a^{3}}-\frac {\ln \left (a -x \right )}{4 a^{3}}\) | \(33\) |
risch | \(\frac {\ln \left (a +x \right )}{4 a^{3}}-\frac {\ln \left (-a +x \right )}{4 a^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{6} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} \,a^{4}+x \right )\right )}{4}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.77, size = 32, normalized size = 1.19 \begin {gather*} \frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{3}} + \frac {\log \left (a + x\right )}{4 \, a^{3}} - \frac {\log \left (-a + x\right )}{4 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 26, normalized size = 0.96 \begin {gather*} \frac {2 \, \arctan \left (\frac {x}{a}\right ) + \log \left (a + x\right ) - \log \left (-a + x\right )}{4 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.05, size = 37, normalized size = 1.37 \begin {gather*} - \frac {\frac {\log {\left (- a + x \right )}}{4} - \frac {\log {\left (a + x \right )}}{4} + \frac {i \log {\left (- i a + x \right )}}{4} - \frac {i \log {\left (i a + x \right )}}{4}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 34, normalized size = 1.26 \begin {gather*} \frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{3}} + \frac {\log \left ({\left | a + x \right |}\right )}{4 \, a^{3}} - \frac {\log \left ({\left | -a + x \right |}\right )}{4 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 18, normalized size = 0.67 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x}{a}\right )+\mathrm {atanh}\left (\frac {x}{a}\right )}{2\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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