Optimal. Leaf size=37 \[ \frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^7}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^7}-\frac {1}{3 a^4 x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {325, 212, 206, 203} \[ -\frac {1}{3 a^4 x^3}+\frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^7}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^7} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx &=-\frac {1}{3 a^4 x^3}+\frac {\int \frac {1}{a^4-x^4} \, dx}{a^4}\\ &=-\frac {1}{3 a^4 x^3}+\frac {\int \frac {1}{a^2-x^2} \, dx}{2 a^6}+\frac {\int \frac {1}{a^2+x^2} \, dx}{2 a^6}\\ &=-\frac {1}{3 a^4 x^3}+\frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^7}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^7}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 48, normalized size = 1.30 \[ -\frac {\log (a-x)}{4 a^7}+\frac {\log (a+x)}{4 a^7}+\frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^7}-\frac {1}{3 a^4 x^3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 37, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^7}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^7}-\frac {1}{3 a^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 45, normalized size = 1.22 \[ \frac {6 \, x^{3} \arctan \left (\frac {x}{a}\right ) + 3 \, x^{3} \log \left (a + x\right ) - 3 \, x^{3} \log \left (-a + x\right ) - 4 \, a^{3}}{12 \, a^{7} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 42, normalized size = 1.14 \[ \frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{7}} + \frac {\log \left ({\left | a + x \right |}\right )}{4 \, a^{7}} - \frac {\log \left ({\left | -a + x \right |}\right )}{4 \, a^{7}} - \frac {1}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 41, normalized size = 1.11
method | result | size |
default | \(\frac {\arctan \left (\frac {x}{a}\right )}{2 a^{7}}-\frac {1}{3 a^{4} x^{3}}+\frac {\ln \left (a +x \right )}{4 a^{7}}-\frac {\ln \left (a -x \right )}{4 a^{7}}\) | \(41\) |
risch | \(-\frac {1}{3 a^{4} x^{3}}-\frac {\ln \left (a -x \right )}{4 a^{7}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{14} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4} a^{28}+4\right ) x -a^{8} \textit {\_R} \right )\right )}{4}+\frac {\ln \left (-a -x \right )}{4 a^{7}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 40, normalized size = 1.08 \[ \frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{7}} + \frac {\log \left (a + x\right )}{4 \, a^{7}} - \frac {\log \left (-a + x\right )}{4 \, a^{7}} - \frac {1}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 31, normalized size = 0.84 \[ \frac {\mathrm {atan}\left (\frac {x}{a}\right )}{2\,a^7}+\frac {\mathrm {atanh}\left (\frac {x}{a}\right )}{2\,a^7}-\frac {1}{3\,a^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.23, size = 48, normalized size = 1.30 \[ - \frac {1}{3 a^{4} x^{3}} - \frac {\frac {\log {\left (- a + x \right )}}{4} - \frac {\log {\left (a + x \right )}}{4} + \frac {i \log {\left (- i a + x \right )}}{4} - \frac {i \log {\left (i a + x \right )}}{4}}{a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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