Integrand size = 11, antiderivative size = 27 \[ \int \frac {1}{a^4-x^4} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{2 a^3}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^3} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {218, 212, 209} \[ \int \frac {1}{a^4-x^4} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{2 a^3}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^3} \]
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Rule 209
Rule 212
Rule 218
Rubi steps \begin{align*} \text {integral}& = \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 {\arctan \left (\frac {x}{a}\right )}{2 a^3}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {1}{a^4-x^4} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{2 a^3}-\frac {\log (a-x)}{4 a^3}+\frac {\log (a+x)}{4 a^3} \]
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22
method | result | size |
default | \(-\frac {\ln \left (a -x \right )}{4 a^{3}}+\frac {\arctan \left (\frac {x}{a}\right )}{2 a^{3}}+\frac {\ln \left (a +x \right )}{4 a^{3}}\) | \(33\) |
parallelrisch | \(-\frac {i \ln \left (-i a +x \right )-i \ln \left (i a +x \right )+\ln \left (-a +x \right )-\ln \left (a +x \right )}{4 a^{3}}\) | \(39\) |
risch | \(\frac {\ln \left (a +x \right )}{4 a^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{6} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} \,a^{4}+x \right )\right )}{4}-\frac {\ln \left (-a +x \right )}{4 a^{3}}\) | \(47\) |
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none
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {1}{a^4-x^4} \, dx=\frac {2 \, \arctan \left (\frac {x}{a}\right ) + \log \left (a + x\right ) - \log \left (-a + x\right )}{4 \, a^{3}} \]
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Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {1}{a^4-x^4} \, dx=- \frac {\frac {\log {\left (- a + x \right )}}{4} - \frac {\log {\left (a + x \right )}}{4} + \frac {i \log {\left (- i a + x \right )}}{4} - \frac {i \log {\left (i a + x \right )}}{4}}{a^{3}} \]
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none
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {1}{a^4-x^4} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{3}} + \frac {\log \left (a + x\right )}{4 \, a^{3}} - \frac {\log \left (-a + x\right )}{4 \, a^{3}} \]
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none
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {1}{a^4-x^4} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{3}} + \frac {\log \left ({\left | a + x \right |}\right )}{4 \, a^{3}} - \frac {\log \left ({\left | -a + x \right |}\right )}{4 \, a^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {1}{a^4-x^4} \, dx=\frac {\mathrm {atan}\left (\frac {x}{a}\right )+\mathrm {atanh}\left (\frac {x}{a}\right )}{2\,a^3} \]
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