Integrand size = 17, antiderivative size = 45 \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{4},\frac {5-m}{4},\frac {x^4}{a^4}\right )}{a^4 (1-m)} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {371} \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{4},\frac {5-m}{4},\frac {x^4}{a^4}\right )}{a^4 (1-m)} \]
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Rule 371
Rubi steps \begin{align*} \text {integral}& = \frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{4},\frac {5-m}{4},\frac {x^4}{a^4}\right )}{a^4 (1-m)} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=-\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4}-\frac {m}{4},\frac {5}{4}-\frac {m}{4},\frac {x^4}{a^4}\right )}{a^4 (-1+m)} \]
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\[\int \frac {x^{-m}}{a^{4}-x^{4}}d x\]
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\[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\int { \frac {1}{{\left (a^{4} - x^{4}\right )} x^{m}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.11 \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=- \frac {m x^{1 - m} \Phi \left (\frac {x^{4} e^{2 i \pi }}{a^{4}}, 1, \frac {1}{4} - \frac {m}{4}\right ) \Gamma \left (\frac {1}{4} - \frac {m}{4}\right )}{16 a^{4} \Gamma \left (\frac {5}{4} - \frac {m}{4}\right )} + \frac {x^{1 - m} \Phi \left (\frac {x^{4} e^{2 i \pi }}{a^{4}}, 1, \frac {1}{4} - \frac {m}{4}\right ) \Gamma \left (\frac {1}{4} - \frac {m}{4}\right )}{16 a^{4} \Gamma \left (\frac {5}{4} - \frac {m}{4}\right )} \]
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\[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\int { \frac {1}{{\left (a^{4} - x^{4}\right )} x^{m}} \,d x } \]
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\[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\int { \frac {1}{{\left (a^{4} - x^{4}\right )} x^{m}} \,d x } \]
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Timed out. \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\int \frac {1}{x^m\,\left (a^4-x^4\right )} \,d x \]
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