3.133 \(\int \frac{x^{-m}}{a^4-x^4} \, dx\)

Optimal. Leaf size=45 \[ \frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{4},\frac{5-m}{4},\frac{x^4}{a^4}\right )}{a^4 (1-m)} \]

[Out]

(x^(1 - m)*Hypergeometric2F1[1, (1 - m)/4, (5 - m)/4, x^4/a^4])/(a^4*(1 - m))

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Rubi [A]  time = 0.026629, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{x^{1-m} \, _2F_1\left (1,\frac{1-m}{4};\frac{5-m}{4};\frac{x^4}{a^4}\right )}{a^4 (1-m)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^m*(a^4 - x^4)),x]

[Out]

(x^(1 - m)*Hypergeometric2F1[1, (1 - m)/4, (5 - m)/4, x^4/a^4])/(a^4*(1 - m))

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Rubi in Sympy [A]  time = 2.28221, size = 29, normalized size = 0.64 \[ \frac{x^{- m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{m}{4} + \frac{1}{4} \\ - \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{x^{4}}{a^{4}}} \right )}}{a^{4} \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**m)/(a**4-x**4),x)

[Out]

x**(-m + 1)*hyper((1, -m/4 + 1/4), (-m/4 + 5/4,), x**4/a**4)/(a**4*(-m + 1))

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Mathematica [A]  time = 0.0271733, size = 46, normalized size = 1.02 \[ -\frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{4},\frac{1-m}{4}+1,\frac{x^4}{a^4}\right )}{a^4 (m-1)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^m*(a^4 - x^4)),x]

[Out]

-((x^(1 - m)*Hypergeometric2F1[1, (1 - m)/4, 1 + (1 - m)/4, x^4/a^4])/(a^4*(-1 +
m)))

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{m} \left ({a}^{4}-{x}^{4} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^m)/(a^4-x^4),x)

[Out]

int(1/(x^m)/(a^4-x^4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-m}}{a^{4} - x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^4 - x^4)*x^m),x, algorithm="maxima")

[Out]

integrate(x^(-m)/(a^4 - x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (a^{4} - x^{4}\right )} x^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^4 - x^4)*x^m),x, algorithm="fricas")

[Out]

integral(1/((a^4 - x^4)*x^m), x)

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Sympy [A]  time = 1.51561, size = 95, normalized size = 2.11 \[ - \frac{m x x^{- m} \Phi \left (\frac{x^{4} e^{2 i \pi }}{a^{4}}, 1, - \frac{m}{4} + \frac{1}{4}\right ) \Gamma \left (- \frac{m}{4} + \frac{1}{4}\right )}{16 a^{4} \Gamma \left (- \frac{m}{4} + \frac{5}{4}\right )} + \frac{x x^{- m} \Phi \left (\frac{x^{4} e^{2 i \pi }}{a^{4}}, 1, - \frac{m}{4} + \frac{1}{4}\right ) \Gamma \left (- \frac{m}{4} + \frac{1}{4}\right )}{16 a^{4} \Gamma \left (- \frac{m}{4} + \frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**m)/(a**4-x**4),x)

[Out]

-m*x*x**(-m)*lerchphi(x**4*exp_polar(2*I*pi)/a**4, 1, -m/4 + 1/4)*gamma(-m/4 + 1
/4)/(16*a**4*gamma(-m/4 + 5/4)) + x*x**(-m)*lerchphi(x**4*exp_polar(2*I*pi)/a**4
, 1, -m/4 + 1/4)*gamma(-m/4 + 1/4)/(16*a**4*gamma(-m/4 + 5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a^{4} - x^{4}\right )} x^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^4 - x^4)*x^m),x, algorithm="giac")

[Out]

integrate(1/((a^4 - x^4)*x^m), x)