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 = 0.0084131, 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, Rules used = {364} \[ \frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{4},\frac{5-m}{4},\frac{x^4}{a^4}\right )}{a^4 (1-m)} \]

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))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{x^{-m}}{a^4-x^4} \, dx &=\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)}\\ \end{align*}

Mathematica [A]  time = 0.0088513, size = 44, normalized size = 0.98 \[ -\frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1}{4}-\frac{m}{4},\frac{5}{4}-\frac{m}{4},\frac{x^4}{a^4}\right )}{a^4 (m-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^m*(a^4 - x^4)),x]

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{m} \left ({a}^{4}-{x}^{4} \right ) }}\, dx \end{align*}

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. \begin{align*} \int \frac{1}{{\left (a^{4} - x^{4}\right )} x^{m}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a^4 - x^4)*x^m), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^4 - x^4)*x^m), x)

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Sympy [C]  time = 0.878606, size = 95, normalized size = 2.11 \begin{align*} - \frac{m x x^{- 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 x^{- 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 )} \end{align*}

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{4} - x^{4}\right )} x^{m}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a^4 - x^4)*x^m), x)