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

Optimal. Leaf size=45 \[ \frac {x^{1-m} \, _2F_1\left (1,\frac {1-m}{4};\frac {5-m}{4};\frac {x^4}{a^4}\right )}{a^4 (1-m)} \]

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 44, normalized size = 0.98 \[ -\frac {x^{1-m} \, _2F_1\left (1,\frac {1}{4}-\frac {m}{4};\frac {5}{4}-\frac {m}{4};\frac {x^4}{a^4}\right )}{a^4 (m-1)} \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-m}}{a^4-x^4} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [F]  time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} - x^{4}\right )} x^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^4 - x^4)*x^m), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{4} - x^{4}\right )} x^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a^4 - x^4)*x^m), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{-m}}{a^{4}-x^{4}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{4} - x^{4}\right )} x^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a^4 - x^4)*x^m), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^m\,\left (a^4-x^4\right )} \,d x \]

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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sympy [C]  time = 1.01, size = 95, normalized size = 2.11 \[ - \frac {m x x^{- m} \Phi \left (\frac {x^{4} e^{2 i \pi }}{a^{4}}, 1, \frac {1}{4} - \frac {m}{4}\right ) \Gamma \left (\frac {1}{4} - \frac {m}{4}\right )}{16 a^{4} \Gamma \left (\frac {5}{4} - \frac {m}{4}\right )} + \frac {x x^{- m} \Phi \left (\frac {x^{4} e^{2 i \pi }}{a^{4}}, 1, \frac {1}{4} - \frac {m}{4}\right ) \Gamma \left (\frac {1}{4} - \frac {m}{4}\right )}{16 a^{4} \Gamma \left (\frac {5}{4} - \frac {m}{4}\right )} \]

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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