3.2.33 \(\int \frac {x^{-m}}{a^4-x^4} \, dx\) [133]

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

[Out]

x^(1-m)*hypergeom([1, 1/4-1/4*m],[5/4-1/4*m],x^4/a^4)/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 = {371} \begin {gather*} \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)} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], 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 = 0.16, size = 44, normalized size = 0.98 \begin {gather*} -\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 (-1+m)} \end {gather*}

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 = 0.03, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.44, size = 95, normalized size = 2.11 \begin {gather*} - \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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