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

Optimal result

Integrand size = 17, antiderivative size = 45 \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{4},\frac {5-m}{4},\frac {x^4}{a^4}\right )}{a^4 (1-m)} \]

[Out]

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

Rubi [A] (verified)

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} \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{4},\frac {5-m}{4},\frac {x^4}{a^4}\right )}{a^4 (1-m)} \]

[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*} \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*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=-\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4}-\frac {m}{4},\frac {5}{4}-\frac {m}{4},\frac {x^4}{a^4}\right )}{a^4 (-1+m)} \]

[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)))

Maple [F]

\[\int \frac {x^{-m}}{a^{4}-x^{4}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(1/((a^4 - x^4)*x^m), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.11 \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=- \frac {m x^{1 - m} \Phi \left (\frac {x^{4} e^{2 i \pi }}{a^{4}}, 1, \frac {1}{4} - \frac {m}{4}\right ) \Gamma \left (\frac {1}{4} - \frac {m}{4}\right )}{16 a^{4} \Gamma \left (\frac {5}{4} - \frac {m}{4}\right )} + \frac {x^{1 - m} \Phi \left (\frac {x^{4} e^{2 i \pi }}{a^{4}}, 1, \frac {1}{4} - \frac {m}{4}\right ) \Gamma \left (\frac {1}{4} - \frac {m}{4}\right )}{16 a^{4} \Gamma \left (\frac {5}{4} - \frac {m}{4}\right )} \]

[In]

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

[Out]

-m*x**(1 - 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**(1 - 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))

Maxima [F]

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

[In]

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

[Out]

integrate(1/((a^4 - x^4)*x^m), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/((a^4 - x^4)*x^m), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-m}}{a^4-x^4} \, dx=\int \frac {1}{x^m\,\left (a^4-x^4\right )} \,d x \]

[In]

int(1/(x^m*(a^4 - x^4)),x)

[Out]

int(1/(x^m*(a^4 - x^4)), x)