\(\int \frac {x^{-m}}{a^3+x^3} \, dx\) [126]

Optimal result

Integrand size = 15, antiderivative size = 46 \[ \int \frac {x^{-m}}{a^3+x^3} \, dx=\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{3},\frac {4-m}{3},-\frac {x^3}{a^3}\right )}{a^3 (1-m)} \]

[Out]

x^(1-m)*hypergeom([1, 1/3-1/3*m],[4/3-1/3*m],-x^3/a^3)/a^3/(1-m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {371} \[ \int \frac {x^{-m}}{a^3+x^3} \, dx=\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{3},\frac {4-m}{3},-\frac {x^3}{a^3}\right )}{a^3 (1-m)} \]

[In]

Int[1/(x^m*(a^3 + x^3)),x]

[Out]

(x^(1 - m)*Hypergeometric2F1[1, (1 - m)/3, (4 - m)/3, -(x^3/a^3)])/(a^3*(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}{3},\frac {4-m}{3},-\frac {x^3}{a^3}\right )}{a^3 (1-m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {x^{-m}}{a^3+x^3} \, dx=-\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{3}-\frac {m}{3},\frac {4}{3}-\frac {m}{3},-\frac {x^3}{a^3}\right )}{a^3 (-1+m)} \]

[In]

Integrate[1/(x^m*(a^3 + x^3)),x]

[Out]

-((x^(1 - m)*Hypergeometric2F1[1, 1/3 - m/3, 4/3 - m/3, -(x^3/a^3)])/(a^3*(-1 + m)))

Maple [F]

\[\int \frac {x^{-m}}{a^{3}+x^{3}}d x\]

[In]

int(1/(x^m)/(a^3+x^3),x)

[Out]

int(1/(x^m)/(a^3+x^3),x)

Fricas [F]

\[ \int \frac {x^{-m}}{a^3+x^3} \, dx=\int { \frac {1}{{\left (a^{3} + x^{3}\right )} x^{m}} \,d x } \]

[In]

integrate(1/(x^m)/(a^3+x^3),x, algorithm="fricas")

[Out]

integral(1/((a^3 + x^3)*x^m), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.00 \[ \int \frac {x^{-m}}{a^3+x^3} \, dx=- \frac {m x^{1 - m} \Phi \left (\frac {x^{3} e^{i \pi }}{a^{3}}, 1, \frac {1}{3} - \frac {m}{3}\right ) \Gamma \left (\frac {1}{3} - \frac {m}{3}\right )}{9 a^{3} \Gamma \left (\frac {4}{3} - \frac {m}{3}\right )} + \frac {x^{1 - m} \Phi \left (\frac {x^{3} e^{i \pi }}{a^{3}}, 1, \frac {1}{3} - \frac {m}{3}\right ) \Gamma \left (\frac {1}{3} - \frac {m}{3}\right )}{9 a^{3} \Gamma \left (\frac {4}{3} - \frac {m}{3}\right )} \]

[In]

integrate(1/(x**m)/(a**3+x**3),x)

[Out]

-m*x**(1 - m)*lerchphi(x**3*exp_polar(I*pi)/a**3, 1, 1/3 - m/3)*gamma(1/3 - m/3)/(9*a**3*gamma(4/3 - m/3)) + x
**(1 - m)*lerchphi(x**3*exp_polar(I*pi)/a**3, 1, 1/3 - m/3)*gamma(1/3 - m/3)/(9*a**3*gamma(4/3 - m/3))

Maxima [F]

\[ \int \frac {x^{-m}}{a^3+x^3} \, dx=\int { \frac {1}{{\left (a^{3} + x^{3}\right )} x^{m}} \,d x } \]

[In]

integrate(1/(x^m)/(a^3+x^3),x, algorithm="maxima")

[Out]

integrate(1/((a^3 + x^3)*x^m), x)

Giac [F]

\[ \int \frac {x^{-m}}{a^3+x^3} \, dx=\int { \frac {1}{{\left (a^{3} + x^{3}\right )} x^{m}} \,d x } \]

[In]

integrate(1/(x^m)/(a^3+x^3),x, algorithm="giac")

[Out]

integrate(1/((a^3 + x^3)*x^m), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-m}}{a^3+x^3} \, dx=\int \frac {1}{x^m\,\left (a^3+x^3\right )} \,d x \]

[In]

int(1/(x^m*(a^3 + x^3)),x)

[Out]

int(1/(x^m*(a^3 + x^3)), x)