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

Optimal. Leaf size=46 \[ \frac {x^{1-m} \, _2F_1\left (1,\frac {1-m}{3};\frac {4-m}{3};-\frac {x^3}{a^3}\right )}{a^3 (1-m)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {371} \begin {gather*} \frac {x^{1-m} \text {Hypergeometric2F1}\left (1,\frac {1-m}{3},\frac {4-m}{3},-\frac {x^3}{a^3}\right )}{a^3 (1-m)} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {x^{-m}}{a^3+x^3} \, dx &=\frac {x^{1-m} \, _2F_1\left (1,\frac {1-m}{3};\frac {4-m}{3};-\frac {x^3}{a^3}\right )}{a^3 (1-m)}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 45, normalized size = 0.98 \begin {gather*} -\frac {x^{1-m} \, _2F_1\left (1,\frac {1}{3}-\frac {m}{3};\frac {4}{3}-\frac {m}{3};-\frac {x^3}{a^3}\right )}{a^3 (-1+m)} \end {gather*}

Antiderivative was successfully verified.

[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)))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{-m}}{a^{3}+x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(1/((a^3 + x^3)*x^m), x)

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Sympy [C] Result contains complex when optimal does not.
time = 1.23, size = 92, normalized size = 2.00 \begin {gather*} - \frac {m x x^{- 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 x^{- 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^3+x^3),x, algorithm="giac")

[Out]

integrate(1/((a^3 + x^3)*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^3+x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^m*(a^3 + x^3)),x)

[Out]

int(1/(x^m*(a^3 + x^3)), x)

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