3.2.93 \(\int \frac {x}{(1-x^2)^{9/8}} \, dx\) [193]

Optimal. Leaf size=13 \[ \frac {4}{\sqrt [8]{1-x^2}} \]

[Out]

4/(-x^2+1)^(1/8)

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \begin {gather*} \frac {4}{\sqrt [8]{1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(1 - x^2)^(9/8),x]

[Out]

4/(1 - x^2)^(1/8)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x}{\left (1-x^2\right )^{9/8}} \, dx &=\frac {4}{\sqrt [8]{1-x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {4}{\sqrt [8]{1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(1 - x^2)^(9/8),x]

[Out]

4/(1 - x^2)^(1/8)

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Maple [A]
time = 0.05, size = 12, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {4}{\left (-x^{2}+1\right )^{\frac {1}{8}}}\)	\(12\)
default	\(\frac {4}{\left (-x^{2}+1\right )^{\frac {1}{8}}}\)	\(12\)
risch	\(\frac {4}{\left (-x^{2}+1\right )^{\frac {1}{8}}}\)	\(12\)
meijerg	\(\frac {x^{2} \hypergeom \left (\left [1, \frac {9}{8}\right ], \left [2\right ], x^{2}\right )}{2}\)	\(15\)
gosper	\(-\frac {4 \left (-1+x \right ) \left (1+x \right )}{\left (-x^{2}+1\right )^{\frac {9}{8}}}\)	\(18\)
trager	\(-\frac {4 \left (-x^{2}+1\right )^{\frac {7}{8}}}{x^{2}-1}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^2+1)^(9/8),x,method=_RETURNVERBOSE)

[Out]

4/(-x^2+1)^(1/8)

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Maxima [A]
time = 1.48, size = 11, normalized size = 0.85 \begin {gather*} \frac {4}{{\left (-x^{2} + 1\right )}^{\frac {1}{8}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^(9/8),x, algorithm="maxima")

[Out]

4/(-x^2 + 1)^(1/8)

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Fricas [A]
time = 0.76, size = 18, normalized size = 1.38 \begin {gather*} -\frac {4 \, {\left (-x^{2} + 1\right )}^{\frac {7}{8}}}{x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^(9/8),x, algorithm="fricas")

[Out]

-4*(-x^2 + 1)^(7/8)/(x^2 - 1)

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Sympy [A]
time = 0.28, size = 8, normalized size = 0.62 \begin {gather*} \frac {4}{\sqrt [8]{1 - x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**2+1)**(9/8),x)

[Out]

4/(1 - x**2)**(1/8)

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Giac [A]
time = 0.49, size = 11, normalized size = 0.85 \begin {gather*} \frac {4}{{\left (-x^{2} + 1\right )}^{\frac {1}{8}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^(9/8),x, algorithm="giac")

[Out]

4/(-x^2 + 1)^(1/8)

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Mupad [B]
time = 0.35, size = 11, normalized size = 0.85 \begin {gather*} \frac {4}{{\left (1-x^2\right )}^{1/8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1 - x^2)^(9/8),x)

[Out]

4/(1 - x^2)^(1/8)

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