3.10.1 \(\int \frac {x}{(1-x^4)^{3/2}} \, dx\) [901]

Optimal. Leaf size=18 \[ \frac {x^2}{2 \sqrt {1-x^4}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, 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 = {270} \begin {gather*} \frac {x^2}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {x^2}{2 \sqrt {1-x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4])

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Maple [A]
time = 0.15, size = 15, normalized size = 0.83

method	result	size
default	\(\frac {x^{2}}{2 \sqrt {-x^{4}+1}}\)	\(15\)
meijerg	\(\frac {x^{2}}{2 \sqrt {-x^{4}+1}}\)	\(15\)
risch	\(\frac {x^{2}}{2 \sqrt {-x^{4}+1}}\)	\(15\)
elliptic	\(\frac {x^{2}}{2 \sqrt {-x^{4}+1}}\)	\(15\)
trager	\(-\frac {x^{2} \sqrt {-x^{4}+1}}{2 \left (x^{4}-1\right )}\)	\(22\)
gosper	\(-\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) x^{2}}{2 \left (-x^{4}+1\right )^{\frac {3}{2}}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.78 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {-x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

1/2*x^2/sqrt(-x^4 + 1)

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Fricas [A]
time = 0.36, size = 21, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {-x^{4} + 1} x^{2}}{2 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-x^4 + 1)*x^2/(x^4 - 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.29, size = 32, normalized size = 1.78 \begin {gather*} \begin {cases} - \frac {i x^{2}}{2 \sqrt {x^{4} - 1}} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {x^{2}}{2 \sqrt {1 - x^{4}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**4+1)**(3/2),x)

[Out]

Piecewise((-I*x**2/(2*sqrt(x**4 - 1)), Abs(x**4) > 1), (x**2/(2*sqrt(1 - x**4)), True))

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Giac [A]
time = 1.00, size = 21, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {-x^{4} + 1} x^{2}}{2 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-x^4 + 1)*x^2/(x^4 - 1)

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Mupad [B]
time = 1.05, size = 14, normalized size = 0.78 \begin {gather*} \frac {x^2}{2\,\sqrt {1-x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1 - x^4)^(3/2),x)

[Out]

x^2/(2*(1 - x^4)^(1/2))

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