3.9.96 \(\int \frac {x^3}{(1-x^4)^{3/2}} \, dx\) [896]

Optimal. Leaf size=15 \[ \frac {1}{2 \sqrt {1-x^4}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, 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 = {267} \begin {gather*} \frac {1}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4])

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

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Mathematica [A]
time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4])

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Maple [A]
time = 0.15, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {1}{2 \sqrt {-x^{4}+1}}\)	\(12\)
default	\(\frac {1}{2 \sqrt {-x^{4}+1}}\)	\(12\)
risch	\(\frac {1}{2 \sqrt {-x^{4}+1}}\)	\(12\)
elliptic	\(\frac {1}{2 \sqrt {-x^{4}+1}}\)	\(12\)
trager	\(-\frac {\sqrt {-x^{4}+1}}{2 \left (x^{4}-1\right )}\)	\(19\)
gosper	\(-\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right )}{2 \left (-x^{4}+1\right )^{\frac {3}{2}}}\)	\(23\)
meijerg	\(-\frac {\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-x^{4}+1}}}{2 \sqrt {\pi }}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{2 \, \sqrt {-x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/sqrt(-x^4 + 1)

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Fricas [A]
time = 0.36, size = 18, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {-x^{4} + 1}}{2 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)/(x^4 - 1)

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Sympy [A]
time = 0.14, size = 10, normalized size = 0.67 \begin {gather*} \frac {1}{2 \sqrt {1 - x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(2*sqrt(1 - x**4))

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Giac [A]
time = 1.14, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{2 \, \sqrt {-x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/sqrt(-x^4 + 1)

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Mupad [B]
time = 1.19, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{2\,\sqrt {1-x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(1 - x^4)^(3/2),x)

[Out]

1/(2*(1 - x^4)^(1/2))

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