3.896 \(\int \frac{x^3}{(1-x^4)^{3/2}} \, dx\)

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

[Out]

1/(2*Sqrt[1 - x^4])

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Rubi [A]  time = 0.0036682, antiderivative size = 15, normalized size of antiderivative = 1., 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 = {261} \[ \frac{1}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4])

Rule 261

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*}

Mathematica [A]  time = 0.0022778, size = 15, normalized size = 1. \[ \frac{1}{2 \sqrt{1-x^4}} \]

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 = 0.005, size = 23, normalized size = 1.5 \begin{align*} -{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) }{2} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-1+x)*(1+x)*(x^2+1)/(-x^4+1)^(3/2)

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Maxima [A]  time = 1.03397, size = 15, normalized size = 1. \begin{align*} \frac{1}{2 \, \sqrt{-x^{4} + 1}} \end{align*}

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 = 1.5437, size = 42, normalized size = 2.8 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{2 \,{\left (x^{4} - 1\right )}} \end{align*}

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 = 0.418224, size = 10, normalized size = 0.67 \begin{align*} \frac{1}{2 \sqrt{1 - x^{4}}} \end{align*}

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 = 1.11264, size = 15, normalized size = 1. \begin{align*} \frac{1}{2 \, \sqrt{-x^{4} + 1}} \end{align*}

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)