3.879 \(\int \frac{x^5}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{4} \sin ^{-1}\left (x^2\right )-\frac{1}{4} x^2 \sqrt{1-x^4} \]

[Out]

-(x^2*Sqrt[1 - x^4])/4 + ArcSin[x^2]/4

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Rubi [A]  time = 0.0116118, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {275, 321, 216} \[ \frac{1}{4} \sin ^{-1}\left (x^2\right )-\frac{1}{4} x^2 \sqrt{1-x^4} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[1 - x^4],x]

[Out]

-(x^2*Sqrt[1 - x^4])/4 + ArcSin[x^2]/4

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 321

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x^5}{\sqrt{1-x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{4} x^2 \sqrt{1-x^4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{4} x^2 \sqrt{1-x^4}+\frac{1}{4} \sin ^{-1}\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0054867, size = 27, normalized size = 1. \[ \frac{1}{4} \sin ^{-1}\left (x^2\right )-\frac{1}{4} x^2 \sqrt{1-x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[1 - x^4],x]

[Out]

-(x^2*Sqrt[1 - x^4])/4 + ArcSin[x^2]/4

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Maple [A]  time = 0.008, size = 22, normalized size = 0.8 \begin{align*}{\frac{\arcsin \left ({x}^{2} \right ) }{4}}-{\frac{{x}^{2}}{4}\sqrt{-{x}^{4}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [B]  time = 1.49624, size = 59, normalized size = 2.19 \begin{align*} \frac{\sqrt{-x^{4} + 1}}{4 \, x^{2}{\left (\frac{x^{4} - 1}{x^{4}} - 1\right )}} - \frac{1}{4} \, \arctan \left (\frac{\sqrt{-x^{4} + 1}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.55144, size = 86, normalized size = 3.19 \begin{align*} -\frac{1}{4} \, \sqrt{-x^{4} + 1} x^{2} - \frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{4} + 1} - 1}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.10801, size = 61, normalized size = 2.26 \begin{align*} \begin{cases} - \frac{i x^{2} \sqrt{x^{4} - 1}}{4} - \frac{i \operatorname{acosh}{\left (x^{2} \right )}}{4} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{x^{6}}{4 \sqrt{1 - x^{4}}} - \frac{x^{2}}{4 \sqrt{1 - x^{4}}} + \frac{\operatorname{asin}{\left (x^{2} \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-I*x**2*sqrt(x**4 - 1)/4 - I*acosh(x**2)/4, Abs(x**4) > 1), (x**6/(4*sqrt(1 - x**4)) - x**2/(4*sqrt
(1 - x**4)) + asin(x**2)/4, True))

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Giac [A]  time = 1.2267, size = 28, normalized size = 1.04 \begin{align*} -\frac{1}{4} \, \sqrt{-x^{4} + 1} x^{2} + \frac{1}{4} \, \arcsin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

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