3.1541 \(\int \frac{x^4}{\sqrt{1-x^{10}}} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{5} \sin ^{-1}\left (x^5\right ) \]

[Out]

ArcSin[x^5]/5

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Rubi [A]  time = 0.0040674, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {275, 216} \[ \frac{1}{5} \sin ^{-1}\left (x^5\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[1 - x^10],x]

[Out]

ArcSin[x^5]/5

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 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^4}{\sqrt{1-x^{10}}} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,x^5\right )\\ &=\frac{1}{5} \sin ^{-1}\left (x^5\right )\\ \end{align*}

Mathematica [A]  time = 0.0029524, size = 8, normalized size = 1. \[ \frac{1}{5} \sin ^{-1}\left (x^5\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[1 - x^10],x]

[Out]

ArcSin[x^5]/5

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Maple [A]  time = 0.035, size = 7, normalized size = 0.9 \begin{align*}{\frac{\arcsin \left ({x}^{5} \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(-x^10+1)^(1/2),x)

[Out]

1/5*arcsin(x^5)

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Maxima [B]  time = 1.45228, size = 22, normalized size = 2.75 \begin{align*} -\frac{1}{5} \, \arctan \left (\frac{\sqrt{-x^{10} + 1}}{x^{5}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^10+1)^(1/2),x, algorithm="maxima")

[Out]

-1/5*arctan(sqrt(-x^10 + 1)/x^5)

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Fricas [B]  time = 1.53108, size = 54, normalized size = 6.75 \begin{align*} -\frac{2}{5} \, \arctan \left (\frac{\sqrt{-x^{10} + 1} - 1}{x^{5}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^10+1)^(1/2),x, algorithm="fricas")

[Out]

-2/5*arctan((sqrt(-x^10 + 1) - 1)/x^5)

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Sympy [A]  time = 0.98542, size = 19, normalized size = 2.38 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (x^{5} \right )}}{5} & \text{for}\: \left |{x^{10}}\right | > 1 \\\frac{\operatorname{asin}{\left (x^{5} \right )}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(-x**10+1)**(1/2),x)

[Out]

Piecewise((-I*acosh(x**5)/5, Abs(x**10) > 1), (asin(x**5)/5, True))

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Giac [A]  time = 1.24473, size = 8, normalized size = 1. \begin{align*} \frac{1}{5} \, \arcsin \left (x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^10+1)^(1/2),x, algorithm="giac")

[Out]

1/5*arcsin(x^5)