3.7.55 \(\int x (1-x^2)^{3/2} \sin ^{-1}(x) \, dx\) [655]

Optimal. Leaf size=37 \[ \frac {x}{5}-\frac {2 x^3}{15}+\frac {x^5}{25}-\frac {1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x) \]

[Out]

1/5*x-2/15*x^3+1/25*x^5-1/5*(-x^2+1)^(5/2)*arcsin(x)

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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4767, 200} \begin {gather*} -\frac {1}{5} \left (1-x^2\right )^{5/2} \text {ArcSin}(x)+\frac {x^5}{25}-\frac {2 x^3}{15}+\frac {x}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(1 - x^2)^(3/2)*ArcSin[x],x]

[Out]

x/5 - (2*x^3)/15 + x^5/25 - ((1 - x^2)^(5/2)*ArcSin[x])/5

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \left (1-x^2\right )^{3/2} \sin ^{-1}(x) \, dx &=-\frac {1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x)+\frac {1}{5} \int \left (1-x^2\right )^2 \, dx\\ &=-\frac {1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x)+\frac {1}{5} \int \left (1-2 x^2+x^4\right ) \, dx\\ &=\frac {x}{5}-\frac {2 x^3}{15}+\frac {x^5}{25}-\frac {1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.89 \begin {gather*} \frac {1}{75} \left (15 x-10 x^3+3 x^5-15 \left (1-x^2\right )^{5/2} \sin ^{-1}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(1 - x^2)^(3/2)*ArcSin[x],x]

[Out]

(15*x - 10*x^3 + 3*x^5 - 15*(1 - x^2)^(5/2)*ArcSin[x])/75

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Maple [A]
time = 0.11, size = 37, normalized size = 1.00

method	result	size
default	\(-\frac {\left (x^{2}-1\right )^{2} \sqrt {-x^{2}+1}\, \arcsin \left (x \right )}{5}+\frac {\left (3 x^{4}-10 x^{2}+15\right ) x}{75}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(x^2-1)^2*(-x^2+1)^(1/2)*arcsin(x)+1/75*(3*x^4-10*x^2+15)*x

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Maxima [A]
time = 0.82, size = 27, normalized size = 0.73 \begin {gather*} \frac {1}{25} \, x^{5} - \frac {1}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} \arcsin \left (x\right ) - \frac {2}{15} \, x^{3} + \frac {1}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^5 - 1/5*(-x^2 + 1)^(5/2)*arcsin(x) - 2/15*x^3 + 1/5*x

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Fricas [A]
time = 0.48, size = 37, normalized size = 1.00 \begin {gather*} \frac {1}{25} \, x^{5} - \frac {2}{15} \, x^{3} - \frac {1}{5} \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) + \frac {1}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^5 - 2/15*x^3 - 1/5*(x^4 - 2*x^2 + 1)*sqrt(-x^2 + 1)*arcsin(x) + 1/5*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
time = 0.49, size = 63, normalized size = 1.70 \begin {gather*} \frac {x^{5}}{25} - \frac {x^{4} \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{5} - \frac {2 x^{3}}{15} + \frac {2 x^{2} \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{5} + \frac {x}{5} - \frac {\sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5/25 - x**4*sqrt(1 - x**2)*asin(x)/5 - 2*x**3/15 + 2*x**2*sqrt(1 - x**2)*asin(x)/5 + x/5 - sqrt(1 - x**2)*a
sin(x)/5

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Giac [A]
time = 1.05, size = 34, normalized size = 0.92 \begin {gather*} \frac {1}{25} \, x^{5} - \frac {1}{5} \, {\left (x^{2} - 1\right )}^{2} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - \frac {2}{15} \, x^{3} + \frac {1}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^5 - 1/5*(x^2 - 1)^2*sqrt(-x^2 + 1)*arcsin(x) - 2/15*x^3 + 1/5*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x\,\mathrm {asin}\left (x\right )\,{\left (1-x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*asin(x)*(1 - x^2)^(3/2),x)

[Out]

int(x*asin(x)*(1 - x^2)^(3/2), x)

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