3.2.2 \(\int \frac {x^3 \text {ArcSin}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [102]

Optimal. Leaf size=72 \[ \frac {2 x}{3 a^3}+\frac {x^3}{9 a}-\frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a^2} \]

[Out]

2/3*x/a^3+1/9*x^3/a-2/3*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^4-1/3*x^2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^2

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Rubi [A]
time = 0.08, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4795, 4767, 8, 30} \begin {gather*} \frac {2 x}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a^2}-\frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a^4}+\frac {x^3}{9 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

(2*x)/(3*a^3) + x^3/(9*a) - (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3
*a^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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]

Rule 4795

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

Rubi steps

\begin {align*} \int \frac {x^3 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}+\frac {2 \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2}+\frac {\int x^2 \, dx}{3 a}\\ &=\frac {x^3}{9 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}+\frac {2 \int 1 \, dx}{3 a^3}\\ &=\frac {2 x}{3 a^3}+\frac {x^3}{9 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 49, normalized size = 0.68 \begin {gather*} \frac {a x \left (6+a^2 x^2\right )-3 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \text {ArcSin}(a x)}{9 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x*(6 + a^2*x^2) - 3*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x])/(9*a^4)

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Maple [A]
time = 0.07, size = 95, normalized size = 1.32

method	result	size
default	\(-\frac {\left (3 a^{4} x^{4} \arcsin \left (a x \right )+3 a^{2} x^{2} \arcsin \left (a x \right )+a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-6 \arcsin \left (a x \right )+6 a x \sqrt {-a^{2} x^{2}+1}\right ) \sqrt {-a^{2} x^{2}+1}}{9 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9/a^4*(3*a^4*x^4*arcsin(a*x)+3*a^2*x^2*arcsin(a*x)+a^3*x^3*(-a^2*x^2+1)^(1/2)-6*arcsin(a*x)+6*a*x*(-a^2*x^2
+1)^(1/2))*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)

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Maxima [A]
time = 0.48, size = 61, normalized size = 0.85 \begin {gather*} \frac {1}{9} \, a {\left (\frac {x^{3}}{a^{2}} + \frac {6 \, x}{a^{4}}\right )} - \frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arcsin \left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*(x^3/a^2 + 6*x/a^4) - 1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arcsin(a*x)

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Fricas [A]
time = 2.12, size = 44, normalized size = 0.61 \begin {gather*} \frac {a^{3} x^{3} - 3 \, {\left (a^{2} x^{2} + 2\right )} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right ) + 6 \, a x}{9 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(a^3*x^3 - 3*(a^2*x^2 + 2)*sqrt(-a^2*x^2 + 1)*arcsin(a*x) + 6*a*x)/a^4

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Sympy [A]
time = 0.35, size = 65, normalized size = 0.90 \begin {gather*} \begin {cases} \frac {x^{3}}{9 a} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{3 a^{2}} + \frac {2 x}{3 a^{3}} - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{3 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3/(9*a) - x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)/(3*a**2) + 2*x/(3*a**3) - 2*sqrt(-a**2*x**2 + 1)*a
sin(a*x)/(3*a**4), Ne(a, 0)), (0, True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\mathrm {asin}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*asin(a*x))/(1 - a^2*x^2)^(1/2),x)

[Out]

int((x^3*asin(a*x))/(1 - a^2*x^2)^(1/2), x)

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