3.303 \(\int \frac{x^4 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{45 x^2}{128 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}+\frac{9 x^2 \sin ^{-1}(a x)^2}{16 a^3}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}+\frac{45 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{64 a^4}+\frac{3 \sin ^{-1}(a x)^4}{32 a^5}-\frac{45 \sin ^{-1}(a x)^2}{128 a^5}-\frac{3 x^4}{128 a}+\frac{3 x^4 \sin ^{-1}(a x)^2}{16 a} \]

[Out]

(-45*x^2)/(128*a^3) - (3*x^4)/(128*a) + (45*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(64*a^4) + (3*x^3*Sqrt[1 - a^2*x^
2]*ArcSin[a*x])/(32*a^2) - (45*ArcSin[a*x]^2)/(128*a^5) + (9*x^2*ArcSin[a*x]^2)/(16*a^3) + (3*x^4*ArcSin[a*x]^
2)/(16*a) - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(4*a^2) + (3
*ArcSin[a*x]^4)/(32*a^5)

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Rubi [A]  time = 0.46789, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4707, 4641, 4627, 30} \[ -\frac{45 x^2}{128 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}+\frac{9 x^2 \sin ^{-1}(a x)^2}{16 a^3}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}+\frac{45 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{64 a^4}+\frac{3 \sin ^{-1}(a x)^4}{32 a^5}-\frac{45 \sin ^{-1}(a x)^2}{128 a^5}-\frac{3 x^4}{128 a}+\frac{3 x^4 \sin ^{-1}(a x)^2}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*x^2)/(128*a^3) - (3*x^4)/(128*a) + (45*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(64*a^4) + (3*x^3*Sqrt[1 - a^2*x^
2]*ArcSin[a*x])/(32*a^2) - (45*ArcSin[a*x]^2)/(128*a^5) + (9*x^2*ArcSin[a*x]^2)/(16*a^3) + (3*x^4*ArcSin[a*x]^
2)/(16*a) - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(4*a^2) + (3
*ArcSin[a*x]^4)/(32*a^5)

Rule 4707

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

Rule 4641

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

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 30

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

Rubi steps

\begin{align*} \int \frac{x^4 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac{3 \int \frac{x^2 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{4 a^2}+\frac{3 \int x^3 \sin ^{-1}(a x)^2 \, dx}{4 a}\\ &=\frac{3 x^4 \sin ^{-1}(a x)^2}{16 a}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}-\frac{3}{8} \int \frac{x^4 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx+\frac{3 \int \frac{\sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{8 a^4}+\frac{9 \int x \sin ^{-1}(a x)^2 \, dx}{8 a^3}\\ &=\frac{3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}+\frac{9 x^2 \sin ^{-1}(a x)^2}{16 a^3}+\frac{3 x^4 \sin ^{-1}(a x)^2}{16 a}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac{3 \sin ^{-1}(a x)^4}{32 a^5}-\frac{9 \int \frac{x^2 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{32 a^2}-\frac{9 \int \frac{x^2 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{8 a^2}-\frac{3 \int x^3 \, dx}{32 a}\\ &=-\frac{3 x^4}{128 a}+\frac{45 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{64 a^4}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}+\frac{9 x^2 \sin ^{-1}(a x)^2}{16 a^3}+\frac{3 x^4 \sin ^{-1}(a x)^2}{16 a}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac{3 \sin ^{-1}(a x)^4}{32 a^5}-\frac{9 \int \frac{\sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{64 a^4}-\frac{9 \int \frac{\sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{16 a^4}-\frac{9 \int x \, dx}{64 a^3}-\frac{9 \int x \, dx}{16 a^3}\\ &=-\frac{45 x^2}{128 a^3}-\frac{3 x^4}{128 a}+\frac{45 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{64 a^4}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}-\frac{45 \sin ^{-1}(a x)^2}{128 a^5}+\frac{9 x^2 \sin ^{-1}(a x)^2}{16 a^3}+\frac{3 x^4 \sin ^{-1}(a x)^2}{16 a}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac{3 \sin ^{-1}(a x)^4}{32 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0623523, size = 125, normalized size = 0.65 \[ \frac{-3 a^2 x^2 \left (a^2 x^2+15\right )-16 a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+3\right ) \sin ^{-1}(a x)^3+3 \left (8 a^4 x^4+24 a^2 x^2-15\right ) \sin ^{-1}(a x)^2+6 a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+15\right ) \sin ^{-1}(a x)+12 \sin ^{-1}(a x)^4}{128 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^2*x^2*(15 + a^2*x^2) + 6*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2)*ArcSin[a*x] + 3*(-15 + 24*a^2*x^2 + 8*a^
4*x^4)*ArcSin[a*x]^2 - 16*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x]^3 + 12*ArcSin[a*x]^4)/(128*a^5)

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Maple [A]  time = 0.068, size = 159, normalized size = 0.8 \begin{align*}{\frac{1}{128\,{a}^{5}} \left ( -32\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}\sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}+24\,{a}^{4}{x}^{4} \left ( \arcsin \left ( ax \right ) \right ) ^{2}+12\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}-3\,{a}^{4}{x}^{4}-48\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}\sqrt{-{a}^{2}{x}^{2}+1}xa+72\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+12\, \left ( \arcsin \left ( ax \right ) \right ) ^{4}+90\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}xa-45\,{a}^{2}{x}^{2}-45\, \left ( \arcsin \left ( ax \right ) \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(-32*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)*x^3*a^3+24*a^4*x^4*arcsin(a*x)^2+12*arcsin(a*x)*(-a^2*x^2+1)^(1/2)
*x^3*a^3-3*a^4*x^4-48*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)*x*a+72*arcsin(a*x)^2*x^2*a^2+12*arcsin(a*x)^4+90*arcsin
(a*x)*(-a^2*x^2+1)^(1/2)*x*a-45*a^2*x^2-45*arcsin(a*x)^2)/a^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \arcsin \left (a x\right )^{3}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4*arcsin(a*x)^3/sqrt(-a^2*x^2 + 1), x)

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Fricas [A]  time = 1.67244, size = 273, normalized size = 1.43 \begin{align*} -\frac{3 \, a^{4} x^{4} + 45 \, a^{2} x^{2} - 12 \, \arcsin \left (a x\right )^{4} - 3 \,{\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right )^{2} + 2 \, \sqrt{-a^{2} x^{2} + 1}{\left (8 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{3} - 3 \,{\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \arcsin \left (a x\right )\right )}}{128 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/128*(3*a^4*x^4 + 45*a^2*x^2 - 12*arcsin(a*x)^4 - 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*arcsin(a*x)^2 + 2*sqrt(-a^
2*x^2 + 1)*(8*(2*a^3*x^3 + 3*a*x)*arcsin(a*x)^3 - 3*(2*a^3*x^3 + 15*a*x)*arcsin(a*x)))/a^5

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Sympy [A]  time = 8.6755, size = 185, normalized size = 0.97 \begin{align*} \begin{cases} \frac{3 x^{4} \operatorname{asin}^{2}{\left (a x \right )}}{16 a} - \frac{3 x^{4}}{128 a} - \frac{x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{4 a^{2}} + \frac{3 x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{32 a^{2}} + \frac{9 x^{2} \operatorname{asin}^{2}{\left (a x \right )}}{16 a^{3}} - \frac{45 x^{2}}{128 a^{3}} - \frac{3 x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{8 a^{4}} + \frac{45 x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{64 a^{4}} + \frac{3 \operatorname{asin}^{4}{\left (a x \right )}}{32 a^{5}} - \frac{45 \operatorname{asin}^{2}{\left (a x \right )}}{128 a^{5}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*asin(a*x)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((3*x**4*asin(a*x)**2/(16*a) - 3*x**4/(128*a) - x**3*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(4*a**2) + 3*x
**3*sqrt(-a**2*x**2 + 1)*asin(a*x)/(32*a**2) + 9*x**2*asin(a*x)**2/(16*a**3) - 45*x**2/(128*a**3) - 3*x*sqrt(-
a**2*x**2 + 1)*asin(a*x)**3/(8*a**4) + 45*x*sqrt(-a**2*x**2 + 1)*asin(a*x)/(64*a**4) + 3*asin(a*x)**4/(32*a**5
) - 45*asin(a*x)**2/(128*a**5), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.4601, size = 259, normalized size = 1.36 \begin{align*} \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x \arcsin \left (a x\right )^{3}}{4 \, a^{4}} - \frac{5 \, \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{8 \, a^{4}} - \frac{3 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x \arcsin \left (a x\right )}{32 \, a^{4}} + \frac{3 \,{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{16 \, a^{5}} + \frac{3 \, \arcsin \left (a x\right )^{4}}{32 \, a^{5}} + \frac{51 \, \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{64 \, a^{4}} + \frac{15 \,{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{16 \, a^{5}} - \frac{3 \,{\left (a^{2} x^{2} - 1\right )}^{2}}{128 \, a^{5}} + \frac{51 \, \arcsin \left (a x\right )^{2}}{128 \, a^{5}} - \frac{51 \,{\left (a^{2} x^{2} - 1\right )}}{128 \, a^{5}} - \frac{195}{1024 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/4*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^3/a^4 - 5/8*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^4 - 3/32*(-a^2*x^2 + 1
)^(3/2)*x*arcsin(a*x)/a^4 + 3/16*(a^2*x^2 - 1)^2*arcsin(a*x)^2/a^5 + 3/32*arcsin(a*x)^4/a^5 + 51/64*sqrt(-a^2*
x^2 + 1)*x*arcsin(a*x)/a^4 + 15/16*(a^2*x^2 - 1)*arcsin(a*x)^2/a^5 - 3/128*(a^2*x^2 - 1)^2/a^5 + 51/128*arcsin
(a*x)^2/a^5 - 51/128*(a^2*x^2 - 1)/a^5 - 195/1024/a^5