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

Optimal. Leaf size=88 \[ \frac{3 x^2}{16 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}+\frac{3 \sin ^{-1}(a x)^2}{16 a^5}+\frac{x^4}{16 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.151599, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4707, 4641, 30} \[ \frac{3 x^2}{16 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}+\frac{3 \sin ^{-1}(a x)^2}{16 a^5}+\frac{x^4}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^2)/(16*a^3) + x^4/(16*a) - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*
x])/(4*a^2) + (3*ArcSin[a*x]^2)/(16*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 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)}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}+\frac{3 \int \frac{x^2 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{4 a^2}+\frac{\int x^3 \, dx}{4 a}\\ &=\frac{x^4}{16 a}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}+\frac{3 \int \frac{\sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{8 a^4}+\frac{3 \int x \, dx}{8 a^3}\\ &=\frac{3 x^2}{16 a^3}+\frac{x^4}{16 a}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}+\frac{3 \sin ^{-1}(a x)^2}{16 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0340756, size = 64, normalized size = 0.73 \[ \frac{a^2 x^2 \left (a^2 x^2+3\right )-2 a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+3\right ) \sin ^{-1}(a x)+3 \sin ^{-1}(a x)^2}{16 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^2*(3 + a^2*x^2) - 2*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x] + 3*ArcSin[a*x]^2)/(16*a^5)

________________________________________________________________________________________

Maple [A]  time = 0.057, size = 74, normalized size = 0.8 \begin{align*}{\frac{1}{16\,{a}^{5}} \left ( -4\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}+{a}^{4}{x}^{4}-6\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}xa+3\,{a}^{2}{x}^{2}+3\, \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)/(-a^2*x^2+1)^(1/2),x)

[Out]

1/16*(-4*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x^3*a^3+a^4*x^4-6*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x*a+3*a^2*x^2+3*arcsi
n(a*x)^2)/a^5

________________________________________________________________________________________

Maxima [A]  time = 1.62586, size = 140, normalized size = 1.59 \begin{align*} \frac{1}{16} \,{\left (\frac{x^{4}}{a^{2}} + \frac{3 \, x^{2}}{a^{4}} - \frac{3 \, \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )^{2}}{a^{6}}\right )} a - \frac{1}{8} \,{\left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x}{a^{4}} - \frac{3 \, \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{4}}\right )} \arcsin \left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(x^4/a^2 + 3*x^2/a^4 - 3*arcsin(a^2*x/sqrt(a^2))^2/a^6)*a - 1/8*(2*sqrt(-a^2*x^2 + 1)*x^3/a^2 + 3*sqrt(-a
^2*x^2 + 1)*x/a^4 - 3*arcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*a^4))*arcsin(a*x)

________________________________________________________________________________________

Fricas [A]  time = 2.08423, size = 142, normalized size = 1.61 \begin{align*} \frac{a^{4} x^{4} + 3 \, a^{2} x^{2} - 2 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right ) + 3 \, \arcsin \left (a x\right )^{2}}{16 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(a^4*x^4 + 3*a^2*x^2 - 2*(2*a^3*x^3 + 3*a*x)*sqrt(-a^2*x^2 + 1)*arcsin(a*x) + 3*arcsin(a*x)^2)/a^5

________________________________________________________________________________________

Sympy [A]  time = 2.88018, size = 82, normalized size = 0.93 \begin{align*} \begin{cases} \frac{x^{4}}{16 a} - \frac{x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{4 a^{2}} + \frac{3 x^{2}}{16 a^{3}} - \frac{3 x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{8 a^{4}} + \frac{3 \operatorname{asin}^{2}{\left (a x \right )}}{16 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)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.27659, size = 123, normalized size = 1.4 \begin{align*} \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x \arcsin \left (a x\right )}{4 \, a^{4}} - \frac{5 \, \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{8 \, a^{4}} + \frac{{\left (a^{2} x^{2} - 1\right )}^{2}}{16 \, a^{5}} + \frac{3 \, \arcsin \left (a x\right )^{2}}{16 \, a^{5}} + \frac{5 \,{\left (a^{2} x^{2} - 1\right )}}{16 \, a^{5}} + \frac{17}{128 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)/a^4 - 5/8*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)/a^4 + 1/16*(a^2*x^2 - 1)^2/a
^5 + 3/16*arcsin(a*x)^2/a^5 + 5/16*(a^2*x^2 - 1)/a^5 + 17/128/a^5