3.291 \(\int (c-a^2 c x^2) \sin ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=158 \[ -\frac{2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac{40 c \sqrt{1-a^2 x^2}}{9 a}+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{2}{3} c x \sin ^{-1}(a x)^3-\frac{14}{3} c x \sin ^{-1}(a x) \]

[Out]

(-40*c*Sqrt[1 - a^2*x^2])/(9*a) - (2*c*(1 - a^2*x^2)^(3/2))/(27*a) - (14*c*x*ArcSin[a*x])/3 + (2*a^2*c*x^3*Arc
Sin[a*x])/9 + (2*c*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a + (c*(1 - a^2*x^2)^(3/2)*ArcSin[a*x]^2)/(3*a) + (2*c*x*A
rcSin[a*x]^3)/3 + (c*x*(1 - a^2*x^2)*ArcSin[a*x]^3)/3

________________________________________________________________________________________

Rubi [A]  time = 0.210908, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {4649, 4619, 4677, 261, 4645, 444, 43} \[ -\frac{2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac{40 c \sqrt{1-a^2 x^2}}{9 a}+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{2}{3} c x \sin ^{-1}(a x)^3-\frac{14}{3} c x \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)*ArcSin[a*x]^3,x]

[Out]

(-40*c*Sqrt[1 - a^2*x^2])/(9*a) - (2*c*(1 - a^2*x^2)^(3/2))/(27*a) - (14*c*x*ArcSin[a*x])/3 + (2*a^2*c*x^3*Arc
Sin[a*x])/9 + (2*c*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a + (c*(1 - a^2*x^2)^(3/2)*ArcSin[a*x]^2)/(3*a) + (2*c*x*A
rcSin[a*x]^3)/3 + (c*x*(1 - a^2*x^2)*ArcSin[a*x]^3)/3

Rule 4649

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

Rule 4619

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

Rule 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^3 \, dx &=\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{1}{3} (2 c) \int \sin ^{-1}(a x)^3 \, dx-(a c) \int x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \, dx\\ &=\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3-\frac{1}{3} (2 c) \int \left (1-a^2 x^2\right ) \sin ^{-1}(a x) \, dx-(2 a c) \int \frac{x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2}{3} c x \sin ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3-(4 c) \int \sin ^{-1}(a x) \, dx+\frac{1}{3} (2 a c) \int \frac{x \left (1-\frac{a^2 x^2}{3}\right )}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{14}{3} c x \sin ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{1}{3} (a c) \operatorname{Subst}\left (\int \frac{1-\frac{a^2 x}{3}}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )+(4 a c) \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 c \sqrt{1-a^2 x^2}}{a}-\frac{14}{3} c x \sin ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac{1}{3} (a c) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-a^2 x}}+\frac{1}{3} \sqrt{1-a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{40 c \sqrt{1-a^2 x^2}}{9 a}-\frac{2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac{14}{3} c x \sin ^{-1}(a x)+\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \sin ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0896877, size = 101, normalized size = 0.64 \[ \frac{c \left (2 \sqrt{1-a^2 x^2} \left (a^2 x^2-61\right )-9 a x \left (a^2 x^2-3\right ) \sin ^{-1}(a x)^3-9 \sqrt{1-a^2 x^2} \left (a^2 x^2-7\right ) \sin ^{-1}(a x)^2+6 a x \left (a^2 x^2-21\right ) \sin ^{-1}(a x)\right )}{27 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)*ArcSin[a*x]^3,x]

[Out]

(c*(2*Sqrt[1 - a^2*x^2]*(-61 + a^2*x^2) + 6*a*x*(-21 + a^2*x^2)*ArcSin[a*x] - 9*Sqrt[1 - a^2*x^2]*(-7 + a^2*x^
2)*ArcSin[a*x]^2 - 9*a*x*(-3 + a^2*x^2)*ArcSin[a*x]^3))/(27*a)

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 132, normalized size = 0.8 \begin{align*} -{\frac{c}{27\,a} \left ( 9\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}+9\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}{a}^{2}{x}^{2}-27\,ax \left ( \arcsin \left ( ax \right ) \right ) ^{3}-6\,{a}^{3}{x}^{3}\arcsin \left ( ax \right ) -63\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-2\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}+126\,ax\arcsin \left ( ax \right ) +122\,\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)*arcsin(a*x)^3,x)

[Out]

-1/27/a*c*(9*arcsin(a*x)^3*a^3*x^3+9*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^2*x^2-27*a*x*arcsin(a*x)^3-6*a^3*x^3*a
rcsin(a*x)-63*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)-2*a^2*x^2*(-a^2*x^2+1)^(1/2)+126*a*x*arcsin(a*x)+122*(-a^2*x^2+
1)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.58552, size = 173, normalized size = 1.09 \begin{align*} -\frac{1}{3} \,{\left (\sqrt{-a^{2} x^{2} + 1} c x^{2} - \frac{7 \, \sqrt{-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \arcsin \left (a x\right )^{2} - \frac{1}{3} \,{\left (a^{2} c x^{3} - 3 \, c x\right )} \arcsin \left (a x\right )^{3} + \frac{2}{27} \,{\left (\sqrt{-a^{2} x^{2} + 1} c x^{2} + \frac{3 \,{\left (a^{2} c x^{3} - 21 \, c x\right )} \arcsin \left (a x\right )}{a} - \frac{61 \, \sqrt{-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sqrt(-a^2*x^2 + 1)*c*x^2 - 7*sqrt(-a^2*x^2 + 1)*c/a^2)*a*arcsin(a*x)^2 - 1/3*(a^2*c*x^3 - 3*c*x)*arcsin(
a*x)^3 + 2/27*(sqrt(-a^2*x^2 + 1)*c*x^2 + 3*(a^2*c*x^3 - 21*c*x)*arcsin(a*x)/a - 61*sqrt(-a^2*x^2 + 1)*c/a^2)*
a

________________________________________________________________________________________

Fricas [A]  time = 1.61613, size = 225, normalized size = 1.42 \begin{align*} -\frac{9 \,{\left (a^{3} c x^{3} - 3 \, a c x\right )} \arcsin \left (a x\right )^{3} - 6 \,{\left (a^{3} c x^{3} - 21 \, a c x\right )} \arcsin \left (a x\right ) -{\left (2 \, a^{2} c x^{2} - 9 \,{\left (a^{2} c x^{2} - 7 \, c\right )} \arcsin \left (a x\right )^{2} - 122 \, c\right )} \sqrt{-a^{2} x^{2} + 1}}{27 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(9*(a^3*c*x^3 - 3*a*c*x)*arcsin(a*x)^3 - 6*(a^3*c*x^3 - 21*a*c*x)*arcsin(a*x) - (2*a^2*c*x^2 - 9*(a^2*c*
x^2 - 7*c)*arcsin(a*x)^2 - 122*c)*sqrt(-a^2*x^2 + 1))/a

________________________________________________________________________________________

Sympy [A]  time = 2.45054, size = 150, normalized size = 0.95 \begin{align*} \begin{cases} - \frac{a^{2} c x^{3} \operatorname{asin}^{3}{\left (a x \right )}}{3} + \frac{2 a^{2} c x^{3} \operatorname{asin}{\left (a x \right )}}{9} - \frac{a c x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{3} + \frac{2 a c x^{2} \sqrt{- a^{2} x^{2} + 1}}{27} + c x \operatorname{asin}^{3}{\left (a x \right )} - \frac{14 c x \operatorname{asin}{\left (a x \right )}}{3} + \frac{7 c \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{3 a} - \frac{122 c \sqrt{- a^{2} x^{2} + 1}}{27 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c*x**3*asin(a*x)**3/3 + 2*a**2*c*x**3*asin(a*x)/9 - a*c*x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**
2/3 + 2*a*c*x**2*sqrt(-a**2*x**2 + 1)/27 + c*x*asin(a*x)**3 - 14*c*x*asin(a*x)/3 + 7*c*sqrt(-a**2*x**2 + 1)*as
in(a*x)**2/(3*a) - 122*c*sqrt(-a**2*x**2 + 1)/(27*a), Ne(a, 0)), (0, True))

________________________________________________________________________________________

Giac [A]  time = 1.38332, size = 188, normalized size = 1.19 \begin{align*} -\frac{1}{3} \,{\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right )^{3} + \frac{2}{3} \, c x \arcsin \left (a x\right )^{3} + \frac{2}{9} \,{\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right ) + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c \arcsin \left (a x\right )^{2}}{3 \, a} - \frac{40}{9} \, c x \arcsin \left (a x\right ) + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} c \arcsin \left (a x\right )^{2}}{a} - \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c}{27 \, a} - \frac{40 \, \sqrt{-a^{2} x^{2} + 1} c}{9 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a^2*x^2 - 1)*c*x*arcsin(a*x)^3 + 2/3*c*x*arcsin(a*x)^3 + 2/9*(a^2*x^2 - 1)*c*x*arcsin(a*x) + 1/3*(-a^2*x
^2 + 1)^(3/2)*c*arcsin(a*x)^2/a - 40/9*c*x*arcsin(a*x) + 2*sqrt(-a^2*x^2 + 1)*c*arcsin(a*x)^2/a - 2/27*(-a^2*x
^2 + 1)^(3/2)*c/a - 40/9*sqrt(-a^2*x^2 + 1)*c/a