3.159 \(\int x (d-c^2 d x^2) (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=138 \[ \frac{b d x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{3 b d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c}-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{3 d \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^2}+\frac{1}{32} b^2 c^2 d x^4-\frac{5}{32} b^2 d x^2 \]

[Out]

(-5*b^2*d*x^2)/32 + (b^2*c^2*d*x^4)/32 + (3*b*d*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c) + (b*d*x*(1 -
c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(8*c) + (3*d*(a + b*ArcSin[c*x])^2)/(32*c^2) - (d*(1 - c^2*x^2)^2*(a + b*A
rcSin[c*x])^2)/(4*c^2)

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Rubi [A]  time = 0.132724, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4677, 4649, 4647, 4641, 30, 14} \[ \frac{b d x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{3 b d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c}-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{3 d \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^2}+\frac{1}{32} b^2 c^2 d x^4-\frac{5}{32} b^2 d x^2 \]

Antiderivative was successfully verified.

[In]

Int[x*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(-5*b^2*d*x^2)/32 + (b^2*c^2*d*x^4)/32 + (3*b*d*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c) + (b*d*x*(1 -
c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(8*c) + (3*d*(a + b*ArcSin[c*x])^2)/(32*c^2) - (d*(1 - c^2*x^2)^2*(a + b*A
rcSin[c*x])^2)/(4*c^2)

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 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 4647

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

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]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{(b d) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 c}\\ &=\frac{b d x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac{1}{8} \left (b^2 d\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac{(3 b d) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 c}\\ &=\frac{3 b d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c}+\frac{b d x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac{1}{8} \left (b^2 d\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac{1}{16} \left (3 b^2 d\right ) \int x \, dx+\frac{(3 b d) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{16 c}\\ &=-\frac{5}{32} b^2 d x^2+\frac{1}{32} b^2 c^2 d x^4+\frac{3 b d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c}+\frac{b d x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{3 d \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^2}-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}\\ \end{align*}

Mathematica [A]  time = 0.288349, size = 157, normalized size = 1.14 \[ -\frac{d \left (c x \left (8 a^2 c x \left (c^2 x^2-2\right )+2 a b \sqrt{1-c^2 x^2} \left (2 c^2 x^2-5\right )+b^2 c x \left (5-c^2 x^2\right )\right )+2 b \sin ^{-1}(c x) \left (a \left (8 c^4 x^4-16 c^2 x^2+5\right )+b c x \sqrt{1-c^2 x^2} \left (2 c^2 x^2-5\right )\right )+b^2 \left (8 c^4 x^4-16 c^2 x^2+5\right ) \sin ^{-1}(c x)^2\right )}{32 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2,x]

[Out]

-(d*(c*x*(b^2*c*x*(5 - c^2*x^2) + 8*a^2*c*x*(-2 + c^2*x^2) + 2*a*b*Sqrt[1 - c^2*x^2]*(-5 + 2*c^2*x^2)) + 2*b*(
b*c*x*Sqrt[1 - c^2*x^2]*(-5 + 2*c^2*x^2) + a*(5 - 16*c^2*x^2 + 8*c^4*x^4))*ArcSin[c*x] + b^2*(5 - 16*c^2*x^2 +
8*c^4*x^4)*ArcSin[c*x]^2))/(32*c^2)

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Maple [A]  time = 0.078, size = 206, normalized size = 1.5 \begin{align*}{\frac{1}{{c}^{2}} \left ( -d{a}^{2} \left ({\frac{{c}^{4}{x}^{4}}{4}}-{\frac{{c}^{2}{x}^{2}}{2}} \right ) -d{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{4}}-{\frac{\arcsin \left ( cx \right ) }{16} \left ( -2\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}+5\,cx\sqrt{-{c}^{2}{x}^{2}+1}+3\,\arcsin \left ( cx \right ) \right ) }+{\frac{3\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{32}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{32}}+{\frac{3\,{c}^{2}{x}^{2}}{32}}-{\frac{3}{32}} \right ) -2\,dab \left ( 1/4\,{c}^{4}{x}^{4}\arcsin \left ( cx \right ) -1/2\,{c}^{2}{x}^{2}\arcsin \left ( cx \right ) +1/16\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-{\frac{5\,cx\sqrt{-{c}^{2}{x}^{2}+1}}{32}}+{\frac{5\,\arcsin \left ( cx \right ) }{32}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(-d*a^2*(1/4*c^4*x^4-1/2*c^2*x^2)-d*b^2*(1/4*arcsin(c*x)^2*(c^2*x^2-1)^2-1/16*arcsin(c*x)*(-2*c^3*x^3*(-
c^2*x^2+1)^(1/2)+5*c*x*(-c^2*x^2+1)^(1/2)+3*arcsin(c*x))+3/32*arcsin(c*x)^2-1/32*(c^2*x^2-1)^2+3/32*c^2*x^2-3/
32)-2*d*a*b*(1/4*c^4*x^4*arcsin(c*x)-1/2*c^2*x^2*arcsin(c*x)+1/16*c^3*x^3*(-c^2*x^2+1)^(1/2)-5/32*c*x*(-c^2*x^
2+1)^(1/2)+5/32*arcsin(c*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, a^{2} c^{2} d x^{4} - \frac{1}{16} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} a b c^{2} d + \frac{1}{2} \, a^{2} d x^{2} + \frac{1}{2} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} a b d - \frac{1}{4} \,{\left (b^{2} c^{2} d x^{4} - 2 \, b^{2} d x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} - \int \frac{{\left (b^{2} c^{3} d x^{4} - 2 \, b^{2} c d x^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{2 \,{\left (c^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2*c^2*d*x^4 - 1/16*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*
arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*a*b*c^2*d + 1/2*a^2*d*x^2 + 1/2*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*
x^2 + 1)*x/c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*a*b*d - 1/4*(b^2*c^2*d*x^4 - 2*b^2*d*x^2)*arctan2(c
*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - integrate(1/2*(b^2*c^3*d*x^4 - 2*b^2*c*d*x^2)*sqrt(c*x + 1)*sqrt(-c*x +
1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)

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Fricas [A]  time = 1.93991, size = 396, normalized size = 2.87 \begin{align*} -\frac{{\left (8 \, a^{2} - b^{2}\right )} c^{4} d x^{4} -{\left (16 \, a^{2} - 5 \, b^{2}\right )} c^{2} d x^{2} +{\left (8 \, b^{2} c^{4} d x^{4} - 16 \, b^{2} c^{2} d x^{2} + 5 \, b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (8 \, a b c^{4} d x^{4} - 16 \, a b c^{2} d x^{2} + 5 \, a b d\right )} \arcsin \left (c x\right ) + 2 \,{\left (2 \, a b c^{3} d x^{3} - 5 \, a b c d x +{\left (2 \, b^{2} c^{3} d x^{3} - 5 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{32 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*((8*a^2 - b^2)*c^4*d*x^4 - (16*a^2 - 5*b^2)*c^2*d*x^2 + (8*b^2*c^4*d*x^4 - 16*b^2*c^2*d*x^2 + 5*b^2*d)*a
rcsin(c*x)^2 + 2*(8*a*b*c^4*d*x^4 - 16*a*b*c^2*d*x^2 + 5*a*b*d)*arcsin(c*x) + 2*(2*a*b*c^3*d*x^3 - 5*a*b*c*d*x
+ (2*b^2*c^3*d*x^3 - 5*b^2*c*d*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^2

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Sympy [A]  time = 4.00265, size = 269, normalized size = 1.95 \begin{align*} \begin{cases} - \frac{a^{2} c^{2} d x^{4}}{4} + \frac{a^{2} d x^{2}}{2} - \frac{a b c^{2} d x^{4} \operatorname{asin}{\left (c x \right )}}{2} - \frac{a b c d x^{3} \sqrt{- c^{2} x^{2} + 1}}{8} + a b d x^{2} \operatorname{asin}{\left (c x \right )} + \frac{5 a b d x \sqrt{- c^{2} x^{2} + 1}}{16 c} - \frac{5 a b d \operatorname{asin}{\left (c x \right )}}{16 c^{2}} - \frac{b^{2} c^{2} d x^{4} \operatorname{asin}^{2}{\left (c x \right )}}{4} + \frac{b^{2} c^{2} d x^{4}}{32} - \frac{b^{2} c d x^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{8} + \frac{b^{2} d x^{2} \operatorname{asin}^{2}{\left (c x \right )}}{2} - \frac{5 b^{2} d x^{2}}{32} + \frac{5 b^{2} d x \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{16 c} - \frac{5 b^{2} d \operatorname{asin}^{2}{\left (c x \right )}}{32 c^{2}} & \text{for}\: c \neq 0 \\\frac{a^{2} d x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c**2*d*x**4/4 + a**2*d*x**2/2 - a*b*c**2*d*x**4*asin(c*x)/2 - a*b*c*d*x**3*sqrt(-c**2*x**2 +
1)/8 + a*b*d*x**2*asin(c*x) + 5*a*b*d*x*sqrt(-c**2*x**2 + 1)/(16*c) - 5*a*b*d*asin(c*x)/(16*c**2) - b**2*c**2*
d*x**4*asin(c*x)**2/4 + b**2*c**2*d*x**4/32 - b**2*c*d*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/8 + b**2*d*x**2*asi
n(c*x)**2/2 - 5*b**2*d*x**2/32 + 5*b**2*d*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(16*c) - 5*b**2*d*asin(c*x)**2/(32*
c**2), Ne(c, 0)), (a**2*d*x**2/2, True))

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Giac [A]  time = 1.45066, size = 321, normalized size = 2.33 \begin{align*} \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d x \arcsin \left (c x\right )}{8 \, c} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d x}{8 \, c} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right )}{16 \, c} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a b d \arcsin \left (c x\right )}{2 \, c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} a b d x}{16 \, c} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a^{2} d}{4 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d}{32 \, c^{2}} + \frac{3 \, b^{2} d \arcsin \left (c x\right )^{2}}{32 \, c^{2}} - \frac{3 \,{\left (c^{2} x^{2} - 1\right )} b^{2} d}{32 \, c^{2}} + \frac{3 \, a b d \arcsin \left (c x\right )}{16 \, c^{2}} - \frac{15 \, b^{2} d}{256 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(-c^2*x^2 + 1)^(3/2)*b^2*d*x*arcsin(c*x)/c - 1/4*(c^2*x^2 - 1)^2*b^2*d*arcsin(c*x)^2/c^2 + 1/8*(-c^2*x^2 +
1)^(3/2)*a*b*d*x/c + 3/16*sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)/c - 1/2*(c^2*x^2 - 1)^2*a*b*d*arcsin(c*x)/c^
2 + 3/16*sqrt(-c^2*x^2 + 1)*a*b*d*x/c - 1/4*(c^2*x^2 - 1)^2*a^2*d/c^2 + 1/32*(c^2*x^2 - 1)^2*b^2*d/c^2 + 3/32*
b^2*d*arcsin(c*x)^2/c^2 - 3/32*(c^2*x^2 - 1)*b^2*d/c^2 + 3/16*a*b*d*arcsin(c*x)/c^2 - 15/256*b^2*d/c^2