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

Optimal. Leaf size=215 \[ \frac{3 a x^2 \sqrt{c-a^2 c x^2}}{8 \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^4}{8 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{4 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{8 a \sqrt{1-a^2 x^2}}-\frac{3}{4} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.164842, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4647, 4641, 4627, 4707, 30} \[ \frac{3 a x^2 \sqrt{c-a^2 c x^2}}{8 \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^4}{8 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{4 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{8 a \sqrt{1-a^2 x^2}}-\frac{3}{4} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^3,x]

[Out]

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

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 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 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 30

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

Rubi steps

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

Mathematica [A]  time = 0.0573778, size = 114, normalized size = 0.53 \[ \frac{\sqrt{c-a^2 c x^2} \left (3 a^2 x^2+4 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+\left (3-6 a^2 x^2\right ) \sin ^{-1}(a x)^2-6 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+\sin ^{-1}(a x)^4\right )}{8 a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c - a^2*c*x^2]*(3*a^2*x^2 - 6*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + (3 - 6*a^2*x^2)*ArcSin[a*x]^2 + 4*a*x*
Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3 + ArcSin[a*x]^4))/(8*a*Sqrt[1 - a^2*x^2])

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Maple [C]  time = 0.139, size = 260, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{8\,a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{6\,i \left ( \arcsin \left ( ax \right ) \right ) ^{2}+4\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}-3\,i-6\,\arcsin \left ( ax \right ) }{32\,a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( -2\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{2}{a}^{2}+2\,{a}^{3}{x}^{3}+i\sqrt{-{a}^{2}{x}^{2}+1}-2\,ax \right ) }+{\frac{-6\,i \left ( \arcsin \left ( ax \right ) \right ) ^{2}+4\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\,i-6\,\arcsin \left ( ax \right ) }{32\,a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{2}{a}^{2}+2\,{a}^{3}{x}^{3}-i\sqrt{-{a}^{2}{x}^{2}+1}-2\,ax \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/(a^2*x^2-1)*arcsin(a*x)^4+1/32*(-c*(a^2*x^2-1))^(1/2)*(-2*I*(
-a^2*x^2+1)^(1/2)*x^2*a^2+2*a^3*x^3+I*(-a^2*x^2+1)^(1/2)-2*a*x)*(6*I*arcsin(a*x)^2+4*arcsin(a*x)^3-3*I-6*arcsi
n(a*x))/a/(a^2*x^2-1)+1/32*(-c*(a^2*x^2-1))^(1/2)*(2*I*(-a^2*x^2+1)^(1/2)*x^2*a^2+2*a^3*x^3-I*(-a^2*x^2+1)^(1/
2)-2*a*x)*(-6*I*arcsin(a*x)^2+4*arcsin(a*x)^3+3*I-6*arcsin(a*x))/a/(a^2*x^2-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{asin}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*asin(a*x)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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