3.3.97 \(\int \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^3 \, dx\) [297]

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

[Out]

-3/4*x*arcsin(a*x)*(-a^2*c*x^2+c)^(1/2)+1/2*x*arcsin(a*x)^3*(-a^2*c*x^2+c)^(1/2)+3/8*a*x^2*(-a^2*c*x^2+c)^(1/2
)/(-a^2*x^2+1)^(1/2)+3/8*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3/4*a*x^2*arcsin(a*x)^2*(-a^2
*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+1/8*arcsin(a*x)^4*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 215, normalized size of antiderivative = 1.00, 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 = {4741, 4737, 4723, 4795, 30} \begin {gather*} \frac {\text {ArcSin}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \text {ArcSin}(a x)^3 \sqrt {c-a^2 c x^2}-\frac {3 a x^2 \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}+\frac {3 \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}-\frac {3}{4} x \text {ArcSin}(a x) \sqrt {c-a^2 c x^2}+\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {1-a^2 x^2}} \end {gather*}

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 30

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

Rule 4723

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 4737

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

Rule 4741

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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 4795

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

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*}

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Mathematica [A]
time = 0.04, size = 114, normalized size = 0.53 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \left (3 a^2 x^2-6 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)+\left (3-6 a^2 x^2\right ) \text {ArcSin}(a x)^2+4 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3+\text {ArcSin}(a x)^4\right )}{8 a \sqrt {1-a^2 x^2}} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.16, size = 260, normalized size = 1.21

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{4}}{8 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}+i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (6 i \arcsin \left (a x \right )^{2}+4 \arcsin \left (a x \right )^{3}-3 i-6 \arcsin \left (a x \right )\right )}{32 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arcsin \left (a x \right )^{2}+4 \arcsin \left (a x \right )^{3}+3 i-6 \arcsin \left (a x \right )\right )}{32 a \left (a^{2} x^{2}-1\right )}\)	\(260\)

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,method=_RETURNVERBOSE)

[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)*a^2*x^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)*a^2*x^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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}^{3}{\left (a x \right )}\, dx \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {asin}\left (a\,x\right )}^3\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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