\(\int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx\) [64]

Optimal result

Integrand size = 22, antiderivative size = 215 \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx=\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {1-a^2 x^2}}-\frac {3}{4} x \sqrt {c-a^2 c x^2} \arcsin (a x)+\frac {3 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{8 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{4 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \arcsin (a x)^3+\frac {\sqrt {c-a^2 c x^2} \arcsin (a x)^4}{8 a \sqrt {1-a^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.53 \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx=\frac {\sqrt {c-a^2 c x^2} \left (3 a^2 x^2-6 a x \sqrt {1-a^2 x^2} \arcsin (a x)+\left (3-6 a^2 x^2\right ) \arcsin (a x)^2+4 a x \sqrt {1-a^2 x^2} \arcsin (a x)^3+\arcsin (a x)^4\right )}{8 a \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(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 + ArcS 
in[a*x]^4))/(8*a*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5156, 5138, 5152, 5210, 15, 5152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \arcsin (a x)^3 \sqrt {c-a^2 c x^2} \, dx\)

\(\Big \downarrow \) 5156

\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \int x \arcsin (a x)^2dx}{2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5138

\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5210

\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}-\frac {3 a \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

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

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[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 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[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 5156

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcSin[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] - Simp[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 5210

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] + (Simp[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] + S 
imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.21

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

Input:

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

Output:

-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*arcsin( 
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)
 

Fricas [F]

\[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx=\int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}^{3}{\left (a x \right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx=\int {\mathrm {asin}\left (a\,x\right )}^3\,\sqrt {c-a^2\,c\,x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx=\sqrt {c}\, \left (\int \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )^{3}d x \right ) \] Input:

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

Output:

sqrt(c)*int(sqrt( - a**2*x**2 + 1)*asin(a*x)**3,x)