3.3.74 \(\int \frac {\arcsin (a x)^2}{(c-a^2 c x^2)^{5/2}} \, dx\) [274]

3.3.74.1 Optimal result

Integrand size = 22, antiderivative size = 283 \[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {x}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\arcsin (a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \arcsin (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \arcsin (a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}} \]

output

1/3*x*arcsin(a*x)^2/c/(-a^2*c*x^2+c)^(3/2)+1/3*x/c^2/(-a^2*c*x^2+c)^(1/2)+ 
2/3*x*arcsin(a*x)^2/c^2/(-a^2*c*x^2+c)^(1/2)-1/3*arcsin(a*x)/a/c^2/(-a^2*x 
^2+1)^(1/2)/(-a^2*c*x^2+c)^(1/2)-2/3*I*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a/ 
c^2/(-a^2*c*x^2+c)^(1/2)+4/3*arcsin(a*x)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2 
)*(-a^2*x^2+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)-2/3*I*polylog(2,-(I*a*x+(- 
a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)
 

3.3.74.2 Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.53 \[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {a x+\left (-2 i \sqrt {1-a^2 x^2}+a x \left (2+\frac {1}{1-a^2 x^2}\right )\right ) \arcsin (a x)^2+\frac {\arcsin (a x) \left (-1+\left (4-4 a^2 x^2\right ) \log \left (1+e^{2 i \arcsin (a x)}\right )\right )}{\sqrt {1-a^2 x^2}}-2 i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}} \]

input

Integrate[ArcSin[a*x]^2/(c - a^2*c*x^2)^(5/2),x]
 

output

(a*x + ((-2*I)*Sqrt[1 - a^2*x^2] + a*x*(2 + (1 - a^2*x^2)^(-1)))*ArcSin[a* 
x]^2 + (ArcSin[a*x]*(-1 + (4 - 4*a^2*x^2)*Log[1 + E^((2*I)*ArcSin[a*x])])) 
/Sqrt[1 - a^2*x^2] - (2*I)*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[a 
*x])])/(3*a*c^2*Sqrt[c - a^2*c*x^2])
 

3.3.74.3 Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5162, 5160, 5180, 3042, 4202, 2620, 2715, 2838, 5182, 208}

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.

\(\displaystyle \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 5160

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

\(\Big \downarrow \) 5180

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {2 a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)}{\left (1-a^2 x^2\right )^2}dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (\frac {1}{2} i \arcsin (a x)^2-2 i \left (\frac {1}{4} \int e^{-2 i \arcsin (a x)} \log \left (1+e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}-\frac {1}{2} i \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {2 a \sqrt {1-a^2 x^2} \int \frac {x \arcsin (a x)}{\left (1-a^2 x^2\right )^2}dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (\frac {1}{2} i \arcsin (a x)^2-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5182

\(\displaystyle -\frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\arcsin (a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx}{2 a}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (\frac {1}{2} i \arcsin (a x)^2-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 208

\(\displaystyle -\frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\arcsin (a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {x}{2 a \sqrt {1-a^2 x^2}}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arcsin (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (\frac {1}{2} i \arcsin (a x)^2-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arcsin (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\)

input

Int[ArcSin[a*x]^2/(c - a^2*c*x^2)^(5/2),x]
 

output

(x*ArcSin[a*x]^2)/(3*c*(c - a^2*c*x^2)^(3/2)) - (2*a*Sqrt[1 - a^2*x^2]*(-1 
/2*x/(a*Sqrt[1 - a^2*x^2]) + ArcSin[a*x]/(2*a^2*(1 - a^2*x^2))))/(3*c^2*Sq 
rt[c - a^2*c*x^2]) + (2*((x*ArcSin[a*x]^2)/(c*Sqrt[c - a^2*c*x^2]) - (2*Sq 
rt[1 - a^2*x^2]*((I/2)*ArcSin[a*x]^2 - (2*I)*((-1/2*I)*ArcSin[a*x]*Log[1 + 
 E^((2*I)*ArcSin[a*x])] - PolyLog[2, -E^((2*I)*ArcSin[a*x])]/4)))/(a*c*Sqr 
t[c - a^2*c*x^2])))/(3*c)
 

3.3.74.3.1 Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5160

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSin[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 0]
 

rule 5162

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 
))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b*Ar 
cSin[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 
*x^2)^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] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 5180

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

rule 5182

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] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(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]
 

3.3.74.4 Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.29

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-2 i \sqrt {-a^{2} x^{2}+1}-3 a x \right ) \left (-2 i \arcsin \left (a x \right ) a^{4} x^{4}-2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+i \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-a^{4} x^{4}+3 \arcsin \left (a x \right )^{2} a^{2} x^{2}+4 i \arcsin \left (a x \right ) a^{2} x^{2}+3 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x -i \sqrt {-a^{2} x^{2}+1}\, a x +3 a^{2} x^{2}-4 \arcsin \left (a x \right )^{2}-2 i \arcsin \left (a x \right )-2\right )}{3 c^{3} \left (3 a^{6} x^{6}-10 a^{4} x^{4}+11 a^{2} x^{2}-4\right ) a}+\frac {2 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (a x \right ) \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (a x \right )^{2}+\operatorname {polylog}\left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )}{3 c^{3} \left (a^{2} x^{2}-1\right ) a}\)	\(365\)

input

int(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
 

output

-1/3*(-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^3-2*I* 
(-a^2*x^2+1)^(1/2)-3*a*x)*(-2*I*arcsin(a*x)*a^4*x^4-2*arcsin(a*x)*(-a^2*x^ 
2+1)^(1/2)*a^3*x^3+I*(-a^2*x^2+1)^(1/2)*a^3*x^3-a^4*x^4+3*arcsin(a*x)^2*a^ 
2*x^2+4*I*arcsin(a*x)*a^2*x^2+3*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x-I*(-a^2 
*x^2+1)^(1/2)*a*x+3*a^2*x^2-4*arcsin(a*x)^2-2*I*arcsin(a*x)-2)/c^3/(3*a^6* 
x^6-10*a^4*x^4+11*a^2*x^2-4)/a+2/3*I*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^( 
1/2)*(2*I*arcsin(a*x)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)+2*arcsin(a*x)^2+p 
olylog(2,-(I*a*x+(-a^2*x^2+1)^(1/2))^2))/c^3/(a^2*x^2-1)/a
 

3.3.74.5 Fricas [F]

\[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")
 

output

integral(-sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^2/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 
+ 3*a^2*c^3*x^2 - c^3), x)
 

3.3.74.6 Sympy [F]

\[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

input

integrate(asin(a*x)**2/(-a**2*c*x**2+c)**(5/2),x)
 

output

Integral(asin(a*x)**2/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)
 

3.3.74.7 Maxima [F]

\[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")
 

output

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(5/2), x)
 

3.3.74.8 Giac [F]

\[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")
 

output

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(5/2), x)
 

3.3.74.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

input

int(asin(a*x)^2/(c - a^2*c*x^2)^(5/2),x)
 

output

int(asin(a*x)^2/(c - a^2*c*x^2)^(5/2), x)