3.3.73 \(\int \frac {\arcsin (a x)^2}{(c-a^2 c x^2)^{3/2}} \, dx\) [273]

3.3.73.1 Optimal result

Integrand size = 22, antiderivative size = 179 \[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arcsin (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (1+e^{2 i \arcsin (a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )}{a c \sqrt {c-a^2 c x^2}} \]

output

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

3.3.73.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.60 \[ \int \frac {\arcsin (a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\arcsin (a x) \left (a x \arcsin (a x)+\sqrt {1-a^2 x^2} \left (-i \arcsin (a x)+2 \log \left (1+e^{2 i \arcsin (a x)}\right )\right )\right )-i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a x)}\right )}{a c \sqrt {c \left (1-a^2 x^2\right )}} \]

input

Integrate[ArcSin[a*x]^2/(c - a^2*c*x^2)^(3/2),x]
 

output

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

3.3.73.3 Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.69, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5160, 5180, 3042, 4202, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5160

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

\(\Big \downarrow \) 5180

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x \arcsin (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \int \arcsin (a x) \tan (\arcsin (a x))d\arcsin (a x)}{a c \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[ArcSin[a*x]^2/(c - a^2*c*x^2)^(3/2),x]
 

output

(x*ArcSin[a*x]^2)/(c*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[1 - a^2*x^2]*((I/2)*Ar 
cSin[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*Sqrt[c - a^2*c*x^2])
 

3.3.73.3.1 Defintions of rubi rules used

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 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]
 

3.3.73.4 Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}+a x \right ) \arcsin \left (a x \right )^{2}}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (a x \right ) \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (a x \right )^{2}+\operatorname {polylog}\left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}\)	\(169\)

input

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

output

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

3.3.73.5 Fricas [F]

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

input

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

output

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

3.3.73.6 Sympy [F]

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

input

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

output

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

3.3.73.7 Maxima [F]

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

input

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

output

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

3.3.73.8 Giac [F]

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

input

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

output

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

3.3.73.9 Mupad [F(-1)]

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

input

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

output

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