3.3.15 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2} \, dx\) [215]

3.3.15.1 Optimal result

Integrand size = 29, antiderivative size = 227 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x}-\frac {i c \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{3 b \sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]

output

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

3.3.15.2 Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2} \, dx=-\frac {a^2 \sqrt {d-c^2 d x^2}}{x}+a^2 c \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\frac {a b \sqrt {d-c^2 d x^2} \left (2 \sqrt {1-c^2 x^2} \arcsin (c x)+c x \arcsin (c x)^2-2 c x \log (c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {b^2 c \sqrt {d-c^2 d x^2} \left (\arcsin (c x) \left (\left (3 i+\frac {3 \sqrt {1-c^2 x^2}}{c x}\right ) \arcsin (c x)+\arcsin (c x)^2-6 \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{3 \sqrt {1-c^2 x^2}} \]

input

Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x^2,x]
 

output

-((a^2*Sqrt[d - c^2*d*x^2])/x) + a^2*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d* 
x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - (a*b*Sqrt[d - c^2*d*x^2]*(2*Sqrt[1 - c^2 
*x^2]*ArcSin[c*x] + c*x*ArcSin[c*x]^2 - 2*c*x*Log[c*x]))/(x*Sqrt[1 - c^2*x 
^2]) - (b^2*c*Sqrt[d - c^2*d*x^2]*(ArcSin[c*x]*((3*I + (3*Sqrt[1 - c^2*x^2 
])/(c*x))*ArcSin[c*x] + ArcSin[c*x]^2 - 6*Log[1 - E^((2*I)*ArcSin[c*x])]) 
+ (3*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(3*Sqrt[1 - c^2*x^2])
 

3.3.15.3 Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.80, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5196, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838, 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.

\(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x^2} \, dx\)

\(\Big \downarrow \) 5196

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5152

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

input

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x^2,x]
 

output

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

3.3.15.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5136

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( 
a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
 

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 5196

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

3.3.15.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (225 ) = 450\).

Time = 0.23 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.49

method	result	size
default	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}-a^{2} c^{2} x \sqrt {-c^{2} d \,x^{2}+d}-\frac {a^{2} c^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} c}{3 c^{2} x^{2}-3}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) x}+\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c}{c^{2} x^{2}-1}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} c}{2 c^{2} x^{2}-2}+\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{c^{2} x^{2}-1}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{x \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{c^{2} x^{2}-1}\right )\)	\(565\)
parts	\(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}-a^{2} c^{2} x \sqrt {-c^{2} d \,x^{2}+d}-\frac {a^{2} c^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} c}{3 c^{2} x^{2}-3}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) x}+\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c}{c^{2} x^{2}-1}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} c}{2 c^{2} x^{2}-2}+\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{c^{2} x^{2}-1}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{x \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{c^{2} x^{2}-1}\right )\)	\(565\)

input

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

output

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

3.3.15.5 Fricas [F]

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

input

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/x^2,x, algorithm="frica 
s")
 

output

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

3.3.15.6 Sympy [F]

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

input

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

output

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**2/x**2, x)
 

3.3.15.7 Maxima [F]

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

input

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/x^2,x, algorithm="maxim 
a")
 

output

-(c*sqrt(d)*arcsin(c*x) + sqrt(-c^2*d*x^2 + d)/x)*a^2 + sqrt(d)*integrate( 
(b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqr 
t(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/x^2, x)
 

3.3.15.8 Giac [F(-2)]

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

input

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/x^2,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
 

3.3.15.9 Mupad [F(-1)]

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

input

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

output

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