3.2.62 \(\int \frac {(d-c^2 d x^2) (a+b \arcsin (c x))^2}{x^2} \, dx\) [162]

3.2.62.1 Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^2} \, dx=2 b^2 c^2 d x-2 b c d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-2 c^2 d x (a+b \arcsin (c x))^2-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}-4 b c d (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )+2 i b^2 c d \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i b^2 c d \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]

output

2*b^2*c^2*d*x-2*c^2*d*x*(a+b*arcsin(c*x))^2-d*(-c^2*x^2+1)*(a+b*arcsin(c*x 
))^2/x-4*b*c*d*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))+2*I*b^2 
*c*d*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*b^2*c*d*polylog(2,I*c*x+(-c^ 
2*x^2+1)^(1/2))-2*b*c*d*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)
 

3.2.62.2 Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.36 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^2} \, dx=-\frac {d \left (a^2+a^2 c^2 x^2+2 a b c x \left (\sqrt {1-c^2 x^2}+c x \arcsin (c x)\right )+b^2 c x \left (2 \sqrt {1-c^2 x^2} \arcsin (c x)+c x \left (-2+\arcsin (c x)^2\right )\right )+2 a b \left (\arcsin (c x)+c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-i b^2 \left (i \arcsin (c x) \left (\arcsin (c x)+2 c x \left (-\log \left (1-e^{i \arcsin (c x)}\right )+\log \left (1+e^{i \arcsin (c x)}\right )\right )\right )+2 c x \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 c x \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )\right )}{x} \]

input

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

output

-((d*(a^2 + a^2*c^2*x^2 + 2*a*b*c*x*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]) 
+ b^2*c*x*(2*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + c*x*(-2 + ArcSin[c*x]^2)) + 2 
*a*b*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]) - I*b^2*(I*ArcSin[c*x] 
*(ArcSin[c*x] + 2*c*x*(-Log[1 - E^(I*ArcSin[c*x])] + Log[1 + E^(I*ArcSin[c 
*x])])) + 2*c*x*PolyLog[2, -E^(I*ArcSin[c*x])] - 2*c*x*PolyLog[2, E^(I*Arc 
Sin[c*x])])))/x)
 

3.2.62.3 Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5200, 5130, 5182, 24, 5198, 24, 5218, 3042, 4671, 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.

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

\(\Big \downarrow \) 5200

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

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5198

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.2.62.3.1 Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 4671

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

rule 5130

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar 
cSin[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - 
 c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[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]
 

rule 5198

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 + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] - Simp[b*c*(n/(f*(m + 2)))*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]) /; FreeQ[{a, b, c, d, e, 
 f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5200

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

rule 5218

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

3.2.62.4 Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.68

method	result	size
derivativedivides	\(c \left (-d \,a^{2} \left (c x +\frac {1}{c x}\right )-2 d \,b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-d \,b^{2} \arcsin \left (c x \right )^{2} c x +2 d \,b^{2} c x -\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{c x}-2 d \,b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d \,b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i d \,b^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d \,b^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d a b \left (c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(250\)
default	\(c \left (-d \,a^{2} \left (c x +\frac {1}{c x}\right )-2 d \,b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-d \,b^{2} \arcsin \left (c x \right )^{2} c x +2 d \,b^{2} c x -\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{c x}-2 d \,b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d \,b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i d \,b^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d \,b^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d a b \left (c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(250\)
parts	\(-d \,a^{2} \left (c^{2} x +\frac {1}{x}\right )-2 d \,b^{2} c \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-d \,b^{2} c^{2} \arcsin \left (c x \right )^{2} x +2 b^{2} c^{2} d x -\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{x}-2 d \,b^{2} c \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d \,b^{2} c \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i b^{2} c d \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i b^{2} c d \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d a b c \left (c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(253\)

input

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

output

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

3.2.62.5 Fricas [F]

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

input

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

output

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

3.2.62.6 Sympy [F]

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

input

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

output

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

3.2.62.7 Maxima [F]

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

input

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

output

-b^2*c^2*d*x*arcsin(c*x)^2 + 2*b^2*c^2*d*(x - sqrt(-c^2*x^2 + 1)*arcsin(c* 
x)/c) - a^2*c^2*d*x - 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c*d - 2 
*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*a*b*d - ( 
2*c*x*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sq 
rt(-c*x + 1))/(c^2*x^3 - x), x) + arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1 
))^2)*b^2*d/x - a^2*d/x
 

3.2.62.8 Giac [F]

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

input

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

output

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

3.2.62.9 Mupad [F(-1)]

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

input

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

output

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