\(\int \frac {a+b \arcsin (c x)}{(d-c^2 d x^2)^3} \, dx\) [6]

Optimal result

Integrand size = 22, antiderivative size = 196 \[ \int \frac {a+b \arcsin (c x)}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \arcsin (c x))}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{4 c d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{8 c d^3}-\frac {3 i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{8 c d^3} \] Output:

-1/12*b/c/d^3/(-c^2*x^2+1)^(3/2)-3/8*b/c/d^3/(-c^2*x^2+1)^(1/2)+1/4*x*(a+b 
*arcsin(c*x))/d^3/(-c^2*x^2+1)^2+3/8*x*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)- 
3/4*I*(a+b*arcsin(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/c/d^3+3/8*I*b*pol 
ylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c/d^3-3/8*I*b*polylog(2,I*(I*c*x+(-c 
^2*x^2+1)^(1/2)))/c/d^3
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(196)=392\).

Time = 1.13 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.56 \[ \int \frac {a+b \arcsin (c x)}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {\frac {2 b \sqrt {1-c^2 x^2}}{3 c (-1+c x)^2}-\frac {b x \sqrt {1-c^2 x^2}}{3 (-1+c x)^2}+\frac {2 b \sqrt {1-c^2 x^2}}{3 c (1+c x)^2}+\frac {b x \sqrt {1-c^2 x^2}}{3 (1+c x)^2}+\frac {3 b \sqrt {1-c^2 x^2}}{c-c^2 x}+\frac {3 b \sqrt {1-c^2 x^2}}{c+c^2 x}-\frac {4 a x}{\left (-1+c^2 x^2\right )^2}+\frac {6 a x}{-1+c^2 x^2}+\frac {3 i b \pi \arcsin (c x)}{c}-\frac {b \arcsin (c x)}{c (-1+c x)^2}+\frac {b \arcsin (c x)}{c (1+c x)^2}-\frac {3 b \arcsin (c x)}{c-c^2 x}+\frac {3 b \arcsin (c x)}{c+c^2 x}-\frac {3 b \pi \log \left (1-i e^{i \arcsin (c x)}\right )}{c}-\frac {6 b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )}{c}-\frac {3 b \pi \log \left (1+i e^{i \arcsin (c x)}\right )}{c}+\frac {6 b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )}{c}+\frac {3 a \log (1-c x)}{c}-\frac {3 a \log (1+c x)}{c}+\frac {3 b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}+\frac {3 b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {6 i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c}+\frac {6 i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c}}{16 d^3} \] Input:

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

Output:

-1/16*((2*b*Sqrt[1 - c^2*x^2])/(3*c*(-1 + c*x)^2) - (b*x*Sqrt[1 - c^2*x^2] 
)/(3*(-1 + c*x)^2) + (2*b*Sqrt[1 - c^2*x^2])/(3*c*(1 + c*x)^2) + (b*x*Sqrt 
[1 - c^2*x^2])/(3*(1 + c*x)^2) + (3*b*Sqrt[1 - c^2*x^2])/(c - c^2*x) + (3* 
b*Sqrt[1 - c^2*x^2])/(c + c^2*x) - (4*a*x)/(-1 + c^2*x^2)^2 + (6*a*x)/(-1 
+ c^2*x^2) + ((3*I)*b*Pi*ArcSin[c*x])/c - (b*ArcSin[c*x])/(c*(-1 + c*x)^2) 
 + (b*ArcSin[c*x])/(c*(1 + c*x)^2) - (3*b*ArcSin[c*x])/(c - c^2*x) + (3*b* 
ArcSin[c*x])/(c + c^2*x) - (3*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c - (6*b* 
ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])])/c - (3*b*Pi*Log[1 + I*E^(I*ArcSi 
n[c*x])])/c + (6*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c + (3*a*Log[ 
1 - c*x])/c - (3*a*Log[1 + c*x])/c + (3*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x]) 
/4]])/c + (3*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((6*I)*b*PolyLog[2 
, (-I)*E^(I*ArcSin[c*x])])/c + ((6*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/c 
)/d^3
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5162, 27, 241, 5162, 241, 5164, 3042, 4669, 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.

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

\(\Big \downarrow \) 5162

\(\displaystyle \frac {3 \int \frac {a+b \arcsin (c x)}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 d}-\frac {b c \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}}dx}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}-\frac {b c \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}}dx}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {3 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5162

\(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx-\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5164

\(\displaystyle \frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 4669

\(\displaystyle \frac {3 \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {3 \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {3 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{4 d^3}+\frac {x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}\)

Input:

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

Output:

-1/12*b/(c*d^3*(1 - c^2*x^2)^(3/2)) + (x*(a + b*ArcSin[c*x]))/(4*d^3*(1 - 
c^2*x^2)^2) + (3*(-1/2*b/(c*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/( 
2*(1 - c^2*x^2)) + ((-2*I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])] + 
 I*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - I*b*PolyLog[2, I*E^(I*ArcSin[c*x 
])])/(2*c)))/(4*d^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

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 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 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 5164

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \arcsin \left (c x \right )-9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arcsin \left (c x \right )+11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c}\)	\(262\)
default	\(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \arcsin \left (c x \right )-9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arcsin \left (c x \right )+11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c}\)	\(262\)
parts	\(-\frac {a \left (-\frac {1}{16 c \left (c x -1\right )^{2}}+\frac {3}{16 c \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c}+\frac {1}{16 c \left (c x +1\right )^{2}}+\frac {3}{16 c \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \arcsin \left (c x \right )-9 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-15 c x \arcsin \left (c x \right )+11 \sqrt {-c^{2} x^{2}+1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3} c}\)	\(279\)

Input:

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

Output:

1/c*(-a/d^3*(1/16/(c*x+1)^2+3/16/(c*x+1)-3/16*ln(c*x+1)-1/16/(c*x-1)^2+3/1 
6/(c*x-1)+3/16*ln(c*x-1))-b/d^3*(1/24*(9*c^3*x^3*arcsin(c*x)-9*c^2*x^2*(-c 
^2*x^2+1)^(1/2)-15*c*x*arcsin(c*x)+11*(-c^2*x^2+1)^(1/2))/(c^4*x^4-2*c^2*x 
^2+1)+3/8*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/8*arcsin(c*x)*l 
n(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/8*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2 
)))+3/8*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/16*a*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) - 3*log(c 
*x + 1)/(c*d^3) + 3*log(c*x - 1)/(c*d^3)) + 1/16*(3*(c^4*x^4 - 2*c^2*x^2 + 
 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 3*(c^4*x^4 - 
 2*c^2*x^2 + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 
 2*(3*c^3*x^3 - 5*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 16*(c^ 
5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3)*integrate(-1/16*(6*c^3*x^3 - 10*c*x - 3 
*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) + 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log( 
-c*x + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c 
^2*d^3*x^2 - d^3), x))*b/(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-16 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{5} x^{4}+32 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{3} x^{2}-16 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b c -3 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{4} x^{4}+6 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{2} x^{2}-3 \,\mathrm {log}\left (c^{2} x -c \right ) a +3 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{4} x^{4}-6 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{2} x^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a -6 a \,c^{3} x^{3}+10 a c x}{16 c \,d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 16*int(asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b*c**5 
*x**4 + 32*int(asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b* 
c**3*x**2 - 16*int(asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x 
)*b*c - 3*log(c**2*x - c)*a*c**4*x**4 + 6*log(c**2*x - c)*a*c**2*x**2 - 3* 
log(c**2*x - c)*a + 3*log(c**2*x + c)*a*c**4*x**4 - 6*log(c**2*x + c)*a*c* 
*2*x**2 + 3*log(c**2*x + c)*a - 6*a*c**3*x**3 + 10*a*c*x)/(16*c*d**3*(c**4 
*x**4 - 2*c**2*x**2 + 1))