Integrand size = 26, antiderivative size = 150 \[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^2}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^2}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^2}+\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^2} \]
-x*(-c^2*x^2+1)/b/c/(a+b*arcsin(c*x))+1/4*Ci((a+b*arcsin(c*x))/b)*cos(a/b) /b^2/c^2+3/4*Ci(3*(a+b*arcsin(c*x))/b)*cos(3*a/b)/b^2/c^2+1/4*Si((a+b*arcs in(c*x))/b)*sin(a/b)/b^2/c^2+3/4*Si(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/ c^2
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83 \[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\frac {-\frac {4 b c x}{a+b \arcsin (c x)}+\frac {4 b c^3 x^3}{a+b \arcsin (c x)}+\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )+3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{4 b^2 c^2} \]
((-4*b*c*x)/(a + b*ArcSin[c*x]) + (4*b*c^3*x^3)/(a + b*ArcSin[c*x]) + Cos[ a/b]*CosIntegral[a/b + ArcSin[c*x]] + 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 3*Sin[(3*a)/b]*S inIntegral[3*(a/b + ArcSin[c*x])])/(4*b^2*c^2)
Time = 0.97 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.22, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5214, 5134, 3042, 3784, 25, 3042, 3780, 3783, 5146, 4906, 2009}
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.
| \(\displaystyle \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx\) |
| \(\Big \downarrow \) 5214 |
| \(\displaystyle -\frac {3 c \int \frac {x^2}{a+b \arcsin (c x)}dx}{b}+\frac {\int \frac {1}{a+b \arcsin (c x)}dx}{b c}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 5134 |
| \(\displaystyle \frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {3 c \int \frac {x^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {3 c \int \frac {x^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {3 c \int \frac {x^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {3 c \int \frac {x^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {3 c \int \frac {x^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c^2}-\frac {3 c \int \frac {x^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle -\frac {3 c \int \frac {x^2}{a+b \arcsin (c x)}dx}{b}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle -\frac {3 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle -\frac {3 \int \left (\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^2}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c^2}-\frac {3 \left (\frac {1}{4} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )}{b c (a+b \arcsin (c x))}\) |
-((x*(1 - c^2*x^2))/(b*c*(a + b*ArcSin[c*x]))) + (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b] + Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c^ 2) - (3*((Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/4 - (Cos[(3*a)/b]*C osIntegral[(3*(a + b*ArcSin[c*x]))/b])/4 + (Sin[a/b]*SinIntegral[(a + b*Ar cSin[c*x])/b])/4 - (Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/4 ))/(b^2*c^2)
3.4.83.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) Su bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 /(b*c^(m + 1)) Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p* ((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 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] + Simp[c*((m + 2*p + 1)/(b*f*(n + 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] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 , 0] && IGtQ[m, -3]
Time = 0.19 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(\frac {3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c -\sin \left (3 \arcsin \left (c x \right )\right ) b}{4 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(223\) |
1/4/c^2*(3*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b+3*arcsin(c*x)* Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b+arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a /b)*b+arcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+3*Si(3*arcsin(c*x)+3*a/b) *sin(3*a/b)*a+3*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a+Si(arcsin(c*x)+a/b)*s in(a/b)*a+Ci(arcsin(c*x)+a/b)*cos(a/b)*a-x*b*c-sin(3*arcsin(c*x))*b)/(a+b* arcsin(c*x))/b^2
\[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
(c^2*x^3 - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*inte grate((3*c^2*x^2 - 1)/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c), x) - x)/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)
Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (140) = 280\).
Time = 0.40 (sec) , antiderivative size = 608, normalized size of antiderivative = 4.05 \[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\frac {3 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {3 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {3 \, a \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {3 \, a \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b c x}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {9 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}\right )}} + \frac {b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}\right )}} - \frac {3 \, b \arcsin \left (c x\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}\right )}} + \frac {b \arcsin \left (c x\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}\right )}} - \frac {9 \, a \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}\right )}} + \frac {a \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}\right )}} - \frac {3 \, a \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}\right )}} + \frac {a \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, {\left (b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}\right )}} \]
3*b*arcsin(c*x)*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*ar csin(c*x) + a*b^2*c^2) + 3*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral( 3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 3*a*cos(a/b)^3* cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 3* a*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin( c*x) + a*b^2*c^2) + (c^2*x^2 - 1)*b*c*x/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 9/4*b*arcsin(c*x)*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2* arcsin(c*x) + a*b^2*c^2) + 1/4*b*arcsin(c*x)*cos(a/b)*cos_integral(a/b + a rcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 3/4*b*arcsin(c*x)*sin(a/b) *sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 1 /4*b*arcsin(c*x)*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin( c*x) + a*b^2*c^2) - 9/4*a*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^ 3*c^2*arcsin(c*x) + a*b^2*c^2) + 1/4*a*cos(a/b)*cos_integral(a/b + arcsin( c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 3/4*a*sin(a/b)*sin_integral(3*a/ b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 1/4*a*sin(a/b)*sin_ integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2)
Timed out. \[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x\,\sqrt {1-c^2\,x^2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]