Integrand size = 26, antiderivative size = 214 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^2 c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^2}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^2 c^2}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^2}+\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^2} \]
-x*(-c^2*x^2+1)^2/b/c/(a+b*arcsin(c*x))+1/8*Ci((a+b*arcsin(c*x))/b)*cos(a/ b)/b^2/c^2+9/16*Ci(3*(a+b*arcsin(c*x))/b)*cos(3*a/b)/b^2/c^2+5/16*Ci(5*(a+ b*arcsin(c*x))/b)*cos(5*a/b)/b^2/c^2+1/8*Si((a+b*arcsin(c*x))/b)*sin(a/b)/ b^2/c^2+9/16*Si(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/c^2+5/16*Si(5*(a+b*a rcsin(c*x))/b)*sin(5*a/b)/b^2/c^2
Time = 0.46 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\frac {-16 b c x+32 b c^3 x^3-16 b c^5 x^5+2 (a+b \arcsin (c x)) \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )+9 (a+b \arcsin (c x)) \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+5 a \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+5 b \arcsin (c x) \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+2 a \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+2 b \arcsin (c x) \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+9 a \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+9 b \arcsin (c x) \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+5 a \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+5 b \arcsin (c x) \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{16 b^2 c^2 (a+b \arcsin (c x))} \]
(-16*b*c*x + 32*b*c^3*x^3 - 16*b*c^5*x^5 + 2*(a + b*ArcSin[c*x])*Cos[a/b]* CosIntegral[a/b + ArcSin[c*x]] + 9*(a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosInt egral[3*(a/b + ArcSin[c*x])] + 5*a*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSi n[c*x])] + 5*b*ArcSin[c*x]*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 2*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 2*b*ArcSin[c*x]*Sin[a/b]* SinIntegral[a/b + ArcSin[c*x]] + 9*a*Sin[(3*a)/b]*SinIntegral[3*(a/b + Arc Sin[c*x])] + 9*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x] )] + 5*a*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 5*b*ArcSin[c*x] *Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b^2*c^2*(a + b*ArcSi n[c*x]))
Time = 1.16 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.36, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5214, 5168, 3042, 3793, 2009, 5224, 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.
\(\displaystyle \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx\) |
\(\Big \downarrow \) 5214 |
\(\displaystyle \frac {\int \frac {1-c^2 x^2}{a+b \arcsin (c x)}dx}{b c}-\frac {5 c \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 5168 |
\(\displaystyle \frac {\int \frac {\cos ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {5 c \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )^3}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {5 c \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\int \left (\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 (a+b \arcsin (c x))}+\frac {3 \cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^2}-\frac {5 c \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5 c \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \arcsin (c x)}dx}{b}+\frac {\frac {3}{4} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {3}{4} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {5 \int \frac {\cos ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}+\frac {\frac {3}{4} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {3}{4} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {5 \int \left (-\frac {\cos \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 (a+b \arcsin (c x))}+\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{8 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^2}+\frac {\frac {3}{4} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {3}{4} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3}{4} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {3}{4} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {5 \left (\frac {1}{8} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{16} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {1}{16} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{16} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {1}{16} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^2}{b c (a+b \arcsin (c x))}\) |
-((x*(1 - c^2*x^2)^2)/(b*c*(a + b*ArcSin[c*x]))) + ((3*Cos[a/b]*CosIntegra l[(a + b*ArcSin[c*x])/b])/4 + (Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c *x]))/b])/4 + (3*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/4 + (Sin[(3* a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/4)/(b^2*c^2) - (5*((Cos[a/b] *CosIntegral[(a + b*ArcSin[c*x])/b])/8 - (Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/16 - (Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]) )/b])/16 + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/8 - (Sin[(3*a)/b] *SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/16 - (Sin[(5*a)/b]*SinIntegral[(5 *(a + b*ArcSin[c*x]))/b])/16))/(b^2*c^2)
3.4.92.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
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_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Subst[Int[ x^n*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b , c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.14 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.59
method | result | size |
default | \(\frac {5 \arcsin \left (c x \right ) \operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +5 \arcsin \left (c x \right ) \operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b +9 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +9 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b +2 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +2 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +5 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +5 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a +9 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +9 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +2 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +2 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -2 x b c -\sin \left (5 \arcsin \left (c x \right )\right ) b -3 \sin \left (3 \arcsin \left (c x \right )\right ) b}{16 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(341\) |
1/16/c^2*(5*arcsin(c*x)*Si(5*arcsin(c*x)+5*a/b)*sin(5*a/b)*b+5*arcsin(c*x) *Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*b+9*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b )*sin(3*a/b)*b+9*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b+2*arcsin (c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+2*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos (a/b)*b+5*Si(5*arcsin(c*x)+5*a/b)*sin(5*a/b)*a+5*Ci(5*arcsin(c*x)+5*a/b)*c os(5*a/b)*a+9*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a+9*Ci(3*arcsin(c*x)+3*a/ b)*cos(3*a/b)*a+2*Si(arcsin(c*x)+a/b)*sin(a/b)*a+2*Ci(arcsin(c*x)+a/b)*cos (a/b)*a-2*x*b*c-sin(5*arcsin(c*x))*b-3*sin(3*arcsin(c*x))*b)/(a+b*arcsin(c *x))/b^2
\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
integral(-(c^2*x^3 - x)*sqrt(-c^2*x^2 + 1)/(b^2*arcsin(c*x)^2 + 2*a*b*arcs in(c*x) + a^2), x)
\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
-(c^4*x^5 - 2*c^2*x^3 - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate((5*c^4*x^4 - 6*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. 1215 vs. \(2 (201) = 402\).
Time = 0.41 (sec) , antiderivative size = 1215, normalized size of antiderivative = 5.68 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \]
5*b*arcsin(c*x)*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*ar csin(c*x) + a*b^2*c^2) + 5*b*arcsin(c*x)*cos(a/b)^4*sin(a/b)*sin_integral( 5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 5*a*cos(a/b)^5* cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 5* a*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin( c*x) + a*b^2*c^2) - 25/4*b*arcsin(c*x)*cos(a/b)^3*cos_integral(5*a/b + 5*a rcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 9/4*b*arcsin(c*x)*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) - 15/4*b*arcsin(c*x)*cos(a/b)^2*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x) )/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 9/4*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) - (c^2*x^2 - 1)^2*b*c*x/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) - 25/4*a*cos(a/b)^ 3*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 9/4*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) - 15/4*a*cos(a/b)^2*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c* x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 9/4*a*cos(a/b)^2*sin(a/b)*sin_inte gral(3*a/b + 3*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/16*b*ar csin(c*x)*cos(a/b)*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x ) + a*b^2*c^2) - 27/16*b*arcsin(c*x)*cos(a/b)*cos_integral(3*a/b + 3*arcsi n(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 1/8*b*arcsin(c*x)*cos(a/b)*...
Timed out. \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x\,{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]