Integrand size = 27, antiderivative size = 131 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx=-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d} \]
-2*(a+b*arcsin(c*x))^2*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)/d+I*b*(a+b*ar csin(c*x))*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d-I*b*(a+b*arcsin(c*x) )*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d-1/2*b^2*polylog(3,-(I*c*x+(-c^ 2*x^2+1)^(1/2))^2)/d+1/2*b^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(453\) vs. \(2(131)=262\).
Time = 0.48 (sec) , antiderivative size = 453, normalized size of antiderivative = 3.46 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx=\frac {-i b^2 \pi ^3-48 i a b \pi \arcsin (c x)+16 i b^2 \arcsin (c x)^3-96 a b \pi \log \left (1+e^{-i \arcsin (c x)}\right )-24 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-48 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )+24 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )-48 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+24 b^2 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+48 a b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-24 b^2 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )+24 a^2 \log (c x)-12 a^2 \log \left (1-c^2 x^2\right )+96 a b \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-24 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+24 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+48 i a b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+48 i a b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+24 i b^2 \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )+24 i b^2 \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-24 i a b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+12 b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )-12 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{24 d} \]
((-I)*b^2*Pi^3 - (48*I)*a*b*Pi*ArcSin[c*x] + (16*I)*b^2*ArcSin[c*x]^3 - 96 *a*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 24*a*b*Pi*Log[1 - I*E^(I*ArcSin[c* x])] - 48*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 24*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] - 48*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 24*b^2*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 48*a*b*ArcSin[c*x]* Log[1 - E^((2*I)*ArcSin[c*x])] - 24*b^2*ArcSin[c*x]^2*Log[1 + E^((2*I)*Arc Sin[c*x])] + 24*a^2*Log[c*x] - 12*a^2*Log[1 - c^2*x^2] + 96*a*b*Pi*Log[Cos [ArcSin[c*x]/2]] - 24*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 24*a*b*Pi *Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (48*I)*a*b*PolyLog[2, (-I)*E^(I*ArcSin [c*x])] + (48*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c*x])] + (24*I)*b^2*ArcSin[c *x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + (24*I)*b^2*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (24*I)*a*b*PolyLog[2, E^((2*I)*ArcSin[c*x])] + 12*b^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] - 12*b^2*PolyLog[3, -E^((2*I)*Ar cSin[c*x])])/(24*d)
Time = 0.57 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5184, 4919, 3042, 4671, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c x))^2}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)}{d}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)}{d}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {2 \left (-b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+b \int (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 \left (-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )}{d}\) |
(2*(-((a + b*ArcSin[c*x])^2*ArcTanh[E^((2*I)*ArcSin[c*x])]) + b*((I/2)*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (b*PolyLog[3, -E^((2 *I)*ArcSin[c*x])])/4) - b*((I/2)*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*A rcSin[c*x])] - (b*PolyLog[3, E^((2*I)*ArcSin[c*x])])/4)))/d
3.2.88.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
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]
Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSi n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (175 ) = 350\).
Time = 0.20 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.45
| method | result | size |
| parts | \(-\frac {a^{2} \left (-\ln \left (x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\) | \(452\) |
| derivativedivides | \(-\frac {a^{2} \left (-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\) | \(454\) |
| default | \(-\frac {a^{2} \left (-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\) | \(454\) |
-a^2/d*(-ln(x)+1/2*ln(c*x-1)+1/2*ln(c*x+1))-b^2/d*(-arcsin(c*x)^2*ln(1+I*c *x+(-c^2*x^2+1)^(1/2))+2*I*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2) )-2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)^2*ln(1+(I*c*x+(-c^2*x ^2+1)^(1/2))^2)-I*arcsin(c*x)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/2 *polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x ^2+1)^(1/2))+2*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-2*polylog (3,I*c*x+(-c^2*x^2+1)^(1/2)))-2*a*b/d*(-arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1 )^(1/2))+I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)*ln(1+(I*c*x+(- c^2*x^2+1)^(1/2))^2)-1/2*I*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-arcsin (c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)) )
\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x} \,d x } \]
\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a^{2}}{c^{2} x^{3} - x}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{3} - x}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{3} - x}\, dx}{d} \]
-(Integral(a**2/(c**2*x**3 - x), x) + Integral(b**2*asin(c*x)**2/(c**2*x** 3 - x), x) + Integral(2*a*b*asin(c*x)/(c**2*x**3 - x), x))/d
\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x} \,d x } \]
-1/2*a^2*(log(c*x + 1)/d + log(c*x - 1)/d - 2*log(x)/d) - integrate((b^2*a rctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*d*x^3 - d*x), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x\,\left (d-c^2\,d\,x^2\right )} \,d x \]