Integrand size = 29, antiderivative size = 378 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x} \, dx=-2 b^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]
-2*b^2*(-c^2*d*x^2+d)^(1/2)+(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2-2*a*b *c*x*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*b^2*c*x*arcsin(c*x)*(-c^2*d *x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2 *x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2*I*b*(a+b*arcsin(c *x))*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1 )^(1/2)-2*I*b*(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2* d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2*b^2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2 ))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2*b^2*polylog(3,I*c*x+(-c^2*x^2 +1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 1.23 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x} \, dx=a^2 \sqrt {d-c^2 d x^2}+a^2 \sqrt {d} \log (c x)-a^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {1-c^2 x^2} \arcsin (c x)+\arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-\arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \left (-2 \sqrt {1-c^2 x^2}-2 c x \arcsin (c x)+\sqrt {1-c^2 x^2} \arcsin (c x)^2+\arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )-\arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )+2 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-2 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+2 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}} \]
a^2*Sqrt[d - c^2*d*x^2] + a^2*Sqrt[d]*Log[c*x] - a^2*Sqrt[d]*Log[d + Sqrt[ d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^ 2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]* Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[ 2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (b^2*Sqrt[d - c^2*d*x^2]*(-2*S qrt[1 - c^2*x^2] - 2*c*x*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + A rcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]^2*Log[1 + E^(I*ArcSi n[c*x])] + (2*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*ArcSin [c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 2*PolyLog[3, -E^(I*ArcSin[c*x])] + 2 *PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]
Time = 0.97 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.62, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {5198, 2009, 5218, 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.
| \(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle -\frac {2 b c \sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{c x}d\arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))^2 \csc (\arcsin (c x))d\arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{\sqrt {1-c^2 x^2}}\) |
Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (2*b*c*Sqrt[d - c^2*d*x^2]*(a* x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/Sqrt[1 - c^2*x^2] + (Sqrt[ d - c^2*d*x^2]*(-2*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])] + 2*b* (I*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])] - b*PolyLog[3, -E^(I *ArcSin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])] - b*PolyLog[3, E^(I*ArcSin[c*x])])))/Sqrt[1 - c^2*x^2]
3.3.14.3.1 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[((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])
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]
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]
Time = 0.21 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(-\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right ) \sqrt {d}\, a^{2}+\sqrt {-c^{2} d \,x^{2}+d}\, a^{2}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{2} x^{2}-2}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} x^{2}-2}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{2} x^{2}-1}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} x^{2}-2}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} x^{2}-2}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{2} x^{2}-1}\right )\) | \(659\) |
| parts | \(-\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right ) \sqrt {d}\, a^{2}+\sqrt {-c^{2} d \,x^{2}+d}\, a^{2}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{2} x^{2}-2}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} x^{2}-2}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{2} x^{2}-1}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} x^{2}-2}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} x^{2}-2}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{2} x^{2}-1}\right )\) | \(659\) |
-ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)*d^(1/2)*a^2+(-c^2*d*x^2+d)^(1/ 2)*a^2+b^2*(1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1 )*(arcsin(c*x)^2-2+2*I*arcsin(c*x))/(c^2*x^2-1)+1/2*(-d*(c^2*x^2-1))^(1/2) *(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))/(c ^2*x^2-1)+(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(arcsin(c*x)^2*ln(1+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*a rcsin(c*x)*polylog(2,-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*polylog (3,I*c*x+(-c^2*x^2+1)^(1/2)))/(c^2*x^2-1))+2*a*b*(1/2*(-d*(c^2*x^2-1))^(1/ 2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)/(c^2*x^2-1)+1/2*(- d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)-I)/ (c^2*x^2-1)-I*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*(I*arc sin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2 +1)^(1/2))-polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+polylog(2,-I*c*x-(-c^2*x^2+ 1)^(1/2))))
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x}\, dx \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
-(sqrt(d)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - sqrt(- c^2*d*x^2 + d))*a^2 + sqrt(d)*integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sq rt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c *x + 1)*sqrt(-c*x + 1)/x, x)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}}{x} \,d x \]