Integrand size = 25, antiderivative size = 178 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x} \, dx=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} d (a+b \arcsin (c x))^2+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i d (a+b \arcsin (c x))^3}{3 b}+d (a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )-i b d (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+\frac {1}{2} b^2 d \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right ) \]
1/4*b^2*c^2*d*x^2-1/4*d*(a+b*arcsin(c*x))^2+1/2*d*(-c^2*x^2+1)*(a+b*arcsin (c*x))^2-1/3*I*d*(a+b*arcsin(c*x))^3/b+d*(a+b*arcsin(c*x))^2*ln(1-(I*c*x+( -c^2*x^2+1)^(1/2))^2)-I*b*d*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1 )^(1/2))^2)+1/2*b^2*d*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*b*c*d*x* (a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)
Time = 0.41 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.44 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x} \, dx=\frac {1}{2} d \left (-a^2 c^2 x^2-2 a b c^2 x^2 \arcsin (c x)-a b \left (c x \sqrt {1-c^2 x^2}-2 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )+\frac {1}{4} b^2 \left (-1+2 \arcsin (c x)^2\right ) \cos (2 \arcsin (c x))+4 a b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+2 a^2 \log (x)-2 i a b \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )+\frac {1}{12} b^2 \left (-i \pi ^3+8 i \arcsin (c x)^3+24 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+24 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )\right )-\frac {1}{2} b^2 \arcsin (c x) \sin (2 \arcsin (c x))\right ) \]
(d*(-(a^2*c^2*x^2) - 2*a*b*c^2*x^2*ArcSin[c*x] - a*b*(c*x*Sqrt[1 - c^2*x^2 ] - 2*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]) + (b^2*(-1 + 2*ArcSin[c*x]^2 )*Cos[2*ArcSin[c*x]])/4 + 4*a*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*a^2*Log[x] - (2*I)*a*b*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c* x])]) + (b^2*((-I)*Pi^3 + (8*I)*ArcSin[c*x]^3 + 24*ArcSin[c*x]^2*Log[1 - E ^((-2*I)*ArcSin[c*x])] + (24*I)*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c* x])] + 12*PolyLog[3, E^((-2*I)*ArcSin[c*x])]))/12 - (b^2*ArcSin[c*x]*Sin[2 *ArcSin[c*x]])/2))/2
Time = 1.08 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5202, 5136, 3042, 25, 4200, 25, 2620, 3011, 2720, 5156, 15, 5152, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+d \int \frac {(a+b \arcsin (c x))^2}{x}dx+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c x}d\arcsin (c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+d \int -(a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-d \int (a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+d \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+d \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -b c d \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -b c d \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -b c d \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )+d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-b c d \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-b c d \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )+d \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )\) |
(d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/2 - b*c*d*(-1/4*(b*c*x^2) + (x*Sqr t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 + (a + b*ArcSin[c*x])^2/(4*b*c)) + d *(((-1/3*I)*(a + b*ArcSin[c*x])^3)/b - (2*I)*((I/2)*(a + b*ArcSin[c*x])^2* Log[1 - E^((2*I)*ArcSin[c*x])] - I*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)))
3.2.61.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^ m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
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 = 0.20 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.07
| method | result | size |
| parts | \(-d \,a^{2} \left (\frac {c^{2} x^{2}}{2}-\ln \left (x \right )\right )-d \,b^{2} \left (\frac {i \arcsin \left (c x \right )^{3}}{3}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\left (2 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {\arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{4}\right )-2 d a b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}+\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )\) | \(368\) |
| derivativedivides | \(-d \,a^{2} \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-d \,b^{2} \left (\frac {i \arcsin \left (c x \right )^{3}}{3}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\left (2 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {\arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{4}\right )-2 d a b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}+\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )\) | \(370\) |
| default | \(-d \,a^{2} \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-d \,b^{2} \left (\frac {i \arcsin \left (c x \right )^{3}}{3}-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\left (2 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {\arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{4}\right )-2 d a b \left (\frac {i \arcsin \left (c x \right )^{2}}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{4}+\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )\) | \(370\) |
-d*a^2*(1/2*c^2*x^2-ln(x))-d*b^2*(1/3*I*arcsin(c*x)^3-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*polylo g(3,I*c*x+(-c^2*x^2+1)^(1/2))-1/8*(2*arcsin(c*x)^2-1)*cos(2*arcsin(c*x))+1 /4*arcsin(c*x)*sin(2*arcsin(c*x)))-2*d*a*b*(1/2*I*arcsin(c*x)^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))-ar csin(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))-1/4*arcsin(c*x)*cos(2*arcsin(c*x))+1/8*sin(2*arcsin(c*x)))
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arcsin(c*x))/x, x)
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x} \, dx=- d \left (\int \left (- \frac {a^{2}}{x}\right )\, dx + \int a^{2} c^{2} x\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int b^{2} c^{2} x \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{2} x \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
-d*(Integral(-a**2/x, x) + Integral(a**2*c**2*x, x) + Integral(-b**2*asin( c*x)**2/x, x) + Integral(-2*a*b*asin(c*x)/x, x) + Integral(b**2*c**2*x*asi n(c*x)**2, x) + Integral(2*a*b*c**2*x*asin(c*x), x))
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
-1/2*a^2*c^2*d*x^2 + a^2*d*log(x) - integrate(((b^2*c^2*d*x^2 - b^2*d)*arc tan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arct an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/x, x)
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right )}{x} \,d x \]