Integrand size = 27, antiderivative size = 233 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b (a+b \arcsin (c x))}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {i (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{c^3 d^2}+\frac {b^2 \text {arctanh}(c x)}{c^3 d^2}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^3 d^2}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^3 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{c^3 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{c^3 d^2} \]
1/2*x*(a+b*arcsin(c*x))^2/c^2/d^2/(-c^2*x^2+1)+I*(a+b*arcsin(c*x))^2*arcta n(I*c*x+(-c^2*x^2+1)^(1/2))/c^3/d^2+b^2*arctanh(c*x)/c^3/d^2-I*b*(a+b*arcs in(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^2+I*b*(a+b*arcsin( c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^2+b^2*polylog(3,-I*(I* c*x+(-c^2*x^2+1)^(1/2)))/c^3/d^2-b^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2) ))/c^3/d^2-b*(a+b*arcsin(c*x))/c^3/d^2/(-c^2*x^2+1)^(1/2)
Time = 2.03 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.96 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {2 a b \sqrt {1-c^2 x^2}}{-1+c x}-\frac {2 a b \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a^2 c x}{-1+c^2 x^2}+2 i a b \pi \arcsin (c x)-\frac {2 a b \arcsin (c x)}{-1+c x}-\frac {2 a b \arcsin (c x)}{1+c x}-\frac {4 b^2 \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 c x \arcsin (c x)^2}{1-c^2 x^2}+4 i b^2 \arcsin (c x)^2 \arctan \left (e^{i \arcsin (c x)}\right )+4 b^2 \text {arctanh}(c x)-2 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-4 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-2 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+4 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+a^2 \log (1-c x)-a^2 \log (1+c x)+2 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-4 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+4 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+4 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-4 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 c^3 d^2} \]
((2*a*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) - (2*a*b*Sqrt[1 - c^2*x^2])/(1 + c*x ) - (2*a^2*c*x)/(-1 + c^2*x^2) + (2*I)*a*b*Pi*ArcSin[c*x] - (2*a*b*ArcSin[ c*x])/(-1 + c*x) - (2*a*b*ArcSin[c*x])/(1 + c*x) - (4*b^2*ArcSin[c*x])/Sqr t[1 - c^2*x^2] + (2*b^2*c*x*ArcSin[c*x]^2)/(1 - c^2*x^2) + (4*I)*b^2*ArcSi n[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] + 4*b^2*ArcTanh[c*x] - 2*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 4*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2 *a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 4*a*b*ArcSin[c*x]*Log[1 + I*E^(I*Ar cSin[c*x])] + a^2*Log[1 - c*x] - a^2*Log[1 + c*x] + 2*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*a*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (4*I)*b *(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (4*I)*b*(a + b*A rcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] + 4*b^2*PolyLog[3, (-I)*E^(I*A rcSin[c*x])] - 4*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*c^3*d^2)
Time = 1.12 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5206, 27, 5164, 3042, 4669, 3011, 2720, 5182, 219, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}-\frac {\int \frac {(a+b \arcsin (c x))^2}{d \left (1-c^2 x^2\right )}dx}{2 c^2 d}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}-\frac {\int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{2 c^2 d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle -\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}-\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c^3 d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c^3 d^2}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3 d^2}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3 d^2}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3 d^2}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3 d^2}-\frac {b \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )}{c d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3 d^2}-\frac {b \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )}{c d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{2 c^3 d^2}-\frac {b \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )}{c d^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\) |
(x*(a + b*ArcSin[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) - (b*((a + b*ArcSin[c* x])/(c^2*Sqrt[1 - c^2*x^2]) - (b*ArcTanh[c*x])/c^2))/(c*d^2) - ((-2*I)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])] + 2*b*(I*(a + b*ArcSin[c*x])* PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - b*PolyLog[3, (-I)*E^(I*ArcSin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] - b*PolyLog[ 3, I*E^(I*ArcSin[c*x])]))/(2*c^3*d^2)
3.2.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 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[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp [b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[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 = 0.33 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.91
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arcsin \left (c x \right ) \left (c x \arcsin \left (c x \right )-2 \sqrt {-c^{2} x^{2}+1}\right )}{2 \left (c^{2} x^{2}-1\right )}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {c x \arcsin \left (c x \right )-\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}+\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}}{c^{3}}\) | \(446\) |
| default | \(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arcsin \left (c x \right ) \left (c x \arcsin \left (c x \right )-2 \sqrt {-c^{2} x^{2}+1}\right )}{2 \left (c^{2} x^{2}-1\right )}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {c x \arcsin \left (c x \right )-\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}+\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}}{c^{3}}\) | \(446\) |
| parts | \(\frac {a^{2} \left (-\frac {1}{4 c^{3} \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{4 c^{3}}-\frac {1}{4 c^{3} \left (c x +1\right )}-\frac {\ln \left (c x +1\right )}{4 c^{3}}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arcsin \left (c x \right ) \left (c x \arcsin \left (c x \right )-2 \sqrt {-c^{2} x^{2}+1}\right )}{2 \left (c^{2} x^{2}-1\right )}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{3}}+\frac {2 a b \left (-\frac {c x \arcsin \left (c x \right )-\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}+\frac {\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2} c^{3}}\) | \(460\) |
1/c^3*(a^2/d^2*(-1/4/(c*x-1)+1/4*ln(c*x-1)-1/4/(c*x+1)-1/4*ln(c*x+1))+b^2/ d^2*(-1/2/(c^2*x^2-1)*arcsin(c*x)*(c*x*arcsin(c*x)-2*(-c^2*x^2+1)^(1/2))+1 /2*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-I*arcsin(c*x)*polylog( 2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))- 1/2*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I*arcsin(c*x)*polylog (2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2 *I*arctan(I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-1/2*(c*x*arcsin(c*x)-(-c^ 2*x^2+1)^(1/2))/(c^2*x^2-1)+1/2*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/ 2)))-1/2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-1/2*I*dilog(1+I*(I *c*x+(-c^2*x^2+1)^(1/2)))+1/2*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
integral((b^2*x^2*arcsin(c*x)^2 + 2*a*b*x^2*arcsin(c*x) + a^2*x^2)/(c^4*d^ 2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2} x^{2}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
(Integral(a**2*x**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b**2*x**2 *asin(c*x)**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(2*a*b*x**2*asin (c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d**2
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
-1/4*a^2*(2*x/(c^4*d^2*x^2 - c^2*d^2) + log(c*x + 1)/(c^3*d^2) - log(c*x - 1)/(c^3*d^2)) - 1/4*(2*b^2*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) ^2 + (b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log( c*x + 1) - (b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^ 2*log(-c*x + 1) + 4*(c^5*d^2*x^2 - c^3*d^2)*integrate(-1/2*(4*a*b*c^2*x^2* arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (2*b^2*c*x*arctan2(c*x, sqrt( c*x + 1)*sqrt(-c*x + 1)) + (b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)* sqrt(-c*x + 1))*log(c*x + 1) - (b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^6*d^2* x^4 - 2*c^4*d^2*x^2 + c^2*d^2), x))/(c^5*d^2*x^2 - c^3*d^2)
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]