Integrand size = 25, antiderivative size = 193 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^3} \, dx=-\frac {b c d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}-\frac {1}{2} c^2 d (a+b \arcsin (c x))^2-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}+\frac {i c^2 d (a+b \arcsin (c x))^3}{3 b}-c^2 d (a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )+b^2 c^2 d \log (x)+i b c^2 d (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )-\frac {1}{2} b^2 c^2 d \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right ) \]
-1/2*c^2*d*(a+b*arcsin(c*x))^2-1/2*d*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/x^2+ 1/3*I*c^2*d*(a+b*arcsin(c*x))^3/b-c^2*d*(a+b*arcsin(c*x))^2*ln(1-(I*c*x+(- c^2*x^2+1)^(1/2))^2)+b^2*c^2*d*ln(x)+I*b*c^2*d*(a+b*arcsin(c*x))*polylog(2 ,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*b^2*c^2*d*polylog(3,(I*c*x+(-c^2*x^2+1) ^(1/2))^2)-b*c*d*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/x
Time = 0.34 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^3} \, dx=\frac {1}{2} d \left (-\frac {a^2}{x^2}-\frac {2 a b \left (c x \sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{x^2}-2 a^2 c^2 \log (x)-\frac {b^2 \left (2 c x \sqrt {1-c^2 x^2} \arcsin (c x)+\arcsin (c x)^2-2 c^2 x^2 \log (c x)\right )}{x^2}+2 i a b c^2 \left (\arcsin (c x) \left (\arcsin (c x)+2 i \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )+\frac {1}{12} i b^2 c^2 \left (\pi ^3-8 \arcsin (c x)^3+24 i \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )-24 \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )+12 i \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )\right )\right ) \]
(d*(-(a^2/x^2) - (2*a*b*(c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 - 2*a^2 *c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 + (2*I)*a*b*c^2*(ArcSin[c*x]*(ArcSin[c*x] + (2*I) *Log[1 - E^((2*I)*ArcSin[c*x])]) + PolyLog[2, E^((2*I)*ArcSin[c*x])]) + (I /12)*b^2*c^2*(Pi^3 - 8*ArcSin[c*x]^3 + (24*I)*ArcSin[c*x]^2*Log[1 - E^((-2 *I)*ArcSin[c*x])] - 24*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + (1 2*I)*PolyLog[3, E^((-2*I)*ArcSin[c*x])])))/2
Time = 1.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5200, 5136, 3042, 25, 4200, 25, 2620, 3011, 2720, 5196, 14, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx+c^2 (-d) \int \frac {(a+b \arcsin (c x))^2}{x}dx-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle c^2 (-d) \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c x}d\arcsin (c x)+b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx+c^2 (-d) \int -(a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx+c^2 d \int (a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx+c^2 (-d) \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx+c^2 (-d) \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx+c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^2}dx-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5196 |
| \(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+b c d \left (c^2 \left (-\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx\right )+b c \int \frac {1}{x}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+b c d \left (c^2 \left (-\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}+b c \log (x)\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}+b c d \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}-\frac {c (a+b \arcsin (c x))^2}{2 b}+b c \log (x)\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle c^2 (-d) \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 x^2}+b c d \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}-\frac {c (a+b \arcsin (c x))^2}{2 b}+b c \log (x)\right )\) |
-1/2*(d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/x^2 + b*c*d*(-((Sqrt[1 - c^2* x^2]*(a + b*ArcSin[c*x]))/x) - (c*(a + b*ArcSin[c*x])^2)/(2*b) + b*c*Log[x ]) - c^2*d*(((-1/3*I)*(a + b*ArcSin[c*x])^3)/b - (2*I)*((I/2)*(a + b*ArcSi n[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)))
3.2.63.3.1 Defintions of rubi rules used
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_.)*((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 + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*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] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int [(f*x)^(m + 2)*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[ {a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
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 + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 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] && 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (211 ) = 422\).
Time = 0.26 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.37
| method | result | size |
| derivativedivides | \(c^{2} \left (-d \,a^{2} \left (\frac {1}{2 c^{2} x^{2}}+\ln \left (c x \right )\right )-d \,b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 d a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )\) | \(458\) |
| default | \(c^{2} \left (-d \,a^{2} \left (\frac {1}{2 c^{2} x^{2}}+\ln \left (c x \right )\right )-d \,b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 d a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )\) | \(458\) |
| parts | \(-\frac {d \,a^{2}}{2 x^{2}}-d \,a^{2} c^{2} \ln \left (x \right )-d \,b^{2} c^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 d a b \,c^{2} \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(461\) |
c^2*(-d*a^2*(1/2/c^2/x^2+ln(c*x))-d*b^2*(-1/3*I*arcsin(c*x)^3+1/2*arcsin(c *x)*(-2*I*c^2*x^2+2*c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x))/c^2/x^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))+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))-ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)+2*l n(I*c*x+(-c^2*x^2+1)^(1/2))-ln(1+I*c*x+(-c^2*x^2+1)^(1/2)))-2*d*a*b*(-1/2* I*arcsin(c*x)^2+1/2*(-I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x))/c^2/x^ 2+arcsin(c*x)*ln(1+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))-I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-I*polylog(2,I*c*x+(-c ^2*x^2+1)^(1/2))))
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^3} \, dx=- d \left (\int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {a^{2} c^{2}}{x}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx\right ) \]
-d*(Integral(-a**2/x**3, x) + Integral(a**2*c**2/x, x) + Integral(-b**2*as in(c*x)**2/x**3, x) + Integral(-2*a*b*asin(c*x)/x**3, x) + Integral(b**2*c **2*asin(c*x)**2/x, x) + Integral(2*a*b*c**2*asin(c*x)/x, x))
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
-a^2*c^2*d*log(x) - a*b*d*(sqrt(-c^2*x^2 + 1)*c/x + arcsin(c*x)/x^2) - 1/2 *a^2*d/x^2 - integrate((2*a*b*c^2*d*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c *x + 1)) + (b^2*c^2*d*x^2 - b^2*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2)/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right )}{x^3} \,d x \]