Integrand size = 23, antiderivative size = 169 \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=-\frac {i (a+b \arcsin (c+d x))^4}{4 b d e}+\frac {(a+b \arcsin (c+d x))^3 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )}{2 d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,e^{2 i \arcsin (c+d x)}\right )}{4 d e} \]
-1/4*I*(a+b*arcsin(d*x+c))^4/b/d/e+(a+b*arcsin(d*x+c))^3*ln(1-(I*(d*x+c)+( 1-(d*x+c)^2)^(1/2))^2)/d/e-3/2*I*b*(a+b*arcsin(d*x+c))^2*polylog(2,(I*(d*x +c)+(1-(d*x+c)^2)^(1/2))^2)/d/e+3/2*b^2*(a+b*arcsin(d*x+c))*polylog(3,(I*( d*x+c)+(1-(d*x+c)^2)^(1/2))^2)/d/e+3/4*I*b^3*polylog(4,(I*(d*x+c)+(1-(d*x+ c)^2)^(1/2))^2)/d/e
Time = 0.47 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=-\frac {i \left (8 a b^2 \pi ^3+b^3 \pi ^4+96 a^2 b \arcsin (c+d x)^2-64 a b^2 \arcsin (c+d x)^3-16 b^3 \arcsin (c+d x)^4+192 i a b^2 \arcsin (c+d x)^2 \log \left (1-e^{-2 i \arcsin (c+d x)}\right )+64 i b^3 \arcsin (c+d x)^3 \log \left (1-e^{-2 i \arcsin (c+d x)}\right )+192 i a^2 b \arcsin (c+d x) \log \left (1-e^{2 i \arcsin (c+d x)}\right )+64 i a^3 \log (c+d x)-96 b^2 \arcsin (c+d x) (2 a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c+d x)}\right )+96 a^2 b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )+96 i a b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c+d x)}\right )+96 i b^3 \arcsin (c+d x) \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c+d x)}\right )+48 b^3 \operatorname {PolyLog}\left (4,e^{-2 i \arcsin (c+d x)}\right )\right )}{64 d e} \]
((-1/64*I)*(8*a*b^2*Pi^3 + b^3*Pi^4 + 96*a^2*b*ArcSin[c + d*x]^2 - 64*a*b^ 2*ArcSin[c + d*x]^3 - 16*b^3*ArcSin[c + d*x]^4 + (192*I)*a*b^2*ArcSin[c + d*x]^2*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (64*I)*b^3*ArcSin[c + d*x]^3* Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (192*I)*a^2*b*ArcSin[c + d*x]*Log[1 - E^((2*I)*ArcSin[c + d*x])] + (64*I)*a^3*Log[c + d*x] - 96*b^2*ArcSin[c + d*x]*(2*a + b*ArcSin[c + d*x])*PolyLog[2, E^((-2*I)*ArcSin[c + d*x])] + 9 6*a^2*b*PolyLog[2, E^((2*I)*ArcSin[c + d*x])] + (96*I)*a*b^2*PolyLog[3, E^ ((-2*I)*ArcSin[c + d*x])] + (96*I)*b^3*ArcSin[c + d*x]*PolyLog[3, E^((-2*I )*ArcSin[c + d*x])] + 48*b^3*PolyLog[4, E^((-2*I)*ArcSin[c + d*x])]))/(d*e )
Time = 0.70 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5304, 27, 5136, 3042, 25, 4200, 25, 2620, 3011, 7163, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^3}{e (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^3}{c+d x}d(c+d x)}{d e}\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{c+d x}d\arcsin (c+d x)}{d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -(a+b \arcsin (c+d x))^3 \tan \left (\arcsin (c+d x)+\frac {\pi }{2}\right )d\arcsin (c+d x)}{d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (a+b \arcsin (c+d x))^3 \tan \left (\arcsin (c+d x)+\frac {\pi }{2}\right )d\arcsin (c+d x)}{d e}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle \frac {2 i \int -\frac {e^{2 i \arcsin (c+d x)} (a+b \arcsin (c+d x))^3}{1-e^{2 i \arcsin (c+d x)}}d\arcsin (c+d x)-\frac {i (a+b \arcsin (c+d x))^4}{4 b}}{d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-2 i \int \frac {e^{2 i \arcsin (c+d x)} (a+b \arcsin (c+d x))^3}{1-e^{2 i \arcsin (c+d x)}}d\arcsin (c+d x)-\frac {i (a+b \arcsin (c+d x))^4}{4 b}}{d e}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3-\frac {3}{2} i b \int (a+b \arcsin (c+d x))^2 \log \left (1-e^{2 i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )-\frac {i (a+b \arcsin (c+d x))^4}{4 b}}{d e}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3-\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-i b \int (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )\right )-\frac {i (a+b \arcsin (c+d x))^4}{4 b}}{d e}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3-\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-i b \left (\frac {1}{2} i b \int \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )d\arcsin (c+d x)-\frac {1}{2} i \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )\right )-\frac {i (a+b \arcsin (c+d x))^4}{4 b}}{d e}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3-\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-i b \left (\frac {1}{4} b \int e^{-2 i \arcsin (c+d x)} \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )de^{2 i \arcsin (c+d x)}-\frac {1}{2} i \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )\right )-\frac {i (a+b \arcsin (c+d x))^4}{4 b}}{d e}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3-\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (4,e^{2 i \arcsin (c+d x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )\right )-\frac {i (a+b \arcsin (c+d x))^4}{4 b}}{d e}\) |
(((-1/4*I)*(a + b*ArcSin[c + d*x])^4)/b - (2*I)*((I/2)*(a + b*ArcSin[c + d *x])^3*Log[1 - E^((2*I)*ArcSin[c + d*x])] - ((3*I)/2)*b*((I/2)*(a + b*ArcS in[c + d*x])^2*PolyLog[2, E^((2*I)*ArcSin[c + d*x])] - I*b*((-1/2*I)*(a + b*ArcSin[c + d*x])*PolyLog[3, E^((2*I)*ArcSin[c + d*x])] + (b*PolyLog[4, E ^((2*I)*ArcSin[c + d*x])])/4))))/(d*e)
3.3.2.3.1 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[(((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_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (206 ) = 412\).
Time = 0.71 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.86
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(652\) |
| default | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(652\) |
| parts | \(\frac {a^{3} \ln \left (d x +c \right )}{e d}+\frac {b^{3} \left (-\frac {i \arcsin \left (d x +c \right )^{4}}{4}+\arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-3 i \arcsin \left (d x +c \right )^{2} \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 \arcsin \left (d x +c \right ) \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+6 i \operatorname {polylog}\left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {3 a \,b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {3 a^{2} b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}\) | \(660\) |
1/d*(a^3/e*ln(d*x+c)+b^3/e*(-1/4*I*arcsin(d*x+c)^4+arcsin(d*x+c)^3*ln(1-I* (d*x+c)-(1-(d*x+c)^2)^(1/2))-3*I*arcsin(d*x+c)^2*polylog(2,I*(d*x+c)+(1-(d *x+c)^2)^(1/2))+6*arcsin(d*x+c)*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+6 *I*polylog(4,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+arcsin(d*x+c)^3*ln(1+I*(d*x+c) +(1-(d*x+c)^2)^(1/2))-3*I*arcsin(d*x+c)^2*polylog(2,-I*(d*x+c)-(1-(d*x+c)^ 2)^(1/2))+6*arcsin(d*x+c)*polylog(3,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+6*I*po lylog(4,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2)))+3*a*b^2/e*(-1/3*I*arcsin(d*x+c)^3 +arcsin(d*x+c)^2*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-2*I*arcsin(d*x+c)*pol ylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+2*polylog(3,-I*(d*x+c)-(1-(d*x+c)^2 )^(1/2))+arcsin(d*x+c)^2*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-2*I*arcsin(d* x+c)*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+2*polylog(3,I*(d*x+c)+(1-(d* x+c)^2)^(1/2)))+3*a^2*b/e*(-1/2*I*arcsin(d*x+c)^2+arcsin(d*x+c)*ln(1+I*(d* x+c)+(1-(d*x+c)^2)^(1/2))-I*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+arcs in(d*x+c)*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-I*polylog(2,I*(d*x+c)+(1-(d* x+c)^2)^(1/2))))
\[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
integral((b^3*arcsin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b*arcs in(d*x + c) + a^3)/(d*e*x + c*e), x)
\[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\frac {\int \frac {a^{3}}{c + d x}\, dx + \int \frac {b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a^{2} b \operatorname {asin}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
(Integral(a**3/(c + d*x), x) + Integral(b**3*asin(c + d*x)**3/(c + d*x), x ) + Integral(3*a*b**2*asin(c + d*x)**2/(c + d*x), x) + Integral(3*a**2*b*a sin(c + d*x)/(c + d*x), x))/e
\[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
a^3*log(d*e*x + c*e)/(d*e) + integrate((b^3*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 3*a*b^2*arctan2(d*x + c, sqrt(d*x + c + 1)*sq rt(-d*x - c + 1))^2 + 3*a^2*b*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d*e*x + c*e), x)
\[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^3}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{c\,e+d\,e\,x} \,d x \]