Integrand size = 23, antiderivative size = 126 \[ \int \frac {(a+b \arcsin (c+d x))^2}{c e+d e x} \, dx=-\frac {i (a+b \arcsin (c+d x))^3}{3 b d e}+\frac {(a+b \arcsin (c+d x))^2 \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {i b (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{d e}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )}{2 d e} \]
-1/3*I*(a+b*arcsin(d*x+c))^3/b/d/e+(a+b*arcsin(d*x+c))^2*ln(1-(I*(d*x+c)+( 1-(d*x+c)^2)^(1/2))^2)/d/e-I*b*(a+b*arcsin(d*x+c))*polylog(2,(I*(d*x+c)+(1 -(d*x+c)^2)^(1/2))^2)/d/e+1/2*b^2*polylog(3,(I*(d*x+c)+(1-(d*x+c)^2)^(1/2) )^2)/d/e
Time = 0.36 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b \arcsin (c+d x))^2}{c e+d e x} \, dx=\frac {2 a b \arcsin (c+d x) \log \left (1-e^{2 i \arcsin (c+d x)}\right )+a^2 \log (c+d x)-i a b \left (\arcsin (c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )\right )+b^2 \left (-\frac {i \pi ^3}{24}+\frac {1}{3} i \arcsin (c+d x)^3+\arcsin (c+d x)^2 \log \left (1-e^{-2 i \arcsin (c+d x)}\right )+i \arcsin (c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c+d x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c+d x)}\right )\right )}{d e} \]
(2*a*b*ArcSin[c + d*x]*Log[1 - E^((2*I)*ArcSin[c + d*x])] + a^2*Log[c + d* x] - I*a*b*(ArcSin[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c + d*x])]) + b ^2*((-1/24*I)*Pi^3 + (I/3)*ArcSin[c + d*x]^3 + ArcSin[c + d*x]^2*Log[1 - E ^((-2*I)*ArcSin[c + d*x])] + I*ArcSin[c + d*x]*PolyLog[2, E^((-2*I)*ArcSin [c + d*x])] + PolyLog[3, E^((-2*I)*ArcSin[c + d*x])]/2))/(d*e)
Time = 0.57 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5304, 27, 5136, 3042, 25, 4200, 25, 2620, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c+d x))^2}{c e+d e x} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^2}{e (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^2}{c+d x}d(c+d x)}{d e}\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{c+d x}d\arcsin (c+d x)}{d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -(a+b \arcsin (c+d x))^2 \tan \left (\arcsin (c+d x)+\frac {\pi }{2}\right )d\arcsin (c+d x)}{d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (a+b \arcsin (c+d x))^2 \tan \left (\arcsin (c+d x)+\frac {\pi }{2}\right )d\arcsin (c+d x)}{d e}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle \frac {2 i \int -\frac {e^{2 i \arcsin (c+d x)} (a+b \arcsin (c+d x))^2}{1-e^{2 i \arcsin (c+d x)}}d\arcsin (c+d x)-\frac {i (a+b \arcsin (c+d x))^3}{3 b}}{d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-2 i \int \frac {e^{2 i \arcsin (c+d x)} (a+b \arcsin (c+d x))^2}{1-e^{2 i \arcsin (c+d x)}}d\arcsin (c+d x)-\frac {i (a+b \arcsin (c+d x))^3}{3 b}}{d e}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-i b \int (a+b \arcsin (c+d x)) \log \left (1-e^{2 i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )-\frac {i (a+b \arcsin (c+d x))^3}{3 b}}{d e}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )\right )-\frac {i (a+b \arcsin (c+d x))^3}{3 b}}{d e}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-\frac {1}{4} b \int e^{-2 i \arcsin (c+d x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )de^{2 i \arcsin (c+d x)}\right )\right )-\frac {i (a+b \arcsin (c+d x))^3}{3 b}}{d e}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c+d x)}\right )\right )\right )-\frac {i (a+b \arcsin (c+d x))^3}{3 b}}{d e}\) |
(((-1/3*I)*(a + b*ArcSin[c + d*x])^3)/b - (2*I)*((I/2)*(a + b*ArcSin[c + d *x])^2*Log[1 - E^((2*I)*ArcSin[c + d*x])] - I*b*((I/2)*(a + b*ArcSin[c + d *x])*PolyLog[2, E^((2*I)*ArcSin[c + d*x])] - (b*PolyLog[3, E^((2*I)*ArcSin [c + d*x])])/4)))/(d*e)
3.2.93.3.1 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (156 ) = 312\).
Time = 0.65 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.91
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {2 a b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(367\) |
| default | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {2 a b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(367\) |
| parts | \(\frac {a^{2} \ln \left (d x +c \right )}{e d}+\frac {b^{2} \left (-\frac {i \arcsin \left (d x +c \right )^{3}}{3}+\arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \arcsin \left (d x +c \right ) \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {2 a b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}\) | \(372\) |
1/d*(a^2/e*ln(d*x+c)+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)*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))+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)))+2*a*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))+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))))
\[ \int \frac {(a+b \arcsin (c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
\[ \int \frac {(a+b \arcsin (c+d x))^2}{c e+d e x} \, dx=\frac {\int \frac {a^{2}}{c + d x}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
(Integral(a**2/(c + d*x), x) + Integral(b**2*asin(c + d*x)**2/(c + d*x), x ) + Integral(2*a*b*asin(c + d*x)/(c + d*x), x))/e
\[ \int \frac {(a+b \arcsin (c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
a^2*log(d*e*x + c*e)/(d*e) + integrate((b^2*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*a*b*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt (-d*x - c + 1)))/(d*e*x + c*e), x)
\[ \int \frac {(a+b \arcsin (c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^2}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2}{c\,e+d\,e\,x} \,d x \]