Integrand size = 33, antiderivative size = 97 \[ \int \frac {\arcsin (a+b x)^2}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=-\frac {i \arcsin (a+b x)^2}{b}+\frac {(a+b x) \arcsin (a+b x)^2}{b \sqrt {1-(a+b x)^2}}+\frac {2 \arcsin (a+b x) \log \left (1+e^{2 i \arcsin (a+b x)}\right )}{b}-\frac {i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a+b x)}\right )}{b} \]
-I*arcsin(b*x+a)^2/b+2*arcsin(b*x+a)*ln(1+(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))^ 2)/b-I*polylog(2,-(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))^2)/b+(b*x+a)*arcsin(b*x+ a)^2/b/(1-(b*x+a)^2)^(1/2)
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18 \[ \int \frac {\arcsin (a+b x)^2}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\frac {\arcsin (a+b x) \left (\frac {\left (a+b x-i \sqrt {1-a^2-2 a b x-b^2 x^2}\right ) \arcsin (a+b x)}{\sqrt {1-a^2-2 a b x-b^2 x^2}}+2 \log \left (1+e^{2 i \arcsin (a+b x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (a+b x)}\right )}{b} \]
(ArcSin[a + b*x]*(((a + b*x - I*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2])*ArcSin[ a + b*x])/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + 2*Log[1 + E^((2*I)*ArcSin[a + b*x])]) - I*PolyLog[2, -E^((2*I)*ArcSin[a + b*x])])/b
Time = 0.54 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {5306, 5160, 5180, 3042, 4202, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {\arcsin (a+b x)^2}{\left (-a^2-2 a b x-b^2 x^2+1\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5306 |
| \(\displaystyle \frac {\int \frac {\arcsin (a+b x)^2}{\left (1-(a+b x)^2\right )^{3/2}}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle \frac {\frac {(a+b x) \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}-2 \int \frac {(a+b x) \arcsin (a+b x)}{1-(a+b x)^2}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 5180 |
| \(\displaystyle \frac {\frac {(a+b x) \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}-2 \int \frac {(a+b x) \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d\arcsin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(a+b x) \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}-2 \int \arcsin (a+b x) \tan (\arcsin (a+b x))d\arcsin (a+b x)}{b}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\frac {(a+b x) \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}-2 \left (\frac {1}{2} i \arcsin (a+b x)^2-2 i \int \frac {e^{2 i \arcsin (a+b x)} \arcsin (a+b x)}{1+e^{2 i \arcsin (a+b x)}}d\arcsin (a+b x)\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {(a+b x) \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}-2 \left (\frac {1}{2} i \arcsin (a+b x)^2-2 i \left (\frac {1}{2} i \int \log \left (1+e^{2 i \arcsin (a+b x)}\right )d\arcsin (a+b x)-\frac {1}{2} i \arcsin (a+b x) \log \left (1+e^{2 i \arcsin (a+b x)}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {(a+b x) \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}-2 \left (\frac {1}{2} i \arcsin (a+b x)^2-2 i \left (\frac {1}{4} \int e^{-2 i \arcsin (a+b x)} \log \left (1+e^{2 i \arcsin (a+b x)}\right )de^{2 i \arcsin (a+b x)}-\frac {1}{2} i \arcsin (a+b x) \log \left (1+e^{2 i \arcsin (a+b x)}\right )\right )\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {(a+b x) \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}-2 \left (\frac {1}{2} i \arcsin (a+b x)^2-2 i \left (-\frac {1}{4} \operatorname {PolyLog}(2,-a-b x)-\frac {1}{2} i \arcsin (a+b x) \log \left (1+e^{2 i \arcsin (a+b x)}\right )\right )\right )}{b}\) |
(((a + b*x)*ArcSin[a + b*x]^2)/Sqrt[1 - (a + b*x)^2] - 2*((I/2)*ArcSin[a + b*x]^2 - (2*I)*((-1/2*I)*ArcSin[a + b*x]*Log[1 + E^((2*I)*ArcSin[a + b*x] )] - PolyLog[2, -a - b*x]/4)))/b
3.4.34.3.1 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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSin[c*x ])^(n - 1)/(1 - c^2*x^2)), 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_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-e^(-1) Subst[Int[(a + b*x)^n*Tan[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_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + ( C_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/d Subst[Int[(-C/d^2 + (C/d^2)*x^2 )^p*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
Time = 2.88 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.95
| method | result | size |
| default | \(\frac {\left (-x b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+i b^{2} x^{2}-a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+2 i a b x +i a^{2}-i\right ) \arcsin \left (b x +a \right )^{2}}{\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) b}-\frac {i \left (2 i \arcsin \left (b x +a \right ) \ln \left (1+\left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )+2 \arcsin \left (b x +a \right )^{2}+\operatorname {polylog}\left (2, -\left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )\right )}{b}\) | \(189\) |
(-x*b*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+I*b^2*x^2-a*(-b^2*x^2-2*a*b*x-a^2+1)^ (1/2)+2*I*a*b*x+I*a^2-I)/(b^2*x^2+2*a*b*x+a^2-1)/b*arcsin(b*x+a)^2-I*(2*I* arcsin(b*x+a)*ln(1+(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))^2)+2*arcsin(b*x+a)^2+po lylog(2,-(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))^2))/b
\[ \int \frac {\arcsin (a+b x)^2}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{2}}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*arcsin(b*x + a)^2/(b^4*x^4 + 4 *a*b^3*x^3 + 2*(3*a^2 - 1)*b^2*x^2 + a^4 + 4*(a^3 - a)*b*x - 2*a^2 + 1), x )
\[ \int \frac {\arcsin (a+b x)^2}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\arcsin (a+b x)^2}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{2}}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\arcsin (a+b x)^2}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{2}}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a+b x)^2}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}} \,d x \]