Integrand size = 12, antiderivative size = 176 \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=-\frac {x^2 \sqrt {1-(a+b x)^2}}{2 b \arcsin (a+b x)^2}+\frac {a^2 (a+b x)}{2 b^3 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{b^3 \arcsin (a+b x)}+\frac {9 a+b x}{8 b^3 \arcsin (a+b x)}-\frac {\left (1+4 a^2\right ) \operatorname {CosIntegral}(\arcsin (a+b x))}{8 b^3}+\frac {9 \operatorname {CosIntegral}(3 \arcsin (a+b x))}{8 b^3}-\frac {3 \sin (3 \arcsin (a+b x))}{8 b^3 \arcsin (a+b x)}+\frac {2 a \text {Si}(2 \arcsin (a+b x))}{b^3} \]
1/2*a^2*(b*x+a)/b^3/arcsin(b*x+a)-2*a*(b*x+a)^2/b^3/arcsin(b*x+a)+1/8*(b*x +9*a)/b^3/arcsin(b*x+a)-1/8*(4*a^2+1)*Ci(arcsin(b*x+a))/b^3+9/8*Ci(3*arcsi n(b*x+a))/b^3+2*a*Si(2*arcsin(b*x+a))/b^3-3/8*sin(3*arcsin(b*x+a))/b^3/arc sin(b*x+a)-1/2*x^2*(1-(b*x+a)^2)^(1/2)/b/arcsin(b*x+a)^2
Time = 0.61 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.65 \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\frac {\frac {4 b x \left (-b x \sqrt {1-a^2-2 a b x-b^2 x^2}+\left (-2+2 a^2+5 a b x+3 b^2 x^2\right ) \arcsin (a+b x)\right )}{\arcsin (a+b x)^2}-\left (1+4 a^2\right ) \operatorname {CosIntegral}(\arcsin (a+b x))+9 \operatorname {CosIntegral}(3 \arcsin (a+b x))+16 a \text {Si}(2 \arcsin (a+b x))}{8 b^3} \]
((4*b*x*(-(b*x*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]) + (-2 + 2*a^2 + 5*a*b*x + 3*b^2*x^2)*ArcSin[a + b*x]))/ArcSin[a + b*x]^2 - (1 + 4*a^2)*CosIntegral [ArcSin[a + b*x]] + 9*CosIntegral[3*ArcSin[a + b*x]] + 16*a*SinIntegral[2* ArcSin[a + b*x]])/(8*b^3)
Time = 0.63 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5304, 27, 5244, 2009}
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.
| \(\displaystyle \int \frac {x^2}{\arcsin (a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {x^2}{\arcsin (a+b x)^3}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b^2 x^2}{\arcsin (a+b x)^3}d(a+b x)}{b^3}\) |
| \(\Big \downarrow \) 5244 |
| \(\displaystyle \frac {\int \left (\frac {a^2}{\arcsin (a+b x)^3}-\frac {2 (a+b x) a}{\arcsin (a+b x)^3}+\frac {(a+b x)^2}{\arcsin (a+b x)^3}\right )d(a+b x)}{b^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{2} a^2 \operatorname {CosIntegral}(\arcsin (a+b x))+\frac {a^2 (a+b x)}{2 \arcsin (a+b x)}-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 \arcsin (a+b x)^2}-\frac {1}{8} \operatorname {CosIntegral}(\arcsin (a+b x))+\frac {9}{8} \operatorname {CosIntegral}(3 \arcsin (a+b x))+2 a \text {Si}(2 \arcsin (a+b x))+\frac {3 (a+b x)^3}{2 \arcsin (a+b x)}-\frac {2 a (a+b x)^2}{\arcsin (a+b x)}-\frac {\sqrt {1-(a+b x)^2} (a+b x)^2}{2 \arcsin (a+b x)^2}-\frac {a+b x}{\arcsin (a+b x)}+\frac {a \sqrt {1-(a+b x)^2} (a+b x)}{\arcsin (a+b x)^2}+\frac {a}{\arcsin (a+b x)}}{b^3}\) |
(-1/2*(a^2*Sqrt[1 - (a + b*x)^2])/ArcSin[a + b*x]^2 + (a*(a + b*x)*Sqrt[1 - (a + b*x)^2])/ArcSin[a + b*x]^2 - ((a + b*x)^2*Sqrt[1 - (a + b*x)^2])/(2 *ArcSin[a + b*x]^2) + a/ArcSin[a + b*x] - (a + b*x)/ArcSin[a + b*x] + (a^2 *(a + b*x))/(2*ArcSin[a + b*x]) - (2*a*(a + b*x)^2)/ArcSin[a + b*x] + (3*( a + b*x)^3)/(2*ArcSin[a + b*x]) - CosIntegral[ArcSin[a + b*x]]/8 - (a^2*Co sIntegral[ArcSin[a + b*x]])/2 + (9*CosIntegral[3*ArcSin[a + b*x]])/8 + 2*a *SinIntegral[2*ArcSin[a + b*x]])/b^3
3.2.51.3.1 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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_Sy mbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; F reeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
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]
Time = 0.70 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{8 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{8 \arcsin \left (b x +a \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (b x +a \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (b x +a \right )\right )}{8}+\frac {a \left (4 \,\operatorname {Si}\left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+2 \cos \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sin \left (2 \arcsin \left (b x +a \right )\right )\right )}{2 \arcsin \left (b x +a \right )^{2}}-\frac {a^{2} \left (\operatorname {Ci}\left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}-\arcsin \left (b x +a \right ) \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \arcsin \left (b x +a \right )^{2}}}{b^{3}}\) | \(215\) |
| default | \(\frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{8 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{8 \arcsin \left (b x +a \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (b x +a \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (b x +a \right )\right )}{8}+\frac {a \left (4 \,\operatorname {Si}\left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+2 \cos \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sin \left (2 \arcsin \left (b x +a \right )\right )\right )}{2 \arcsin \left (b x +a \right )^{2}}-\frac {a^{2} \left (\operatorname {Ci}\left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}-\arcsin \left (b x +a \right ) \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \arcsin \left (b x +a \right )^{2}}}{b^{3}}\) | \(215\) |
1/b^3*(-1/8/arcsin(b*x+a)^2*(1-(b*x+a)^2)^(1/2)+1/8*(b*x+a)/arcsin(b*x+a)- 1/8*Ci(arcsin(b*x+a))+1/8/arcsin(b*x+a)^2*cos(3*arcsin(b*x+a))-3/8/arcsin( b*x+a)*sin(3*arcsin(b*x+a))+9/8*Ci(3*arcsin(b*x+a))+1/2*a*(4*Si(2*arcsin(b *x+a))*arcsin(b*x+a)^2+2*cos(2*arcsin(b*x+a))*arcsin(b*x+a)+sin(2*arcsin(b *x+a)))/arcsin(b*x+a)^2-1/2*a^2*(Ci(arcsin(b*x+a))*arcsin(b*x+a)^2-arcsin( b*x+a)*(b*x+a)+(1-(b*x+a)^2)^(1/2))/arcsin(b*x+a)^2)
\[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\int { \frac {x^{2}}{\arcsin \left (b x + a\right )^{3}} \,d x } \]
\[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\int \frac {x^{2}}{\operatorname {asin}^{3}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\int { \frac {x^{2}}{\arcsin \left (b x + a\right )^{3}} \,d x } \]
-1/2*(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*b*x^2 + arctan2(b*x + a, sqrt(b *x + a + 1)*sqrt(-b*x - a + 1))^2*integrate((9*b^2*x^2 + 10*a*b*x + 2*a^2 - 2)/arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)), x) - (3*b^2*x ^3 + 5*a*b*x^2 + 2*(a^2 - 1)*x)*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b *x - a + 1)))/(b^2*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^ 2)
Time = 0.32 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.55 \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=-\frac {a^{2} \operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{2 \, b^{3}} + \frac {{\left (b x + a\right )} a^{2}}{2 \, b^{3} \arcsin \left (b x + a\right )} + \frac {2 \, a \operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{b^{3}} + \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )}}{2 \, b^{3} \arcsin \left (b x + a\right )} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a}{b^{3} \arcsin \left (b x + a\right )} + \frac {9 \, \operatorname {Ci}\left (3 \, \arcsin \left (b x + a\right )\right )}{8 \, b^{3}} - \frac {\operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{8 \, b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{b^{3} \arcsin \left (b x + a\right )^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} + \frac {b x + a}{2 \, b^{3} \arcsin \left (b x + a\right )} - \frac {a}{b^{3} \arcsin \left (b x + a\right )} + \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} \]
-1/2*a^2*cos_integral(arcsin(b*x + a))/b^3 + 1/2*(b*x + a)*a^2/(b^3*arcsin (b*x + a)) + 2*a*sin_integral(2*arcsin(b*x + a))/b^3 + 3/2*((b*x + a)^2 - 1)*(b*x + a)/(b^3*arcsin(b*x + a)) - 2*((b*x + a)^2 - 1)*a/(b^3*arcsin(b*x + a)) + 9/8*cos_integral(3*arcsin(b*x + a))/b^3 - 1/8*cos_integral(arcsin (b*x + a))/b^3 + sqrt(-(b*x + a)^2 + 1)*(b*x + a)*a/(b^3*arcsin(b*x + a)^2 ) - 1/2*sqrt(-(b*x + a)^2 + 1)*a^2/(b^3*arcsin(b*x + a)^2) + 1/2*(b*x + a) /(b^3*arcsin(b*x + a)) - a/(b^3*arcsin(b*x + a)) + 1/2*(-(b*x + a)^2 + 1)^ (3/2)/(b^3*arcsin(b*x + a)^2) - 1/2*sqrt(-(b*x + a)^2 + 1)/(b^3*arcsin(b*x + a)^2)
Timed out. \[ \int \frac {x^2}{\arcsin (a+b x)^3} \, dx=\int \frac {x^2}{{\mathrm {asin}\left (a+b\,x\right )}^3} \,d x \]