Integrand size = 12, antiderivative size = 309 \[ \int e^{\arcsin (a+b x)} x^3 \, dx=-\frac {3 a e^{\arcsin (a+b x)} (a+b x)}{8 b^4}-\frac {a^3 e^{\arcsin (a+b x)} (a+b x)}{2 b^4}-\frac {3 a e^{\arcsin (a+b x)} \sqrt {1-(a+b x)^2}}{8 b^4}-\frac {a^3 e^{\arcsin (a+b x)} \sqrt {1-(a+b x)^2}}{2 b^4}-\frac {e^{\arcsin (a+b x)} \cos (2 \arcsin (a+b x))}{10 b^4}-\frac {3 a^2 e^{\arcsin (a+b x)} \cos (2 \arcsin (a+b x))}{5 b^4}+\frac {3 a e^{\arcsin (a+b x)} \cos (3 \arcsin (a+b x))}{40 b^4}+\frac {e^{\arcsin (a+b x)} \cos (4 \arcsin (a+b x))}{34 b^4}+\frac {e^{\arcsin (a+b x)} \sin (2 \arcsin (a+b x))}{20 b^4}+\frac {3 a^2 e^{\arcsin (a+b x)} \sin (2 \arcsin (a+b x))}{10 b^4}+\frac {9 a e^{\arcsin (a+b x)} \sin (3 \arcsin (a+b x))}{40 b^4}-\frac {e^{\arcsin (a+b x)} \sin (4 \arcsin (a+b x))}{136 b^4} \]
-3/8*a*exp(arcsin(b*x+a))*(b*x+a)/b^4-1/2*a^3*exp(arcsin(b*x+a))*(b*x+a)/b ^4-1/10*exp(arcsin(b*x+a))*cos(2*arcsin(b*x+a))/b^4-3/5*a^2*exp(arcsin(b*x +a))*cos(2*arcsin(b*x+a))/b^4+3/40*a*exp(arcsin(b*x+a))*cos(3*arcsin(b*x+a ))/b^4+1/34*exp(arcsin(b*x+a))*cos(4*arcsin(b*x+a))/b^4+1/20*exp(arcsin(b* x+a))*sin(2*arcsin(b*x+a))/b^4+3/10*a^2*exp(arcsin(b*x+a))*sin(2*arcsin(b* x+a))/b^4+9/40*a*exp(arcsin(b*x+a))*sin(3*arcsin(b*x+a))/b^4-1/136*exp(arc sin(b*x+a))*sin(4*arcsin(b*x+a))/b^4-3/8*a*exp(arcsin(b*x+a))*(1-(b*x+a)^2 )^(1/2)/b^4-1/2*a^3*exp(arcsin(b*x+a))*(1-(b*x+a)^2)^(1/2)/b^4
Time = 0.35 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.48 \[ \int e^{\arcsin (a+b x)} x^3 \, dx=\frac {e^{\arcsin (a+b x)} \left (-255 a (a+b x)-340 a^3 (a+b x)-85 a \left (3+4 a^2\right ) \sqrt {1-(a+b x)^2}-68 \left (1+6 a^2\right ) \cos (2 \arcsin (a+b x))+51 a \cos (3 \arcsin (a+b x))+20 \cos (4 \arcsin (a+b x))+34 \sin (2 \arcsin (a+b x))+204 a^2 \sin (2 \arcsin (a+b x))+153 a \sin (3 \arcsin (a+b x))-5 \sin (4 \arcsin (a+b x))\right )}{680 b^4} \]
(E^ArcSin[a + b*x]*(-255*a*(a + b*x) - 340*a^3*(a + b*x) - 85*a*(3 + 4*a^2 )*Sqrt[1 - (a + b*x)^2] - 68*(1 + 6*a^2)*Cos[2*ArcSin[a + b*x]] + 51*a*Cos [3*ArcSin[a + b*x]] + 20*Cos[4*ArcSin[a + b*x]] + 34*Sin[2*ArcSin[a + b*x] ] + 204*a^2*Sin[2*ArcSin[a + b*x]] + 153*a*Sin[3*ArcSin[a + b*x]] - 5*Sin[ 4*ArcSin[a + b*x]]))/(680*b^4)
Time = 0.72 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5335, 25, 7292, 27, 7293, 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.
| \(\displaystyle \int x^3 e^{\arcsin (a+b x)} \, dx\) |
| \(\Big \downarrow \) 5335 |
| \(\displaystyle \frac {\int -e^{\arcsin (a+b x)} \left (\frac {a}{b}-\frac {a+b x}{b}\right )^3 \sqrt {1-(a+b x)^2}d\arcsin (a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int e^{\arcsin (a+b x)} \left (\frac {a}{b}-\frac {a+b x}{b}\right )^3 \sqrt {1-(a+b x)^2}d\arcsin (a+b x)}{b}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {\int -e^{\arcsin (a+b x)} x^3 \sqrt {1-(a+b x)^2}d\arcsin (a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b^3 e^{\arcsin (a+b x)} x^3 \sqrt {1-(a+b x)^2}d\arcsin (a+b x)}{b^4}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (e^{\arcsin (a+b x)} \sqrt {1-(a+b x)^2} a^3-3 e^{\arcsin (a+b x)} (a+b x) \sqrt {1-(a+b x)^2} a^2+3 e^{\arcsin (a+b x)} (a+b x)^2 \sqrt {1-(a+b x)^2} a-e^{\arcsin (a+b x)} (a+b x)^3 \sqrt {1-(a+b x)^2}\right )d\arcsin (a+b x)}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{2} a^3 (a+b x) e^{\arcsin (a+b x)}+\frac {1}{2} a^3 \sqrt {1-(a+b x)^2} e^{\arcsin (a+b x)}-\frac {3}{10} a^2 e^{\arcsin (a+b x)} \sin (2 \arcsin (a+b x))+\frac {3}{5} a^2 e^{\arcsin (a+b x)} \cos (2 \arcsin (a+b x))+\frac {3}{8} a (a+b x) e^{\arcsin (a+b x)}+\frac {3}{8} a \sqrt {1-(a+b x)^2} e^{\arcsin (a+b x)}-\frac {9}{40} a e^{\arcsin (a+b x)} \sin (3 \arcsin (a+b x))-\frac {1}{20} e^{\arcsin (a+b x)} \sin (2 \arcsin (a+b x))+\frac {1}{136} e^{\arcsin (a+b x)} \sin (4 \arcsin (a+b x))-\frac {3}{40} a e^{\arcsin (a+b x)} \cos (3 \arcsin (a+b x))+\frac {1}{10} e^{\arcsin (a+b x)} \cos (2 \arcsin (a+b x))-\frac {1}{34} e^{\arcsin (a+b x)} \cos (4 \arcsin (a+b x))}{b^4}\) |
-(((3*a*E^ArcSin[a + b*x]*(a + b*x))/8 + (a^3*E^ArcSin[a + b*x]*(a + b*x)) /2 + (3*a*E^ArcSin[a + b*x]*Sqrt[1 - (a + b*x)^2])/8 + (a^3*E^ArcSin[a + b *x]*Sqrt[1 - (a + b*x)^2])/2 + (E^ArcSin[a + b*x]*Cos[2*ArcSin[a + b*x]])/ 10 + (3*a^2*E^ArcSin[a + b*x]*Cos[2*ArcSin[a + b*x]])/5 - (3*a*E^ArcSin[a + b*x]*Cos[3*ArcSin[a + b*x]])/40 - (E^ArcSin[a + b*x]*Cos[4*ArcSin[a + b* x]])/34 - (E^ArcSin[a + b*x]*Sin[2*ArcSin[a + b*x]])/20 - (3*a^2*E^ArcSin[ a + b*x]*Sin[2*ArcSin[a + b*x]])/10 - (9*a*E^ArcSin[a + b*x]*Sin[3*ArcSin[ a + b*x]])/40 + (E^ArcSin[a + b*x]*Sin[4*ArcSin[a + b*x]])/136)/b^4)
3.5.51.3.1 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[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Simp[ 1/b Subst[Int[(u /. x -> -a/b + Sin[x]/b)*f^(c*x^n)*Cos[x], x], x, ArcSin [a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
\[\int {\mathrm e}^{\arcsin \left (b x +a \right )} x^{3}d x\]
Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.42 \[ \int e^{\arcsin (a+b x)} x^3 \, dx=\frac {{\left (40 \, b^{4} x^{4} + 7 \, a b^{3} x^{3} - 3 \, {\left (5 \, a^{2} + 2\right )} b^{2} x^{2} + 6 \, a^{4} + 3 \, {\left (8 \, a^{3} + 13 \, a\right )} b x - 57 \, a^{2} + {\left (10 \, b^{3} x^{3} - 21 \, a b^{2} x^{2} - 24 \, a^{3} + 6 \, {\left (5 \, a^{2} + 2\right )} b x - 39 \, a\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} - 12\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{170 \, b^{4}} \]
1/170*(40*b^4*x^4 + 7*a*b^3*x^3 - 3*(5*a^2 + 2)*b^2*x^2 + 6*a^4 + 3*(8*a^3 + 13*a)*b*x - 57*a^2 + (10*b^3*x^3 - 21*a*b^2*x^2 - 24*a^3 + 6*(5*a^2 + 2 )*b*x - 39*a)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1) - 12)*e^(arcsin(b*x + a)) /b^4
Time = 0.61 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.35 \[ \int e^{\arcsin (a+b x)} x^3 \, dx=\begin {cases} \frac {3 a^{4} e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{4}} + \frac {12 a^{3} x e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{3}} - \frac {12 a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{4}} - \frac {3 a^{2} x^{2} e^{\operatorname {asin}{\left (a + b x \right )}}}{34 b^{2}} + \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{17 b^{3}} - \frac {57 a^{2} e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b^{4}} + \frac {7 a x^{3} e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b} - \frac {21 a x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b^{2}} + \frac {39 a x e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b^{3}} - \frac {39 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{170 b^{4}} + \frac {4 x^{4} e^{\operatorname {asin}{\left (a + b x \right )}}}{17} + \frac {x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{17 b} - \frac {3 x^{2} e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{2}} + \frac {6 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{3}} - \frac {6 e^{\operatorname {asin}{\left (a + b x \right )}}}{85 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} e^{\operatorname {asin}{\left (a \right )}}}{4} & \text {otherwise} \end {cases} \]
Piecewise((3*a**4*exp(asin(a + b*x))/(85*b**4) + 12*a**3*x*exp(asin(a + b* x))/(85*b**3) - 12*a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*exp(asin(a + b*x))/(85*b**4) - 3*a**2*x**2*exp(asin(a + b*x))/(34*b**2) + 3*a**2*x*sqr t(-a**2 - 2*a*b*x - b**2*x**2 + 1)*exp(asin(a + b*x))/(17*b**3) - 57*a**2* exp(asin(a + b*x))/(170*b**4) + 7*a*x**3*exp(asin(a + b*x))/(170*b) - 21*a *x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*exp(asin(a + b*x))/(170*b**2) + 39*a*x*exp(asin(a + b*x))/(170*b**3) - 39*a*sqrt(-a**2 - 2*a*b*x - b**2* x**2 + 1)*exp(asin(a + b*x))/(170*b**4) + 4*x**4*exp(asin(a + b*x))/17 + x **3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*exp(asin(a + b*x))/(17*b) - 3*x* *2*exp(asin(a + b*x))/(85*b**2) + 6*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1 )*exp(asin(a + b*x))/(85*b**3) - 6*exp(asin(a + b*x))/(85*b**4), Ne(b, 0)) , (x**4*exp(asin(a))/4, True))
\[ \int e^{\arcsin (a+b x)} x^3 \, dx=\int { x^{3} e^{\left (\arcsin \left (b x + a\right )\right )} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.08 \[ \int e^{\arcsin (a+b x)} x^3 \, dx=-\frac {{\left (b x + a\right )} a^{3} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b^{4}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a^{2} e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{3} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b^{4}} - \frac {9 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} a e^{\left (\arcsin \left (b x + a\right )\right )}}{10 \, b^{4}} + \frac {6 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a^{2} e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} - \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{17 \, b^{4}} + \frac {3 \, {\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} a e^{\left (\arcsin \left (b x + a\right )\right )}}{10 \, b^{4}} + \frac {4 \, {\left ({\left (b x + a\right )}^{2} - 1\right )}^{2} e^{\left (\arcsin \left (b x + a\right )\right )}}{17 \, b^{4}} - \frac {3 \, {\left (b x + a\right )} a e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} + \frac {3 \, a^{2} e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} + \frac {11 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{85 \, b^{4}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a e^{\left (\arcsin \left (b x + a\right )\right )}}{5 \, b^{4}} + \frac {37 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{85 \, b^{4}} + \frac {11 \, e^{\left (\arcsin \left (b x + a\right )\right )}}{85 \, b^{4}} \]
-1/2*(b*x + a)*a^3*e^(arcsin(b*x + a))/b^4 + 3/5*sqrt(-(b*x + a)^2 + 1)*(b *x + a)*a^2*e^(arcsin(b*x + a))/b^4 - 1/2*sqrt(-(b*x + a)^2 + 1)*a^3*e^(ar csin(b*x + a))/b^4 - 9/10*((b*x + a)^2 - 1)*(b*x + a)*a*e^(arcsin(b*x + a) )/b^4 + 6/5*((b*x + a)^2 - 1)*a^2*e^(arcsin(b*x + a))/b^4 - 1/17*(-(b*x + a)^2 + 1)^(3/2)*(b*x + a)*e^(arcsin(b*x + a))/b^4 + 3/10*(-(b*x + a)^2 + 1 )^(3/2)*a*e^(arcsin(b*x + a))/b^4 + 4/17*((b*x + a)^2 - 1)^2*e^(arcsin(b*x + a))/b^4 - 3/5*(b*x + a)*a*e^(arcsin(b*x + a))/b^4 + 3/5*a^2*e^(arcsin(b *x + a))/b^4 + 11/85*sqrt(-(b*x + a)^2 + 1)*(b*x + a)*e^(arcsin(b*x + a))/ b^4 - 3/5*sqrt(-(b*x + a)^2 + 1)*a*e^(arcsin(b*x + a))/b^4 + 37/85*((b*x + a)^2 - 1)*e^(arcsin(b*x + a))/b^4 + 11/85*e^(arcsin(b*x + a))/b^4
Timed out. \[ \int e^{\arcsin (a+b x)} x^3 \, dx=\int x^3\,{\mathrm {e}}^{\mathrm {asin}\left (a+b\,x\right )} \,d x \]