Integrand size = 12, antiderivative size = 119 \[ \int (a+b \arcsin (c+d x))^4 \, dx=24 b^4 x-\frac {24 b^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{d}-\frac {12 b^2 (c+d x) (a+b \arcsin (c+d x))^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^4}{d} \]
24*b^4*x-12*b^2*(d*x+c)*(a+b*arcsin(d*x+c))^2/d+(d*x+c)*(a+b*arcsin(d*x+c) )^4/d-24*b^3*(a+b*arcsin(d*x+c))*(1-(d*x+c)^2)^(1/2)/d+4*b*(a+b*arcsin(d*x +c))^3*(1-(d*x+c)^2)^(1/2)/d
Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.97 \[ \int (a+b \arcsin (c+d x))^4 \, dx=\frac {4 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3+(c+d x) (a+b \arcsin (c+d x))^4-12 b^2 \left (-2 b^2 (c+d x)+2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))+(c+d x) (a+b \arcsin (c+d x))^2\right )}{d} \]
(4*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 + (c + d*x)*(a + b*Ar cSin[c + d*x])^4 - 12*b^2*(-2*b^2*(c + d*x) + 2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (c + d*x)*(a + b*ArcSin[c + d*x])^2))/d
Time = 0.44 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5302, 5130, 5182, 5130, 5182, 24}
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.
| \(\displaystyle \int (a+b \arcsin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 5302 |
| \(\displaystyle \frac {\int (a+b \arcsin (c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^4-4 b \int \frac {(c+d x) (a+b \arcsin (c+d x))^3}{\sqrt {1-(c+d x)^2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^4-4 b \left (3 b \int (a+b \arcsin (c+d x))^2d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )}{d}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^4-4 b \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \int \frac {(c+d x) (a+b \arcsin (c+d x))}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )}{d}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^4-4 b \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b \int 1d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^4-4 b \left (3 b \left ((c+d x) (a+b \arcsin (c+d x))^2-2 b \left (b (c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3\right )}{d}\) |
((c + d*x)*(a + b*ArcSin[c + d*x])^4 - 4*b*(-(Sqrt[1 - (c + d*x)^2]*(a + b *ArcSin[c + d*x])^3) + 3*b*((c + d*x)*(a + b*ArcSin[c + d*x])^2 - 2*b*(b*( c + d*x) - Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])))))/d
3.3.9.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(254\) vs. \(2(115)=230\).
Time = 0.66 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.14
| method | result | size |
| derivativedivides | \(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\arcsin \left (d x +c \right )^{4} \left (d x +c \right )+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+4 a \,b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+6 a^{2} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+4 a^{3} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(255\) |
| default | \(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\arcsin \left (d x +c \right )^{4} \left (d x +c \right )+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+4 a \,b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+6 a^{2} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+4 a^{3} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(255\) |
| parts | \(x \,a^{4}+\frac {b^{4} \left (\arcsin \left (d x +c \right )^{4} \left (d x +c \right )+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )}{d}+\frac {4 a \,b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )}{d}+\frac {6 a^{2} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )}{d}+\frac {4 a^{3} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(259\) |
1/d*((d*x+c)*a^4+b^4*(arcsin(d*x+c)^4*(d*x+c)+4*arcsin(d*x+c)^3*(1-(d*x+c) ^2)^(1/2)-12*arcsin(d*x+c)^2*(d*x+c)+24*d*x+24*c-24*arcsin(d*x+c)*(1-(d*x+ c)^2)^(1/2))+4*a*b^3*(arcsin(d*x+c)^3*(d*x+c)+3*arcsin(d*x+c)^2*(1-(d*x+c) ^2)^(1/2)-6*(1-(d*x+c)^2)^(1/2)-6*(d*x+c)*arcsin(d*x+c))+6*a^2*b^2*(arcsin (d*x+c)^2*(d*x+c)-2*d*x-2*c+2*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2))+4*a^3*b*( (d*x+c)*arcsin(d*x+c)+(1-(d*x+c)^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (115) = 230\).
Time = 0.27 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.96 \[ \int (a+b \arcsin (c+d x))^4 \, dx=\frac {{\left (b^{4} d x + b^{4} c\right )} \arcsin \left (d x + c\right )^{4} + 4 \, {\left (a b^{3} d x + a b^{3} c\right )} \arcsin \left (d x + c\right )^{3} + {\left (a^{4} - 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} d x + 6 \, {\left ({\left (a^{2} b^{2} - 2 \, b^{4}\right )} d x + {\left (a^{2} b^{2} - 2 \, b^{4}\right )} c\right )} \arcsin \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{3} b - 6 \, a b^{3}\right )} d x + {\left (a^{3} b - 6 \, a b^{3}\right )} c\right )} \arcsin \left (d x + c\right ) + 4 \, {\left (b^{4} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{3} \arcsin \left (d x + c\right )^{2} + a^{3} b - 6 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 2 \, b^{4}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \]
((b^4*d*x + b^4*c)*arcsin(d*x + c)^4 + 4*(a*b^3*d*x + a*b^3*c)*arcsin(d*x + c)^3 + (a^4 - 12*a^2*b^2 + 24*b^4)*d*x + 6*((a^2*b^2 - 2*b^4)*d*x + (a^2 *b^2 - 2*b^4)*c)*arcsin(d*x + c)^2 + 4*((a^3*b - 6*a*b^3)*d*x + (a^3*b - 6 *a*b^3)*c)*arcsin(d*x + c) + 4*(b^4*arcsin(d*x + c)^3 + 3*a*b^3*arcsin(d*x + c)^2 + a^3*b - 6*a*b^3 + 3*(a^2*b^2 - 2*b^4)*arcsin(d*x + c))*sqrt(-d^2 *x^2 - 2*c*d*x - c^2 + 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (105) = 210\).
Time = 0.26 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.73 \[ \int (a+b \arcsin (c+d x))^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b c \operatorname {asin}{\left (c + d x \right )}}{d} + 4 a^{3} b x \operatorname {asin}{\left (c + d x \right )} + \frac {4 a^{3} b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {6 a^{2} b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 12 a^{2} b^{2} x + \frac {12 a^{2} b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {4 a b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} c \operatorname {asin}{\left (c + d x \right )}}{d} + 4 a b^{3} x \operatorname {asin}^{3}{\left (c + d x \right )} - 24 a b^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {12 a b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {b^{4} c \operatorname {asin}^{4}{\left (c + d x \right )}}{d} - \frac {12 b^{4} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + b^{4} x \operatorname {asin}^{4}{\left (c + d x \right )} - 12 b^{4} x \operatorname {asin}^{2}{\left (c + d x \right )} + 24 b^{4} x + \frac {4 b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {24 b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((a**4*x + 4*a**3*b*c*asin(c + d*x)/d + 4*a**3*b*x*asin(c + d*x) + 4*a**3*b*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d + 6*a**2*b**2*c*asin(c + d*x)**2/d + 6*a**2*b**2*x*asin(c + d*x)**2 - 12*a**2*b**2*x + 12*a**2*b* *2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/d + 4*a*b**3*c*asin (c + d*x)**3/d - 24*a*b**3*c*asin(c + d*x)/d + 4*a*b**3*x*asin(c + d*x)**3 - 24*a*b**3*x*asin(c + d*x) + 12*a*b**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/d - 24*a*b**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/ d + b**4*c*asin(c + d*x)**4/d - 12*b**4*c*asin(c + d*x)**2/d + b**4*x*asin (c + d*x)**4 - 12*b**4*x*asin(c + d*x)**2 + 24*b**4*x + 4*b**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/d - 24*b**4*sqrt(-c**2 - 2*c*d *x - d**2*x**2 + 1)*asin(c + d*x)/d, Ne(d, 0)), (x*(a + b*asin(c))**4, Tru e))
\[ \int (a+b \arcsin (c+d x))^4 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
b^4*x*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4 + a^4*x + 4 *((d*x + c)*arcsin(d*x + c) + sqrt(-(d*x + c)^2 + 1))*a^3*b/d + integrate( 2*(2*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*b^4*d*x*arctan2(d*x + c, sqrt(d* x + c + 1)*sqrt(-d*x - c + 1))^3 + 2*(a*b^3*d^2*x^2 + 2*a*b^3*c*d*x + a*b^ 3*c^2 - a*b^3)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 3*(a^2*b^2*d^2*x^2 + 2*a^2*b^2*c*d*x + a^2*b^2*c^2 - a^2*b^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2)/(d^2*x^2 + 2*c*d*x + c^2 - 1 ), x)
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (115) = 230\).
Time = 0.30 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.76 \[ \int (a+b \arcsin (c+d x))^4 \, dx=\frac {{\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{4}}{d} + \frac {4 \, {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )^{3}}{d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} \arcsin \left (d x + c\right )^{3}}{d} + \frac {6 \, {\left (d x + c\right )} a^{2} b^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {12 \, {\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{2}}{d} + \frac {12 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3} \arcsin \left (d x + c\right )^{2}}{d} + \frac {4 \, {\left (d x + c\right )} a^{3} b \arcsin \left (d x + c\right )}{d} - \frac {24 \, {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )}{d} + \frac {12 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{2} \arcsin \left (d x + c\right )}{d} - \frac {24 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{4}}{d} - \frac {12 \, {\left (d x + c\right )} a^{2} b^{2}}{d} + \frac {24 \, {\left (d x + c\right )} b^{4}}{d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{3} b}{d} - \frac {24 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3}}{d} \]
(d*x + c)*b^4*arcsin(d*x + c)^4/d + 4*(d*x + c)*a*b^3*arcsin(d*x + c)^3/d + 4*sqrt(-(d*x + c)^2 + 1)*b^4*arcsin(d*x + c)^3/d + 6*(d*x + c)*a^2*b^2*a rcsin(d*x + c)^2/d - 12*(d*x + c)*b^4*arcsin(d*x + c)^2/d + 12*sqrt(-(d*x + c)^2 + 1)*a*b^3*arcsin(d*x + c)^2/d + 4*(d*x + c)*a^3*b*arcsin(d*x + c)/ d - 24*(d*x + c)*a*b^3*arcsin(d*x + c)/d + 12*sqrt(-(d*x + c)^2 + 1)*a^2*b ^2*arcsin(d*x + c)/d - 24*sqrt(-(d*x + c)^2 + 1)*b^4*arcsin(d*x + c)/d + ( d*x + c)*a^4/d - 12*(d*x + c)*a^2*b^2/d + 24*(d*x + c)*b^4/d + 4*sqrt(-(d* x + c)^2 + 1)*a^3*b/d - 24*sqrt(-(d*x + c)^2 + 1)*a*b^3/d
Time = 0.61 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.92 \[ \int (a+b \arcsin (c+d x))^4 \, dx=a^4\,x+\frac {b^4\,\left (c+d\,x\right )\,\left ({\mathrm {asin}\left (c+d\,x\right )}^4-12\,{\mathrm {asin}\left (c+d\,x\right )}^2+24\right )}{d}-\frac {b^4\,\left (24\,\mathrm {asin}\left (c+d\,x\right )-4\,{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d}+\frac {6\,a^2\,b^2\,\left (2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}+\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )\right )}{d}+\frac {4\,a^3\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {4\,a\,b^3\,\left (3\,{\mathrm {asin}\left (c+d\,x\right )}^2-6\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d}-\frac {4\,a\,b^3\,\left (6\,\mathrm {asin}\left (c+d\,x\right )-{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\left (c+d\,x\right )}{d} \]
a^4*x + (b^4*(c + d*x)*(asin(c + d*x)^4 - 12*asin(c + d*x)^2 + 24))/d - (b ^4*(24*asin(c + d*x) - 4*asin(c + d*x)^3)*(1 - (c + d*x)^2)^(1/2))/d + (6* a^2*b^2*(2*asin(c + d*x)*(1 - (c + d*x)^2)^(1/2) + (asin(c + d*x)^2 - 2)*( c + d*x)))/d + (4*a^3*b*((1 - (c + d*x)^2)^(1/2) + asin(c + d*x)*(c + d*x) ))/d + (4*a*b^3*(3*asin(c + d*x)^2 - 6)*(1 - (c + d*x)^2)^(1/2))/d - (4*a* b^3*(6*asin(c + d*x) - asin(c + d*x)^3)*(c + d*x))/d