\(\int (a+b \arcsin (c x))^4 \, dx\) [1]

Optimal result

Integrand size = 10, antiderivative size = 95 \[ \int (a+b \arcsin (c x))^4 \, dx=24 b^4 x-\frac {24 b^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}-12 b^2 x (a+b \arcsin (c x))^2+\frac {4 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{c}+x (a+b \arcsin (c x))^4 \] Output:

24*b^4*x-24*b^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c-12*b^2*x*(a+b*arcsi 
n(c*x))^2+4*b*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^3/c+x*(a+b*arcsin(c*x)) 
^4
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int (a+b \arcsin (c x))^4 \, dx=x (a+b \arcsin (c x))^4+\frac {4 b \left (-3 b c x (a+b \arcsin (c x))^2+\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3+6 b^2 \left (b c x-\sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )\right )}{c} \] Input:

Integrate[(a + b*ArcSin[c*x])^4,x]
 

Output:

x*(a + b*ArcSin[c*x])^4 + (4*b*(-3*b*c*x*(a + b*ArcSin[c*x])^2 + Sqrt[1 - 
c^2*x^2]*(a + b*ArcSin[c*x])^3 + 6*b^2*(b*c*x - Sqrt[1 - c^2*x^2]*(a + b*A 
rcSin[c*x]))))/c
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {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.

\(\displaystyle \int (a+b \arcsin (c x))^4 \, dx\)

\(\Big \downarrow \) 5130

\(\displaystyle x (a+b \arcsin (c x))^4-4 b c \int \frac {x (a+b \arcsin (c x))^3}{\sqrt {1-c^2 x^2}}dx\)

\(\Big \downarrow \) 5182

\(\displaystyle x (a+b \arcsin (c x))^4-4 b c \left (\frac {3 b \int (a+b \arcsin (c x))^2dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{c^2}\right )\)

\(\Big \downarrow \) 5130

\(\displaystyle x (a+b \arcsin (c x))^4-4 b c \left (\frac {3 b \left (x (a+b \arcsin (c x))^2-2 b c \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{c^2}\right )\)

\(\Big \downarrow \) 5182

\(\displaystyle x (a+b \arcsin (c x))^4-4 b c \left (\frac {3 b \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{c^2}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle x (a+b \arcsin (c x))^4-4 b c \left (\frac {3 b \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{c^2}\right )\)

Input:

Int[(a + b*ArcSin[c*x])^4,x]
 

Output:

x*(a + b*ArcSin[c*x])^4 - 4*b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^ 
3)/c^2) + (3*b*(x*(a + b*ArcSin[c*x])^2 - 2*b*c*((b*x)/c - (Sqrt[1 - c^2*x 
^2]*(a + b*ArcSin[c*x]))/c^2)))/c)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 5130

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]
 

rule 5182

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(91)=182\).

Time = 0.36 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.13

method	result	size
derivativedivides	\(\frac {c x \,a^{4}+b^{4} \left (\arcsin \left (c x \right )^{4} c x +4 \arcsin \left (c x \right )^{3} \sqrt {-c^{2} x^{2}+1}-12 \arcsin \left (c x \right )^{2} c x +24 c x -24 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+4 a \,b^{3} \left (\arcsin \left (c x \right )^{3} c x +3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )+6 a^{2} b^{2} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+4 a^{3} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\)	\(202\)
default	\(\frac {c x \,a^{4}+b^{4} \left (\arcsin \left (c x \right )^{4} c x +4 \arcsin \left (c x \right )^{3} \sqrt {-c^{2} x^{2}+1}-12 \arcsin \left (c x \right )^{2} c x +24 c x -24 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+4 a \,b^{3} \left (\arcsin \left (c x \right )^{3} c x +3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )+6 a^{2} b^{2} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+4 a^{3} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\)	\(202\)
parts	\(x \,a^{4}+\frac {b^{4} \left (\arcsin \left (c x \right )^{4} c x +4 \arcsin \left (c x \right )^{3} \sqrt {-c^{2} x^{2}+1}-12 \arcsin \left (c x \right )^{2} c x +24 c x -24 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )}{c}+\frac {4 a \,b^{3} \left (\arcsin \left (c x \right )^{3} c x +3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )}{c}+\frac {6 a^{2} b^{2} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )}{c}+\frac {4 a^{3} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\)	\(209\)
orering	\(x \left (a +b \arcsin \left (c x \right )\right )^{4}+\frac {8 \left (a +b \arcsin \left (c x \right )\right )^{3} b}{c \sqrt {-c^{2} x^{2}+1}}+\frac {\left (5 c^{2} x^{2}-2\right ) x \left (\frac {12 \left (a +b \arcsin \left (c x \right )\right )^{2} b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {4 \left (a +b \arcsin \left (c x \right )\right )^{3} b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{c^{2}}+\frac {\left (c x -1\right ) \left (c x +1\right ) \left (5 c^{2} x^{2}+1\right ) \left (\frac {24 \left (a +b \arcsin \left (c x \right )\right ) b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {36 \left (a +b \arcsin \left (c x \right )\right )^{2} b^{2} c^{4} x}{\left (-c^{2} x^{2}+1\right )^{2}}+\frac {12 \left (a +b \arcsin \left (c x \right )\right )^{3} b \,c^{5} x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {4 \left (a +b \arcsin \left (c x \right )\right )^{3} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{c^{4}}+\frac {x \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (\frac {24 b^{4} c^{4}}{\left (-c^{2} x^{2}+1\right )^{2}}+\frac {144 \left (a +b \arcsin \left (c x \right )\right ) b^{3} c^{5} x}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {180 \left (a +b \arcsin \left (c x \right )\right )^{2} b^{2} c^{6} x^{2}}{\left (-c^{2} x^{2}+1\right )^{3}}+\frac {48 \left (a +b \arcsin \left (c x \right )\right )^{2} b^{2} c^{4}}{\left (-c^{2} x^{2}+1\right )^{2}}+\frac {60 \left (a +b \arcsin \left (c x \right )\right )^{3} b \,c^{7} x^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {36 \left (a +b \arcsin \left (c x \right )\right )^{3} b \,c^{5} x}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{c^{4}}\)	\(452\)

Input:

int((a+b*arcsin(c*x))^4,x,method=_RETURNVERBOSE)
 

Output:

1/c*(c*x*a^4+b^4*(arcsin(c*x)^4*c*x+4*arcsin(c*x)^3*(-c^2*x^2+1)^(1/2)-12* 
arcsin(c*x)^2*c*x+24*c*x-24*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+4*a*b^3*(arcsi 
n(c*x)^3*c*x+3*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)-6*(-c^2*x^2+1)^(1/2)-6*c*x 
*arcsin(c*x))+6*a^2*b^2*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1 
)^(1/2))+4*a^3*b*(c*x*arcsin(c*x)+(-c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.69 \[ \int (a+b \arcsin (c x))^4 \, dx=\frac {b^{4} c x \arcsin \left (c x\right )^{4} + 4 \, a b^{3} c x \arcsin \left (c x\right )^{3} + 6 \, {\left (a^{2} b^{2} - 2 \, b^{4}\right )} c x \arcsin \left (c x\right )^{2} + 4 \, {\left (a^{3} b - 6 \, a b^{3}\right )} c x \arcsin \left (c x\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} c x + 4 \, {\left (b^{4} \arcsin \left (c x\right )^{3} + 3 \, a b^{3} \arcsin \left (c x\right )^{2} + a^{3} b - 6 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 2 \, b^{4}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{c} \] Input:

integrate((a+b*arcsin(c*x))^4,x, algorithm="fricas")
 

Output:

(b^4*c*x*arcsin(c*x)^4 + 4*a*b^3*c*x*arcsin(c*x)^3 + 6*(a^2*b^2 - 2*b^4)*c 
*x*arcsin(c*x)^2 + 4*(a^3*b - 6*a*b^3)*c*x*arcsin(c*x) + (a^4 - 12*a^2*b^2 
 + 24*b^4)*c*x + 4*(b^4*arcsin(c*x)^3 + 3*a*b^3*arcsin(c*x)^2 + a^3*b - 6* 
a*b^3 + 3*(a^2*b^2 - 2*b^4)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (88) = 176\).

Time = 0.22 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.68 \[ \int (a+b \arcsin (c x))^4 \, dx=\begin {cases} a^{4} x + 4 a^{3} b x \operatorname {asin}{\left (c x \right )} + \frac {4 a^{3} b \sqrt {- c^{2} x^{2} + 1}}{c} + 6 a^{2} b^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 12 a^{2} b^{2} x + \frac {12 a^{2} b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + 4 a b^{3} x \operatorname {asin}^{3}{\left (c x \right )} - 24 a b^{3} x \operatorname {asin}{\left (c x \right )} + \frac {12 a b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{c} - \frac {24 a b^{3} \sqrt {- c^{2} x^{2} + 1}}{c} + b^{4} x \operatorname {asin}^{4}{\left (c x \right )} - 12 b^{4} x \operatorname {asin}^{2}{\left (c x \right )} + 24 b^{4} x + \frac {4 b^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (c x \right )}}{c} - \frac {24 b^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} & \text {for}\: c \neq 0 \\a^{4} x & \text {otherwise} \end {cases} \] Input:

integrate((a+b*asin(c*x))**4,x)
 

Output:

Piecewise((a**4*x + 4*a**3*b*x*asin(c*x) + 4*a**3*b*sqrt(-c**2*x**2 + 1)/c 
 + 6*a**2*b**2*x*asin(c*x)**2 - 12*a**2*b**2*x + 12*a**2*b**2*sqrt(-c**2*x 
**2 + 1)*asin(c*x)/c + 4*a*b**3*x*asin(c*x)**3 - 24*a*b**3*x*asin(c*x) + 1 
2*a*b**3*sqrt(-c**2*x**2 + 1)*asin(c*x)**2/c - 24*a*b**3*sqrt(-c**2*x**2 + 
 1)/c + b**4*x*asin(c*x)**4 - 12*b**4*x*asin(c*x)**2 + 24*b**4*x + 4*b**4* 
sqrt(-c**2*x**2 + 1)*asin(c*x)**3/c - 24*b**4*sqrt(-c**2*x**2 + 1)*asin(c* 
x)/c, Ne(c, 0)), (a**4*x, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (91) = 182\).

Time = 0.11 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.42 \[ \int (a+b \arcsin (c x))^4 \, dx=b^{4} x \arcsin \left (c x\right )^{4} + 4 \, a b^{3} x \arcsin \left (c x\right )^{3} + 6 \, a^{2} b^{2} x \arcsin \left (c x\right )^{2} + 12 \, {\left (\frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )^{2}}{c} - \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )}}{c}\right )} a b^{3} + 4 \, {\left (\frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )^{3}}{c} - 3 \, {\left (\frac {x \arcsin \left (c x\right )^{2}}{c} - \frac {2 \, {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )}}{c}\right )} c\right )} b^{4} - 12 \, a^{2} b^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{4} x + \frac {4 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a^{3} b}{c} \] Input:

integrate((a+b*arcsin(c*x))^4,x, algorithm="maxima")
 

Output:

b^4*x*arcsin(c*x)^4 + 4*a*b^3*x*arcsin(c*x)^3 + 6*a^2*b^2*x*arcsin(c*x)^2 
+ 12*(sqrt(-c^2*x^2 + 1)*arcsin(c*x)^2/c - 2*(c*x*arcsin(c*x) + sqrt(-c^2* 
x^2 + 1))/c)*a*b^3 + 4*(sqrt(-c^2*x^2 + 1)*arcsin(c*x)^3/c - 3*(x*arcsin(c 
*x)^2/c - 2*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c)/c)*c)*b^4 - 12*a^2*b^2* 
(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^4*x + 4*(c*x*arcsin(c*x) + sqrt 
(-c^2*x^2 + 1))*a^3*b/c
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (91) = 182\).

Time = 0.12 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.53 \[ \int (a+b \arcsin (c x))^4 \, dx=b^{4} x \arcsin \left (c x\right )^{4} + 4 \, a b^{3} x \arcsin \left (c x\right )^{3} + 6 \, a^{2} b^{2} x \arcsin \left (c x\right )^{2} - 12 \, b^{4} x \arcsin \left (c x\right )^{2} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{4} \arcsin \left (c x\right )^{3}}{c} + 4 \, a^{3} b x \arcsin \left (c x\right ) - 24 \, a b^{3} x \arcsin \left (c x\right ) + \frac {12 \, \sqrt {-c^{2} x^{2} + 1} a b^{3} \arcsin \left (c x\right )^{2}}{c} + a^{4} x - 12 \, a^{2} b^{2} x + 24 \, b^{4} x + \frac {12 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b^{2} \arcsin \left (c x\right )}{c} - \frac {24 \, \sqrt {-c^{2} x^{2} + 1} b^{4} \arcsin \left (c x\right )}{c} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a^{3} b}{c} - \frac {24 \, \sqrt {-c^{2} x^{2} + 1} a b^{3}}{c} \] Input:

integrate((a+b*arcsin(c*x))^4,x, algorithm="giac")
 

Output:

b^4*x*arcsin(c*x)^4 + 4*a*b^3*x*arcsin(c*x)^3 + 6*a^2*b^2*x*arcsin(c*x)^2 
- 12*b^4*x*arcsin(c*x)^2 + 4*sqrt(-c^2*x^2 + 1)*b^4*arcsin(c*x)^3/c + 4*a^ 
3*b*x*arcsin(c*x) - 24*a*b^3*x*arcsin(c*x) + 12*sqrt(-c^2*x^2 + 1)*a*b^3*a 
rcsin(c*x)^2/c + a^4*x - 12*a^2*b^2*x + 24*b^4*x + 12*sqrt(-c^2*x^2 + 1)*a 
^2*b^2*arcsin(c*x)/c - 24*sqrt(-c^2*x^2 + 1)*b^4*arcsin(c*x)/c + 4*sqrt(-c 
^2*x^2 + 1)*a^3*b/c - 24*sqrt(-c^2*x^2 + 1)*a*b^3/c
 

Mupad [B] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.75 \[ \int (a+b \arcsin (c x))^4 \, dx=\left \{\begin {array}{cl} a^4\,x+b^4\,\left (x\,\left ({\mathrm {asin}\left (c\,x\right )}^4-12\,{\mathrm {asin}\left (c\,x\right )}^2+24\right )-\sqrt {\frac {1}{c^2}-x^2}\,\left (24\,\mathrm {asin}\left (c\,x\right )-4\,{\mathrm {asin}\left (c\,x\right )}^3\right )\right )+6\,a^2\,b^2\,\left (x\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )+2\,\mathrm {asin}\left (c\,x\right )\,\sqrt {\frac {1}{c^2}-x^2}\right )-4\,a\,b^3\,\left (x\,\left (6\,\mathrm {asin}\left (c\,x\right )-{\mathrm {asin}\left (c\,x\right )}^3\right )-\sqrt {\frac {1}{c^2}-x^2}\,\left (3\,{\mathrm {asin}\left (c\,x\right )}^2-6\right )\right )+\frac {4\,a^3\,b\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c} & \text {\ if\ \ }0<c\\ a^4\,x+b^4\,x\,\left ({\mathrm {asin}\left (c\,x\right )}^4-12\,{\mathrm {asin}\left (c\,x\right )}^2+24\right )+\frac {4\,a^3\,b\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}+6\,a^2\,b^2\,x\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )+\frac {12\,a\,b^3\,\sqrt {1-c^2\,x^2}\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )}{c}+\frac {4\,b^4\,\mathrm {asin}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}\,\left ({\mathrm {asin}\left (c\,x\right )}^2-6\right )}{c}+4\,a\,b^3\,x\,\mathrm {asin}\left (c\,x\right )\,\left ({\mathrm {asin}\left (c\,x\right )}^2-6\right )+\frac {12\,a^2\,b^2\,\mathrm {asin}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c} & \text {\ if\ \ }\neg 0<c \end {array}\right . \] Input:

int((a + b*asin(c*x))^4,x)
 

Output:

piecewise(0 < c, a^4*x + b^4*(x*(- 12*asin(c*x)^2 + asin(c*x)^4 + 24) - (1 
/c^2 - x^2)^(1/2)*(24*asin(c*x) - 4*asin(c*x)^3)) + 6*a^2*b^2*(x*(asin(c*x 
)^2 - 2) + 2*asin(c*x)*(1/c^2 - x^2)^(1/2)) - 4*a*b^3*(x*(6*asin(c*x) - as 
in(c*x)^3) - (1/c^2 - x^2)^(1/2)*(3*asin(c*x)^2 - 6)) + (4*a^3*b*((- c^2*x 
^2 + 1)^(1/2) + c*x*asin(c*x)))/c, ~0 < c, a^4*x + b^4*x*(- 12*asin(c*x)^2 
 + asin(c*x)^4 + 24) + (4*a^3*b*((- c^2*x^2 + 1)^(1/2) + c*x*asin(c*x)))/c 
 + 6*a^2*b^2*x*(asin(c*x)^2 - 2) + (12*a*b^3*(- c^2*x^2 + 1)^(1/2)*(asin(c 
*x)^2 - 2))/c + (4*b^4*asin(c*x)*(- c^2*x^2 + 1)^(1/2)*(asin(c*x)^2 - 6))/ 
c + 4*a*b^3*x*asin(c*x)*(asin(c*x)^2 - 6) + (12*a^2*b^2*asin(c*x)*(- c^2*x 
^2 + 1)^(1/2))/c)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.41 \[ \int (a+b \arcsin (c x))^4 \, dx=\frac {\mathit {asin} \left (c x \right )^{4} b^{4} c x +4 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{3} b^{4}+4 \mathit {asin} \left (c x \right )^{3} a \,b^{3} c x +12 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} a \,b^{3}+6 \mathit {asin} \left (c x \right )^{2} a^{2} b^{2} c x -12 \mathit {asin} \left (c x \right )^{2} b^{4} c x +12 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a^{2} b^{2}-24 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{4}+4 \mathit {asin} \left (c x \right ) a^{3} b c x -24 \mathit {asin} \left (c x \right ) a \,b^{3} c x +4 \sqrt {-c^{2} x^{2}+1}\, a^{3} b -24 \sqrt {-c^{2} x^{2}+1}\, a \,b^{3}+a^{4} c x -12 a^{2} b^{2} c x +24 b^{4} c x}{c} \] Input:

int((a+b*asin(c*x))^4,x)
 

Output:

(asin(c*x)**4*b**4*c*x + 4*sqrt( - c**2*x**2 + 1)*asin(c*x)**3*b**4 + 4*as 
in(c*x)**3*a*b**3*c*x + 12*sqrt( - c**2*x**2 + 1)*asin(c*x)**2*a*b**3 + 6* 
asin(c*x)**2*a**2*b**2*c*x - 12*asin(c*x)**2*b**4*c*x + 12*sqrt( - c**2*x* 
*2 + 1)*asin(c*x)*a**2*b**2 - 24*sqrt( - c**2*x**2 + 1)*asin(c*x)*b**4 + 4 
*asin(c*x)*a**3*b*c*x - 24*asin(c*x)*a*b**3*c*x + 4*sqrt( - c**2*x**2 + 1) 
*a**3*b - 24*sqrt( - c**2*x**2 + 1)*a*b**3 + a**4*c*x - 12*a**2*b**2*c*x + 
 24*b**4*c*x)/c