Integrand size = 10, antiderivative size = 79 \[ \int (a+b \arcsin (c x))^3 \, dx=-\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^2 x (a+b \arcsin (c x))+\frac {3 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c}+x (a+b \arcsin (c x))^3 \] Output:
-6*b^3*(-c^2*x^2+1)^(1/2)/c-6*b^2*x*(a+b*arcsin(c*x))+3*b*(-c^2*x^2+1)^(1/ 2)*(a+b*arcsin(c*x))^2/c+x*(a+b*arcsin(c*x))^3
Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int (a+b \arcsin (c x))^3 \, dx=x (a+b \arcsin (c x))^3+\frac {3 b \left (\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 b \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arcsin (c x)\right )\right )}{c} \] Input:
Integrate[(a + b*ArcSin[c*x])^3,x]
Output:
x*(a + b*ArcSin[c*x])^3 + (3*b*(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x])))/c
Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5130, 5182, 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.
\(\displaystyle \int (a+b \arcsin (c x))^3 \, dx\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle x (a+b \arcsin (c x))^3-3 b c \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \int (a+b \arcsin (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\) |
Input:
Int[(a + b*ArcSin[c*x])^3,x]
Output:
x*(a + b*ArcSin[c*x])^3 - 3*b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^ 2)/c^2) + (2*b*(a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/c)
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]
Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.67
method | result | size |
derivativedivides | \(\frac {c x \,a^{3}+b^{3} \left (\arcsin \left (c x \right )^{3} c x +3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )+3 a \,b^{2} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(132\) |
default | \(\frac {c x \,a^{3}+b^{3} \left (\arcsin \left (c x \right )^{3} c x +3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )+3 a \,b^{2} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(132\) |
parts | \(x \,a^{3}+\frac {b^{3} \left (\arcsin \left (c x \right )^{3} c x +3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )}{c}+\frac {3 a^{2} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}+\frac {3 a \,b^{2} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(136\) |
orering | \(x \left (a +b \arcsin \left (c x \right )\right )^{3}-\frac {3 \left (c^{2} x^{2}-2\right ) \left (a +b \arcsin \left (c x \right )\right )^{2} b}{c \sqrt {-c^{2} x^{2}+1}}-\frac {2 x \left (c x -1\right ) \left (c x +1\right ) \left (\frac {6 \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {3 \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{c^{2}}-\frac {\left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (\frac {6 b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{4} x}{\left (-c^{2} x^{2}+1\right )^{2}}+\frac {9 \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{5} x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {3 \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{c^{4}}\) | \(253\) |
Input:
int((a+b*arcsin(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c*(c*x*a^3+b^3*(arcsin(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))+3*a*b^2*(arcsin(c*x)^2*c*x-2*c*x+2*ar csin(c*x)*(-c^2*x^2+1)^(1/2))+3*a^2*b*(c*x*arcsin(c*x)+(-c^2*x^2+1)^(1/2)) )
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.37 \[ \int (a+b \arcsin (c x))^3 \, dx=\frac {b^{3} c x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} c x \arcsin \left (c x\right )^{2} + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} c x \arcsin \left (c x\right ) + {\left (a^{3} - 6 \, a b^{2}\right )} c x + 3 \, {\left (b^{3} \arcsin \left (c x\right )^{2} + 2 \, a b^{2} \arcsin \left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{c} \] Input:
integrate((a+b*arcsin(c*x))^3,x, algorithm="fricas")
Output:
(b^3*c*x*arcsin(c*x)^3 + 3*a*b^2*c*x*arcsin(c*x)^2 + 3*(a^2*b - 2*b^3)*c*x *arcsin(c*x) + (a^3 - 6*a*b^2)*c*x + 3*(b^3*arcsin(c*x)^2 + 2*a*b^2*arcsin (c*x) + a^2*b - 2*b^3)*sqrt(-c^2*x^2 + 1))/c
Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (71) = 142\).
Time = 0.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.03 \[ \int (a+b \arcsin (c x))^3 \, dx=\begin {cases} a^{3} x + 3 a^{2} b x \operatorname {asin}{\left (c x \right )} + \frac {3 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{c} + 3 a b^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 6 a b^{2} x + \frac {6 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + b^{3} x \operatorname {asin}^{3}{\left (c x \right )} - 6 b^{3} x \operatorname {asin}{\left (c x \right )} + \frac {3 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{c} - \frac {6 b^{3} \sqrt {- c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\a^{3} x & \text {otherwise} \end {cases} \] Input:
integrate((a+b*asin(c*x))**3,x)
Output:
Piecewise((a**3*x + 3*a**2*b*x*asin(c*x) + 3*a**2*b*sqrt(-c**2*x**2 + 1)/c + 3*a*b**2*x*asin(c*x)**2 - 6*a*b**2*x + 6*a*b**2*sqrt(-c**2*x**2 + 1)*as in(c*x)/c + b**3*x*asin(c*x)**3 - 6*b**3*x*asin(c*x) + 3*b**3*sqrt(-c**2*x **2 + 1)*asin(c*x)**2/c - 6*b**3*sqrt(-c**2*x**2 + 1)/c, Ne(c, 0)), (a**3* x, True))
Time = 0.11 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.78 \[ \int (a+b \arcsin (c x))^3 \, dx=b^{3} x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} x \arcsin \left (c x\right )^{2} + 3 \, {\left (\frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )^{2}}{c} - \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )}}{c}\right )} b^{3} - 6 \, a b^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{3} x + \frac {3 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a^{2} b}{c} \] Input:
integrate((a+b*arcsin(c*x))^3,x, algorithm="maxima")
Output:
b^3*x*arcsin(c*x)^3 + 3*a*b^2*x*arcsin(c*x)^2 + 3*(sqrt(-c^2*x^2 + 1)*arcs in(c*x)^2/c - 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))/c)*b^3 - 6*a*b^2*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^3*x + 3*(c*x*arcsin(c*x) + sqrt(- c^2*x^2 + 1))*a^2*b/c
Time = 0.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.90 \[ \int (a+b \arcsin (c x))^3 \, dx=b^{3} x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} x \arcsin \left (c x\right )^{2} + 3 \, a^{2} b x \arcsin \left (c x\right ) - 6 \, b^{3} x \arcsin \left (c x\right ) + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arcsin \left (c x\right )^{2}}{c} + a^{3} x - 6 \, a b^{2} x + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arcsin \left (c x\right )}{c} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{c} - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{c} \] Input:
integrate((a+b*arcsin(c*x))^3,x, algorithm="giac")
Output:
b^3*x*arcsin(c*x)^3 + 3*a*b^2*x*arcsin(c*x)^2 + 3*a^2*b*x*arcsin(c*x) - 6* b^3*x*arcsin(c*x) + 3*sqrt(-c^2*x^2 + 1)*b^3*arcsin(c*x)^2/c + a^3*x - 6*a *b^2*x + 6*sqrt(-c^2*x^2 + 1)*a*b^2*arcsin(c*x)/c + 3*sqrt(-c^2*x^2 + 1)*a ^2*b/c - 6*sqrt(-c^2*x^2 + 1)*b^3/c
Time = 0.01 (sec) , antiderivative size = 242, normalized size of antiderivative = 3.06 \[ \int (a+b \arcsin (c x))^3 \, dx=\left \{\begin {array}{cl} a^3\,x-b^3\,\left (x\,\left (6\,\mathrm {asin}\left (c\,x\right )-{\mathrm {asin}\left (c\,x\right )}^3\right )-\sqrt {\frac {1}{c^2}-x^2}\,\left (3\,{\mathrm {asin}\left (c\,x\right )}^2-6\right )\right )+3\,a\,b^2\,\left (x\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )+2\,\mathrm {asin}\left (c\,x\right )\,\sqrt {\frac {1}{c^2}-x^2}\right )+\frac {3\,a^2\,b\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c} & \text {\ if\ \ }0<c\\ a^3\,x+\frac {3\,a^2\,b\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}+3\,a\,b^2\,x\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )+b^3\,x\,\mathrm {asin}\left (c\,x\right )\,\left ({\mathrm {asin}\left (c\,x\right )}^2-6\right )+\frac {3\,b^3\,\sqrt {1-c^2\,x^2}\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )}{c}+\frac {6\,a\,b^2\,\mathrm {asin}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c} & \text {\ if\ \ }\neg 0<c \end {array}\right . \] Input:
int((a + b*asin(c*x))^3,x)
Output:
piecewise(0 < c, a^3*x - b^3*(x*(6*asin(c*x) - asin(c*x)^3) - (1/c^2 - x^2 )^(1/2)*(3*asin(c*x)^2 - 6)) + 3*a*b^2*(x*(asin(c*x)^2 - 2) + 2*asin(c*x)* (1/c^2 - x^2)^(1/2)) + (3*a^2*b*((- c^2*x^2 + 1)^(1/2) + c*x*asin(c*x)))/c , ~0 < c, a^3*x + (3*a^2*b*((- c^2*x^2 + 1)^(1/2) + c*x*asin(c*x)))/c + 3* a*b^2*x*(asin(c*x)^2 - 2) + b^3*x*asin(c*x)*(asin(c*x)^2 - 6) + (3*b^3*(- c^2*x^2 + 1)^(1/2)*(asin(c*x)^2 - 2))/c + (6*a*b^2*asin(c*x)*(- c^2*x^2 + 1)^(1/2))/c)
Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.82 \[ \int (a+b \arcsin (c x))^3 \, dx=\frac {\mathit {asin} \left (c x \right )^{3} b^{3} c x +3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} b^{3}+3 \mathit {asin} \left (c x \right )^{2} a \,b^{2} c x +6 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a \,b^{2}+3 \mathit {asin} \left (c x \right ) a^{2} b c x -6 \mathit {asin} \left (c x \right ) b^{3} c x +3 \sqrt {-c^{2} x^{2}+1}\, a^{2} b -6 \sqrt {-c^{2} x^{2}+1}\, b^{3}+a^{3} c x -6 a \,b^{2} c x}{c} \] Input:
int((a+b*asin(c*x))^3,x)
Output:
(asin(c*x)**3*b**3*c*x + 3*sqrt( - c**2*x**2 + 1)*asin(c*x)**2*b**3 + 3*as in(c*x)**2*a*b**2*c*x + 6*sqrt( - c**2*x**2 + 1)*asin(c*x)*a*b**2 + 3*asin (c*x)*a**2*b*c*x - 6*asin(c*x)*b**3*c*x + 3*sqrt( - c**2*x**2 + 1)*a**2*b - 6*sqrt( - c**2*x**2 + 1)*b**3 + a**3*c*x - 6*a*b**2*c*x)/c