\(\int (a+b \arcsin (c x))^3 \, dx\) [155]

Optimal result

Integrand size = 10, antiderivative size = 82 \[ \int (a+b \arcsin (c x))^3 \, dx=-6 a b^2 x-\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^3 x \arcsin (c x)+\frac {3 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c}+x (a+b \arcsin (c x))^3 \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4715, 4767, 267} \[ \int (a+b \arcsin (c x))^3 \, dx=\frac {3 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c}+x (a+b \arcsin (c x))^3-6 a b^2 x-6 b^3 x \arcsin (c x)-\frac {6 b^3 \sqrt {1-c^2 x^2}}{c} \]

[In]

Int[(a + b*ArcSin[c*x])^3,x]

[Out]

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

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[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 4767

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] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(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]

Rubi steps \begin{align*} \text {integral}& = 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 \\ & = \frac {3 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c}+x (a+b \arcsin (c x))^3-\left (6 b^2\right ) \int (a+b \arcsin (c x)) \, dx \\ & = -6 a b^2 x+\frac {3 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c}+x (a+b \arcsin (c x))^3-\left (6 b^3\right ) \int \arcsin (c x) \, dx \\ & = -6 a b^2 x-6 b^3 x \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+\left (6 b^3 c\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx \\ & = -6 a b^2 x-\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^3 x \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 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.94 \[ \int (a+b \arcsin (c x))^3 \, dx=x (a+b \arcsin (c x))^3+\frac {3 b \left (\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 b \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arcsin (c x)\right )\right )}{c} \]

[In]

Integrate[(a + b*ArcSin[c*x])^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.61

method	result	size
derivativedivides	\(\frac {c x \,a^{3}+b^{3} \left (c x \arcsin \left (c x \right )^{3}+3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )+3 a \,b^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\)	\(132\)
default	\(\frac {c x \,a^{3}+b^{3} \left (c x \arcsin \left (c x \right )^{3}+3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )+3 a \,b^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\)	\(132\)
parts	\(x \,a^{3}+\frac {b^{3} \left (c x \arcsin \left (c x \right )^{3}+3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-6 \sqrt {-c^{2} x^{2}+1}-6 c x \arcsin \left (c x \right )\right )}{c}+\frac {3 a \,b^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )}{c}+\frac {3 a^{2} b \left (c x \arcsin \left (c x \right )+\sqrt {-c^{2} x^{2}+1}\right )}{c}\)	\(136\)

[In]

int((a+b*arcsin(c*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/c*(c*x*a^3+b^3*(c*x*arcsin(c*x)^3+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*(c*x*arcsin(c*x)^2-2*c*x+2*arcsin(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
)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.32 \[ \int (a+b \arcsin (c x))^3 \, dx=\frac {b^{3} c x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} c x \arcsin \left (c x\right )^{2} + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} c x \arcsin \left (c x\right ) + {\left (a^{3} - 6 \, a b^{2}\right )} c x + 3 \, {\left (b^{3} \arcsin \left (c x\right )^{2} + 2 \, a b^{2} \arcsin \left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{c} \]

[In]

integrate((a+b*arcsin(c*x))^3,x, algorithm="fricas")

[Out]

(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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (76) = 152\).

Time = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.95 \[ \int (a+b \arcsin (c x))^3 \, dx=\begin {cases} a^{3} x + 3 a^{2} b x \operatorname {asin}{\left (c x \right )} + \frac {3 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{c} + 3 a b^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 6 a b^{2} x + \frac {6 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + b^{3} x \operatorname {asin}^{3}{\left (c x \right )} - 6 b^{3} x \operatorname {asin}{\left (c x \right )} + \frac {3 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{c} - \frac {6 b^{3} \sqrt {- c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\a^{3} x & \text {otherwise} \end {cases} \]

[In]

integrate((a+b*asin(c*x))**3,x)

[Out]

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)*asin(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))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.72 \[ \int (a+b \arcsin (c x))^3 \, dx=b^{3} x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} x \arcsin \left (c x\right )^{2} + 3 \, {\left (\frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )^{2}}{c} - \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )}}{c}\right )} b^{3} - 6 \, a b^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{3} x + \frac {3 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a^{2} b}{c} \]

[In]

integrate((a+b*arcsin(c*x))^3,x, algorithm="maxima")

[Out]

b^3*x*arcsin(c*x)^3 + 3*a*b^2*x*arcsin(c*x)^2 + 3*(sqrt(-c^2*x^2 + 1)*arcsin(c*x)^2/c - 2*(c*x*arcsin(c*x) + s
qrt(-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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.83 \[ \int (a+b \arcsin (c x))^3 \, dx=b^{3} x \arcsin \left (c x\right )^{3} + 3 \, a b^{2} x \arcsin \left (c x\right )^{2} + 3 \, a^{2} b x \arcsin \left (c x\right ) - 6 \, b^{3} x \arcsin \left (c x\right ) + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arcsin \left (c x\right )^{2}}{c} + a^{3} x - 6 \, a b^{2} x + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arcsin \left (c x\right )}{c} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{c} - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{c} \]

[In]

integrate((a+b*arcsin(c*x))^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.95 \[ \int (a+b \arcsin (c x))^3 \, dx=\left \{\begin {array}{cl} a^3\,x-b^3\,\left (x\,\left (6\,\mathrm {asin}\left (c\,x\right )-{\mathrm {asin}\left (c\,x\right )}^3\right )-\sqrt {\frac {1}{c^2}-x^2}\,\left (3\,{\mathrm {asin}\left (c\,x\right )}^2-6\right )\right )+3\,a\,b^2\,\left (x\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )+2\,\mathrm {asin}\left (c\,x\right )\,\sqrt {\frac {1}{c^2}-x^2}\right )+\frac {3\,a^2\,b\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c} & \text {\ if\ \ }0<c\\ a^3\,x+\frac {3\,a^2\,b\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}+3\,a\,b^2\,x\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )+b^3\,x\,\mathrm {asin}\left (c\,x\right )\,\left ({\mathrm {asin}\left (c\,x\right )}^2-6\right )+\frac {3\,b^3\,\sqrt {1-c^2\,x^2}\,\left ({\mathrm {asin}\left (c\,x\right )}^2-2\right )}{c}+\frac {6\,a\,b^2\,\mathrm {asin}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c} & \text {\ if\ \ }\neg 0<c \end {array}\right . \]

[In]

int((a + b*asin(c*x))^3,x)

[Out]

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)