\(\int (a+b \arcsin (c+d x))^3 \, dx\) [201]

Optimal result

Integrand size = 12, antiderivative size = 104 \[ \int (a+b \arcsin (c+d x))^3 \, dx=-6 a b^2 x-\frac {6 b^3 \sqrt {1-(c+d x)^2}}{d}-\frac {6 b^3 (c+d x) \arcsin (c+d x)}{d}+\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^3}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4887, 4715, 4767, 267} \[ \int (a+b \arcsin (c+d x))^3 \, dx=\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^3}{d}-6 a b^2 x-\frac {6 b^3 (c+d x) \arcsin (c+d x)}{d}-\frac {6 b^3 \sqrt {1-(c+d x)^2}}{d} \]

[In]

Int[(a + b*ArcSin[c + d*x])^3,x]

[Out]

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

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]

Rule 4887

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92 \[ \int (a+b \arcsin (c+d x))^3 \, dx=\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+(c+d x) (a+b \arcsin (c+d x))^3-6 b^2 \left (a (c+d x)+b \sqrt {1-(c+d x)^2}+b (c+d x) \arcsin (c+d x)\right )}{d} \]

[In]

Integrate[(a + b*ArcSin[c + d*x])^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+3 a \,b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+3 a^{2} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\)	\(166\)
default	\(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+3 a \,b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+3 a^{2} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\)	\(166\)
parts	\(x \,a^{3}+\frac {b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )}{d}+\frac {3 a^{2} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\)	\(167\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.52 \[ \int (a+b \arcsin (c+d x))^3 \, dx=\frac {{\left (b^{3} d x + b^{3} c\right )} \arcsin \left (d x + c\right )^{3} + {\left (a^{3} - 6 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x + a b^{2} c\right )} \arcsin \left (d x + c\right )^{2} + 3 \, {\left ({\left (a^{2} b - 2 \, b^{3}\right )} d x + {\left (a^{2} b - 2 \, b^{3}\right )} c\right )} \arcsin \left (d x + c\right ) + 3 \, {\left (b^{3} \arcsin \left (d x + c\right )^{2} + 2 \, a b^{2} \arcsin \left (d x + c\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (92) = 184\).

Time = 0.19 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.71 \[ \int (a+b \arcsin (c+d x))^3 \, dx=\begin {cases} a^{3} x + \frac {3 a^{2} b c \operatorname {asin}{\left (c + d x \right )}}{d} + 3 a^{2} b x \operatorname {asin}{\left (c + d x \right )} + \frac {3 a^{2} b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {3 a b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 6 a b^{2} x + \frac {6 a b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {6 b^{3} c \operatorname {asin}{\left (c + d x \right )}}{d} + b^{3} x \operatorname {asin}^{3}{\left (c + d x \right )} - 6 b^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {3 b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {6 b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (a+b \arcsin (c+d x))^3 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

b^3*x*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + a^3*x + 3*((d*x + c)*arcsin(d*x + c) + sqrt(-
(d*x + c)^2 + 1))*a^2*b/d + integrate(3*(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*b^3*d*x*arctan2(d*x + c, sqrt(d*
x + c + 1)*sqrt(-d*x - c + 1))^2 + (a*b^2*d^2*x^2 + 2*a*b^2*c*d*x + a*b^2*c^2 - a*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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (100) = 200\).

Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.00 \[ \int (a+b \arcsin (c+d x))^3 \, dx=\frac {{\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{3}}{d} + \frac {3 \, {\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right )^{2}}{d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} \arcsin \left (d x + c\right )^{2}}{d} + \frac {3 \, {\left (d x + c\right )} a^{2} b \arcsin \left (d x + c\right )}{d} - \frac {6 \, {\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )}{d} + \frac {6 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{2} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{3}}{d} - \frac {6 \, {\left (d x + c\right )} a b^{2}}{d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b}{d} - \frac {6 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3}}{d} \]

[In]

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

[Out]

(d*x + c)*b^3*arcsin(d*x + c)^3/d + 3*(d*x + c)*a*b^2*arcsin(d*x + c)^2/d + 3*sqrt(-(d*x + c)^2 + 1)*b^3*arcsi
n(d*x + c)^2/d + 3*(d*x + c)*a^2*b*arcsin(d*x + c)/d - 6*(d*x + c)*b^3*arcsin(d*x + c)/d + 6*sqrt(-(d*x + c)^2
+ 1)*a*b^2*arcsin(d*x + c)/d + (d*x + c)*a^3/d - 6*(d*x + c)*a*b^2/d + 3*sqrt(-(d*x + c)^2 + 1)*a^2*b/d - 6*s
qrt(-(d*x + c)^2 + 1)*b^3/d

Mupad [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.46 \[ \int (a+b \arcsin (c+d x))^3 \, dx=a^3\,x-\frac {b^3\,\left (6\,\mathrm {asin}\left (c+d\,x\right )-{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\left (c+d\,x\right )}{d}+\frac {3\,a\,b^2\,\left (2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}+\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )\right )}{d}+\frac {3\,a^2\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {b^3\,\left (3\,{\mathrm {asin}\left (c+d\,x\right )}^2-6\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d} \]

[In]

int((a + b*asin(c + d*x))^3,x)

[Out]

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