\(\int \arccos (a+b x)^3 \, dx\) [32]

Optimal result

Integrand size = 8, antiderivative size = 82 \[ \int \arccos (a+b x)^3 \, dx=\frac {6 \sqrt {1-(a+b x)^2}}{b}-\frac {6 (a+b x) \arccos (a+b x)}{b}-\frac {3 \sqrt {1-(a+b x)^2} \arccos (a+b x)^2}{b}+\frac {(a+b x) \arccos (a+b x)^3}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4888, 4716, 4768, 267} \[ \int \arccos (a+b x)^3 \, dx=\frac {(a+b x) \arccos (a+b x)^3}{b}-\frac {3 \sqrt {1-(a+b x)^2} \arccos (a+b x)^2}{b}-\frac {6 (a+b x) \arccos (a+b x)}{b}+\frac {6 \sqrt {1-(a+b x)^2}}{b} \]

[In]

Int[ArcCos[a + b*x]^3,x]

[Out]

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

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 4716

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4768

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcCos[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*ArcCos[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 4888

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCos[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 \arccos (x)^3 \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \arccos (a+b x)^3}{b}+\frac {3 \text {Subst}\left (\int \frac {x \arccos (x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {3 \sqrt {1-(a+b x)^2} \arccos (a+b x)^2}{b}+\frac {(a+b x) \arccos (a+b x)^3}{b}-\frac {6 \text {Subst}(\int \arccos (x) \, dx,x,a+b x)}{b} \\ & = -\frac {6 (a+b x) \arccos (a+b x)}{b}-\frac {3 \sqrt {1-(a+b x)^2} \arccos (a+b x)^2}{b}+\frac {(a+b x) \arccos (a+b x)^3}{b}-\frac {6 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {6 \sqrt {1-(a+b x)^2}}{b}-\frac {6 (a+b x) \arccos (a+b x)}{b}-\frac {3 \sqrt {1-(a+b x)^2} \arccos (a+b x)^2}{b}+\frac {(a+b x) \arccos (a+b x)^3}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int \arccos (a+b x)^3 \, dx=\frac {6 \sqrt {1-(a+b x)^2}-6 (a+b x) \arccos (a+b x)-3 \sqrt {1-(a+b x)^2} \arccos (a+b x)^2+(a+b x) \arccos (a+b x)^3}{b} \]

[In]

Integrate[ArcCos[a + b*x]^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\arccos \left (b x +a \right )^{3} \left (b x +a \right )-3 \arccos \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}+6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arccos \left (b x +a \right )}{b}\)	\(71\)
default	\(\frac {\arccos \left (b x +a \right )^{3} \left (b x +a \right )-3 \arccos \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}+6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arccos \left (b x +a \right )}{b}\)	\(71\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int \arccos (a+b x)^3 \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right )^{3} - 6 \, {\left (b x + a\right )} \arccos \left (b x + a\right ) - 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\arccos \left (b x + a\right )^{2} - 2\right )}}{b} \]

[In]

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

[Out]

((b*x + a)*arccos(b*x + a)^3 - 6*(b*x + a)*arccos(b*x + a) - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(arccos(b*x
+ a)^2 - 2))/b

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.33 \[ \int \arccos (a+b x)^3 \, dx=\begin {cases} \frac {a \operatorname {acos}^{3}{\left (a + b x \right )}}{b} - \frac {6 a \operatorname {acos}{\left (a + b x \right )}}{b} + x \operatorname {acos}^{3}{\left (a + b x \right )} - 6 x \operatorname {acos}{\left (a + b x \right )} - \frac {3 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a + b x \right )}}{b} + \frac {6 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acos}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(acos(b*x+a)**3,x)

[Out]

Piecewise((a*acos(a + b*x)**3/b - 6*a*acos(a + b*x)/b + x*acos(a + b*x)**3 - 6*x*acos(a + b*x) - 3*sqrt(-a**2
- 2*a*b*x - b**2*x**2 + 1)*acos(a + b*x)**2/b + 6*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/b, Ne(b, 0)), (x*acos(
a)**3, True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95 \[ \int \arccos (a+b x)^3 \, dx=\frac {{\left (b x + a\right )} \arccos \left (b x + a\right )^{3}}{b} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arccos \left (b x + a\right )^{2}}{b} - \frac {6 \, {\left (b x + a\right )} \arccos \left (b x + a\right )}{b} + \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]

[In]

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

[Out]

(b*x + a)*arccos(b*x + a)^3/b - 3*sqrt(-(b*x + a)^2 + 1)*arccos(b*x + a)^2/b - 6*(b*x + a)*arccos(b*x + a)/b +
6*sqrt(-(b*x + a)^2 + 1)/b

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.73 \[ \int \arccos (a+b x)^3 \, dx=-\frac {\left (3\,{\mathrm {acos}\left (a+b\,x\right )}^2-6\right )\,\sqrt {1-{\left (a+b\,x\right )}^2}}{b}-\frac {\left (6\,\mathrm {acos}\left (a+b\,x\right )-{\mathrm {acos}\left (a+b\,x\right )}^3\right )\,\left (a+b\,x\right )}{b} \]

[In]

int(acos(a + b*x)^3,x)

[Out]

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