\(\int \arccos (a x)^2 \, dx\) [16]

Optimal result

Integrand size = 6, antiderivative size = 35 \[ \int \arccos (a x)^2 \, dx=-2 x-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)}{a}+x \arccos (a x)^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4716, 4768, 8} \[ \int \arccos (a x)^2 \, dx=-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)}{a}+x \arccos (a x)^2-2 x \]

[In]

Int[ArcCos[a*x]^2,x]

[Out]

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

Rule 8

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

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]

Rubi steps \begin{align*} \text {integral}& = x \arccos (a x)^2+(2 a) \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)}{a}+x \arccos (a x)^2-2 \int 1 \, dx \\ & = -2 x-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)}{a}+x \arccos (a x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \arccos (a x)^2 \, dx=-2 x-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)}{a}+x \arccos (a x)^2 \]

[In]

Integrate[ArcCos[a*x]^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {\arccos \left (a x \right )^{2} a x -2 a x -2 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\)	\(37\)
default	\(\frac {\arccos \left (a x \right )^{2} a x -2 a x -2 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\)	\(37\)

[In]

int(arccos(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \arccos (a x)^2 \, dx=\frac {a x \arccos \left (a x\right )^{2} - 2 \, a x - 2 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \]

[In]

integrate(arccos(a*x)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \arccos (a x)^2 \, dx=\begin {cases} x \operatorname {acos}^{2}{\left (a x \right )} - 2 x - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x}{4} & \text {otherwise} \end {cases} \]

[In]

integrate(acos(a*x)**2,x)

[Out]

Piecewise((x*acos(a*x)**2 - 2*x - 2*sqrt(-a**2*x**2 + 1)*acos(a*x)/a, Ne(a, 0)), (pi**2*x/4, True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \arccos (a x)^2 \, dx=x \arccos \left (a x\right )^{2} - 2 \, x - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \]

[In]

integrate(arccos(a*x)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \arccos (a x)^2 \, dx=x \arccos \left (a x\right )^{2} - 2 \, x - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \]

[In]

integrate(arccos(a*x)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \arccos (a x)^2 \, dx=\left \{\begin {array}{cl} \frac {x\,\pi ^2}{4} & \text {\ if\ \ }a=0\\ x\,\left ({\mathrm {acos}\left (a\,x\right )}^2-2\right )-\frac {2\,\mathrm {acos}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{a} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

[In]

int(acos(a*x)^2,x)

[Out]

piecewise(a == 0, (x*pi^2)/4, a ~= 0, x*(acos(a*x)^2 - 2) - (2*acos(a*x)*(- a^2*x^2 + 1)^(1/2))/a)