\(\int x \arccos (a x)^3 \, dx\) [25]

Optimal result

Integrand size = 8, antiderivative size = 99 \[ \int x \arccos (a x)^3 \, dx=\frac {3 x \sqrt {1-a^2 x^2}}{8 a}-\frac {3}{4} x^2 \arccos (a x)-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 a}-\frac {\arccos (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^3-\frac {3 \arcsin (a x)}{8 a^2} \]

[Out]

-3/4*x^2*arccos(a*x)-1/4*arccos(a*x)^3/a^2+1/2*x^2*arccos(a*x)^3-3/8*arcsin(a*x)/a^2+3/8*x*(-a^2*x^2+1)^(1/2)/
a-3/4*x*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4724, 4796, 4738, 327, 222} \[ \int x \arccos (a x)^3 \, dx=-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 a}-\frac {\arccos (a x)^3}{4 a^2}-\frac {3 \arcsin (a x)}{8 a^2}+\frac {3 x \sqrt {1-a^2 x^2}}{8 a}+\frac {1}{2} x^2 \arccos (a x)^3-\frac {3}{4} x^2 \arccos (a x) \]

[In]

Int[x*ArcCos[a*x]^3,x]

[Out]

(3*x*Sqrt[1 - a^2*x^2])/(8*a) - (3*x^2*ArcCos[a*x])/4 - (3*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(4*a) - ArcCos[a
*x]^3/(4*a^2) + (x^2*ArcCos[a*x]^3)/2 - (3*ArcSin[a*x])/(8*a^2)

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 4724

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

Rule 4738

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

Rule 4796

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \arccos (a x)^3+\frac {1}{2} (3 a) \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 a}+\frac {1}{2} x^2 \arccos (a x)^3-\frac {3}{2} \int x \arccos (a x) \, dx+\frac {3 \int \frac {\arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a} \\ & = -\frac {3}{4} x^2 \arccos (a x)-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 a}-\frac {\arccos (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^3-\frac {1}{4} (3 a) \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 x \sqrt {1-a^2 x^2}}{8 a}-\frac {3}{4} x^2 \arccos (a x)-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 a}-\frac {\arccos (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^3-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a} \\ & = \frac {3 x \sqrt {1-a^2 x^2}}{8 a}-\frac {3}{4} x^2 \arccos (a x)-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 a}-\frac {\arccos (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arccos (a x)^3-\frac {3 \arcsin (a x)}{8 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int x \arccos (a x)^3 \, dx=\frac {3 a x \sqrt {1-a^2 x^2}-6 a^2 x^2 \arccos (a x)-6 a x \sqrt {1-a^2 x^2} \arccos (a x)^2+\left (-2+4 a^2 x^2\right ) \arccos (a x)^3-3 \arcsin (a x)}{8 a^2} \]

[In]

Integrate[x*ArcCos[a*x]^3,x]

[Out]

(3*a*x*Sqrt[1 - a^2*x^2] - 6*a^2*x^2*ArcCos[a*x] - 6*a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2 + (-2 + 4*a^2*x^2)*Ar
cCos[a*x]^3 - 3*ArcSin[a*x])/(8*a^2)

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\arccos \left (a x \right )^{3} a^{2} x^{2}}{2}-\frac {3 \arccos \left (a x \right )^{2} \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{4}-\frac {3 a^{2} x^{2} \arccos \left (a x \right )}{4}+\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arccos \left (a x \right )}{8}+\frac {\arccos \left (a x \right )^{3}}{2}}{a^{2}}\)	\(90\)
default	\(\frac {\frac {\arccos \left (a x \right )^{3} a^{2} x^{2}}{2}-\frac {3 \arccos \left (a x \right )^{2} \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{4}-\frac {3 a^{2} x^{2} \arccos \left (a x \right )}{4}+\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arccos \left (a x \right )}{8}+\frac {\arccos \left (a x \right )^{3}}{2}}{a^{2}}\)	\(90\)

[In]

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

[Out]

1/a^2*(1/2*arccos(a*x)^3*a^2*x^2-3/4*arccos(a*x)^2*(a*x*(-a^2*x^2+1)^(1/2)+arccos(a*x))-3/4*a^2*x^2*arccos(a*x
)+3/8*a*x*(-a^2*x^2+1)^(1/2)+3/8*arccos(a*x)+1/2*arccos(a*x)^3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int x \arccos (a x)^3 \, dx=\frac {2 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )^{3} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right ) - 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x \arccos \left (a x\right )^{2} - a x\right )}}{8 \, a^{2}} \]

[In]

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

[Out]

1/8*(2*(2*a^2*x^2 - 1)*arccos(a*x)^3 - 3*(2*a^2*x^2 - 1)*arccos(a*x) - 3*sqrt(-a^2*x^2 + 1)*(2*a*x*arccos(a*x)
^2 - a*x))/a^2

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int x \arccos (a x)^3 \, dx=\begin {cases} \frac {x^{2} \operatorname {acos}^{3}{\left (a x \right )}}{2} - \frac {3 x^{2} \operatorname {acos}{\left (a x \right )}}{4} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{4 a} + \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{8 a} - \frac {\operatorname {acos}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {acos}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{2}}{16} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**2*acos(a*x)**3/2 - 3*x**2*acos(a*x)/4 - 3*x*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(4*a) + 3*x*sqrt(-
a**2*x**2 + 1)/(8*a) - acos(a*x)**3/(4*a**2) + 3*acos(a*x)/(8*a**2), Ne(a, 0)), (pi**3*x**2/16, True))

Maxima [F]

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

[In]

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

[Out]

1/2*x^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3 - 3*a*integrate(1/2*sqrt(a*x + 1)*sqrt(-a*x + 1)*x^2*arct
an2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2/(a^2*x^2 - 1), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84 \[ \int x \arccos (a x)^3 \, dx=\frac {1}{2} \, x^{2} \arccos \left (a x\right )^{3} - \frac {3}{4} \, x^{2} \arccos \left (a x\right ) - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{2}}{4 \, a} - \frac {\arccos \left (a x\right )^{3}}{4 \, a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a} + \frac {3 \, \arccos \left (a x\right )}{8 \, a^{2}} \]

[In]

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

[Out]

1/2*x^2*arccos(a*x)^3 - 3/4*x^2*arccos(a*x) - 3/4*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)^2/a - 1/4*arccos(a*x)^3/a^2
+ 3/8*sqrt(-a^2*x^2 + 1)*x/a + 3/8*arccos(a*x)/a^2

Mupad [F(-1)]

Timed out. \[ \int x \arccos (a x)^3 \, dx=\int x\,{\mathrm {acos}\left (a\,x\right )}^3 \,d x \]

[In]

int(x*acos(a*x)^3,x)

[Out]

int(x*acos(a*x)^3, x)