\(\int x^3 \arccos (a x)^3 \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 167 \[ \int x^3 \arccos (a x)^3 \, dx=\frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}-\frac {9 x^2 \arccos (a x)}{32 a^2}-\frac {3}{32} x^4 \arccos (a x)-\frac {9 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{16 a}-\frac {3 \arccos (a x)^3}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^3-\frac {45 \arcsin (a x)}{256 a^4} \]

[Out]

-9/32*x^2*arccos(a*x)/a^2-3/32*x^4*arccos(a*x)-3/32*arccos(a*x)^3/a^4+1/4*x^4*arccos(a*x)^3-45/256*arcsin(a*x)
/a^4+45/256*x*(-a^2*x^2+1)^(1/2)/a^3+3/128*x^3*(-a^2*x^2+1)^(1/2)/a-9/32*x*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)/a^
3-3/16*x^3*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4796, 4738, 327, 222} \[ \int x^3 \arccos (a x)^3 \, dx=-\frac {3 \arccos (a x)^3}{32 a^4}-\frac {45 \arcsin (a x)}{256 a^4}-\frac {9 x^2 \arccos (a x)}{32 a^2}-\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{16 a}+\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}-\frac {9 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{32 a^3}+\frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}+\frac {1}{4} x^4 \arccos (a x)^3-\frac {3}{32} x^4 \arccos (a x) \]

[In]

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

[Out]

(45*x*Sqrt[1 - a^2*x^2])/(256*a^3) + (3*x^3*Sqrt[1 - a^2*x^2])/(128*a) - (9*x^2*ArcCos[a*x])/(32*a^2) - (3*x^4
*ArcCos[a*x])/32 - (9*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(32*a^3) - (3*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(1
6*a) - (3*ArcCos[a*x]^3)/(32*a^4) + (x^4*ArcCos[a*x]^3)/4 - (45*ArcSin[a*x])/(256*a^4)

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}{4} x^4 \arccos (a x)^3+\frac {1}{4} (3 a) \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{16 a}+\frac {1}{4} x^4 \arccos (a x)^3-\frac {3}{8} \int x^3 \arccos (a x) \, dx+\frac {9 \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{16 a} \\ & = -\frac {3}{32} x^4 \arccos (a x)-\frac {9 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{16 a}+\frac {1}{4} x^4 \arccos (a x)^3+\frac {9 \int \frac {\arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{32 a^3}-\frac {9 \int x \arccos (a x) \, dx}{16 a^2}-\frac {1}{32} (3 a) \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}-\frac {9 x^2 \arccos (a x)}{32 a^2}-\frac {3}{32} x^4 \arccos (a x)-\frac {9 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{16 a}-\frac {3 \arccos (a x)^3}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^3-\frac {9 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{128 a}-\frac {9 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{32 a} \\ & = \frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}-\frac {9 x^2 \arccos (a x)}{32 a^2}-\frac {3}{32} x^4 \arccos (a x)-\frac {9 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{16 a}-\frac {3 \arccos (a x)^3}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^3-\frac {9 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{256 a^3}-\frac {9 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{64 a^3} \\ & = \frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}-\frac {9 x^2 \arccos (a x)}{32 a^2}-\frac {3}{32} x^4 \arccos (a x)-\frac {9 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{16 a}-\frac {3 \arccos (a x)^3}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^3-\frac {45 \arcsin (a x)}{256 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.69 \[ \int x^3 \arccos (a x)^3 \, dx=\frac {3 a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right )-24 a^2 x^2 \left (3+a^2 x^2\right ) \arccos (a x)-24 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arccos (a x)^2+8 \left (-3+8 a^4 x^4\right ) \arccos (a x)^3-45 \arcsin (a x)}{256 a^4} \]

[In]

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

[Out]

(3*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2) - 24*a^2*x^2*(3 + a^2*x^2)*ArcCos[a*x] - 24*a*x*Sqrt[1 - a^2*x^2]*(3
+ 2*a^2*x^2)*ArcCos[a*x]^2 + 8*(-3 + 8*a^4*x^4)*ArcCos[a*x]^3 - 45*ArcSin[a*x])/(256*a^4)

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\frac {a^{4} x^{4} \arccos \left (a x \right )^{3}}{4}-\frac {3 \arccos \left (a x \right )^{2} \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 a x \sqrt {-a^{2} x^{2}+1}+3 \arccos \left (a x \right )\right )}{32}-\frac {3 a^{4} x^{4} \arccos \left (a x \right )}{32}+\frac {3 a x \left (2 a^{2} x^{2}+3\right ) \sqrt {-a^{2} x^{2}+1}}{256}+\frac {45 \arccos \left (a x \right )}{256}-\frac {9 a^{2} x^{2} \arccos \left (a x \right )}{32}+\frac {9 a x \sqrt {-a^{2} x^{2}+1}}{64}+\frac {3 \arccos \left (a x \right )^{3}}{16}}{a^{4}}\)	\(151\)
default	\(\frac {\frac {a^{4} x^{4} \arccos \left (a x \right )^{3}}{4}-\frac {3 \arccos \left (a x \right )^{2} \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 a x \sqrt {-a^{2} x^{2}+1}+3 \arccos \left (a x \right )\right )}{32}-\frac {3 a^{4} x^{4} \arccos \left (a x \right )}{32}+\frac {3 a x \left (2 a^{2} x^{2}+3\right ) \sqrt {-a^{2} x^{2}+1}}{256}+\frac {45 \arccos \left (a x \right )}{256}-\frac {9 a^{2} x^{2} \arccos \left (a x \right )}{32}+\frac {9 a x \sqrt {-a^{2} x^{2}+1}}{64}+\frac {3 \arccos \left (a x \right )^{3}}{16}}{a^{4}}\)	\(151\)

[In]

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

[Out]

1/a^4*(1/4*a^4*x^4*arccos(a*x)^3-3/32*arccos(a*x)^2*(2*a^3*x^3*(-a^2*x^2+1)^(1/2)+3*a*x*(-a^2*x^2+1)^(1/2)+3*a
rccos(a*x))-3/32*a^4*x^4*arccos(a*x)+3/256*a*x*(2*a^2*x^2+3)*(-a^2*x^2+1)^(1/2)+45/256*arccos(a*x)-9/32*a^2*x^
2*arccos(a*x)+9/64*a*x*(-a^2*x^2+1)^(1/2)+3/16*arccos(a*x)^3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.57 \[ \int x^3 \arccos (a x)^3 \, dx=\frac {8 \, {\left (8 \, a^{4} x^{4} - 3\right )} \arccos \left (a x\right )^{3} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arccos \left (a x\right ) + 3 \, {\left (2 \, a^{3} x^{3} - 8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arccos \left (a x\right )^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{256 \, a^{4}} \]

[In]

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

[Out]

1/256*(8*(8*a^4*x^4 - 3)*arccos(a*x)^3 - 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*arccos(a*x) + 3*(2*a^3*x^3 - 8*(2*a^3
*x^3 + 3*a*x)*arccos(a*x)^2 + 15*a*x)*sqrt(-a^2*x^2 + 1))/a^4

Sympy [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00 \[ \int x^3 \arccos (a x)^3 \, dx=\begin {cases} \frac {x^{4} \operatorname {acos}^{3}{\left (a x \right )}}{4} - \frac {3 x^{4} \operatorname {acos}{\left (a x \right )}}{32} - \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{16 a} + \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1}}{128 a} - \frac {9 x^{2} \operatorname {acos}{\left (a x \right )}}{32 a^{2}} - \frac {9 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{32 a^{3}} + \frac {45 x \sqrt {- a^{2} x^{2} + 1}}{256 a^{3}} - \frac {3 \operatorname {acos}^{3}{\left (a x \right )}}{32 a^{4}} + \frac {45 \operatorname {acos}{\left (a x \right )}}{256 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{4}}{32} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**4*acos(a*x)**3/4 - 3*x**4*acos(a*x)/32 - 3*x**3*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(16*a) + 3*x**
3*sqrt(-a**2*x**2 + 1)/(128*a) - 9*x**2*acos(a*x)/(32*a**2) - 9*x*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(32*a**3)
+ 45*x*sqrt(-a**2*x**2 + 1)/(256*a**3) - 3*acos(a*x)**3/(32*a**4) + 45*acos(a*x)/(256*a**4), Ne(a, 0)), (pi**3
*x**4/32, True))

Maxima [F]

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

[In]

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

[Out]

1/4*x^4*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3 - 3*a*integrate(1/4*sqrt(a*x + 1)*sqrt(-a*x + 1)*x^4*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 = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.84 \[ \int x^3 \arccos (a x)^3 \, dx=\frac {1}{4} \, x^{4} \arccos \left (a x\right )^{3} - \frac {3}{32} \, x^{4} \arccos \left (a x\right ) - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{2}}{16 \, a} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{128 \, a} - \frac {9 \, x^{2} \arccos \left (a x\right )}{32 \, a^{2}} - \frac {9 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{2}}{32 \, a^{3}} - \frac {3 \, \arccos \left (a x\right )^{3}}{32 \, a^{4}} + \frac {45 \, \sqrt {-a^{2} x^{2} + 1} x}{256 \, a^{3}} + \frac {45 \, \arccos \left (a x\right )}{256 \, a^{4}} \]

[In]

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

[Out]

1/4*x^4*arccos(a*x)^3 - 3/32*x^4*arccos(a*x) - 3/16*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)^2/a + 3/128*sqrt(-a^2*x
^2 + 1)*x^3/a - 9/32*x^2*arccos(a*x)/a^2 - 9/32*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)^2/a^3 - 3/32*arccos(a*x)^3/a^
4 + 45/256*sqrt(-a^2*x^2 + 1)*x/a^3 + 45/256*arccos(a*x)/a^4

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^3*acos(a*x)^3, x)