\(\int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx\) [70]

Optimal result

Integrand size = 16, antiderivative size = 8 \[ \int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx=\frac {1}{2 \arccos (x)^2} \]

[Out]

1/2/arccos(x)^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4738} \[ \int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx=\frac {1}{2 \arccos (x)^2} \]

[In]

Int[1/(Sqrt[1 - x^2]*ArcCos[x]^3),x]

[Out]

1/(2*ArcCos[x]^2)

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 \arccos (x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx=\frac {1}{2 \arccos (x)^2} \]

[In]

Integrate[1/(Sqrt[1 - x^2]*ArcCos[x]^3),x]

[Out]

1/(2*ArcCos[x]^2)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {1}{2 \arccos \left (x \right )^{2}}\)	\(7\)
default	\(\frac {1}{2 \arccos \left (x \right )^{2}}\)	\(7\)

[In]

int(1/arccos(x)^3/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/arccos(x)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx=\frac {1}{2 \, \arccos \left (x\right )^{2}} \]

[In]

integrate(1/arccos(x)^3/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2/arccos(x)^2

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx=\frac {1}{2 \operatorname {acos}^{2}{\left (x \right )}} \]

[In]

integrate(1/acos(x)**3/(-x**2+1)**(1/2),x)

[Out]

1/(2*acos(x)**2)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx=\frac {1}{2 \, \arccos \left (x\right )^{2}} \]

[In]

integrate(1/arccos(x)^3/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/2/arccos(x)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx=\frac {1}{2 \, \arccos \left (x\right )^{2}} \]

[In]

integrate(1/arccos(x)^3/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/2/arccos(x)^2

Mupad [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {1-x^2} \arccos (x)^3} \, dx=\frac {1}{2\,{\mathrm {acos}\left (x\right )}^2} \]

[In]

int(1/(acos(x)^3*(1 - x^2)^(1/2)),x)

[Out]

1/(2*acos(x)^2)