\(\int \frac {x \arccos (x)}{(1-x^2)^{3/2}} \, dx\) [662]

Optimal result

Integrand size = 15, antiderivative size = 17 \[ \int \frac {x \arccos (x)}{\left (1-x^2\right )^{3/2}} \, dx=\frac {\arccos (x)}{\sqrt {1-x^2}}+\text {arctanh}(x) \]

[Out]

arctanh(x)+arccos(x)/(-x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4768, 212} \[ \int \frac {x \arccos (x)}{\left (1-x^2\right )^{3/2}} \, dx=\frac {\arccos (x)}{\sqrt {1-x^2}}+\text {arctanh}(x) \]

[In]

Int[(x*ArcCos[x])/(1 - x^2)^(3/2),x]

[Out]

ArcCos[x]/Sqrt[1 - x^2] + ArcTanh[x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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}& = \frac {\arccos (x)}{\sqrt {1-x^2}}+\int \frac {1}{1-x^2} \, dx \\ & = \frac {\arccos (x)}{\sqrt {1-x^2}}+\text {arctanh}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {x \arccos (x)}{\left (1-x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {2 \arccos (x)}{\sqrt {1-x^2}}-\log (1-x)+\log (1+x)\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(15)=30\).

Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.76

method	result	size
default	\(-\frac {\sqrt {-x^{2}+1}\, \arccos \left (x \right )}{x^{2}-1}-\ln \left (-\frac {x}{\sqrt {-x^{2}+1}}+\frac {1}{\sqrt {-x^{2}+1}}\right )\)	\(47\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int \frac {x \arccos (x)}{\left (1-x^2\right )^{3/2}} \, dx=\frac {{\left (x^{2} - 1\right )} \log \left (x + 1\right ) - {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 2 \, \sqrt {-x^{2} + 1} \arccos \left (x\right )}{2 \, {\left (x^{2} - 1\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 4.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {x \arccos (x)}{\left (1-x^2\right )^{3/2}} \, dx=- \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} + \frac {\operatorname {acos}{\left (x \right )}}{\sqrt {1 - x^{2}}} \]

[In]

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

[Out]

-log(x - 1)/2 + log(x + 1)/2 + acos(x)/sqrt(1 - x**2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47 \[ \int \frac {x \arccos (x)}{\left (1-x^2\right )^{3/2}} \, dx=\frac {\arccos \left (x\right )}{\sqrt {-x^{2} + 1}} + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {x \arccos (x)}{\left (1-x^2\right )^{3/2}} \, dx=\frac {\arccos \left (x\right )}{\sqrt {-x^{2} + 1}} + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x \arccos (x)}{\left (1-x^2\right )^{3/2}} \, dx=\int \frac {x\,\mathrm {acos}\left (x\right )}{{\left (1-x^2\right )}^{3/2}} \,d x \]

[In]

int((x*acos(x))/(1 - x^2)^(3/2),x)

[Out]

int((x*acos(x))/(1 - x^2)^(3/2), x)