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

Optimal. Leaf size=17 \[ \frac {\cos ^{-1}(x)}{\sqrt {1-x^2}}+\tanh ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4768, 212} \begin {gather*} \frac {\text {ArcCos}(x)}{\sqrt {1-x^2}}+\tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {x \cos ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx &=\frac {\cos ^{-1}(x)}{\sqrt {1-x^2}}+\int \frac {1}{1-x^2} \, dx\\ &=\frac {\cos ^{-1}(x)}{\sqrt {1-x^2}}+\tanh ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 32, normalized size = 1.88 \begin {gather*} \frac {1}{2} \left (\frac {2 \cos ^{-1}(x)}{\sqrt {1-x^2}}-\log (1-x)+\log (1+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(15)=30\).
time = 0.10, size = 47, normalized size = 2.76

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

Verification of antiderivative is not currently implemented for this CAS.

[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/(-x^2+1)^(1/2))

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Maxima [A]
time = 0.92, size = 25, normalized size = 1.47 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).
time = 0.54, size = 44, normalized size = 2.59 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 4.83, size = 20, normalized size = 1.18 \begin {gather*} \begin {cases} \operatorname {acoth}{\left (x \right )} & \text {for}\: x^{2} > 1 \\\operatorname {atanh}{\left (x \right )} & \text {for}\: x^{2} < 1 \end {cases} + \frac {\operatorname {acos}{\left (x \right )}}{\sqrt {1 - x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((acoth(x), x**2 > 1), (atanh(x), x**2 < 1)) + acos(x)/sqrt(1 - x**2)

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Giac [A]
time = 1.39, size = 27, normalized size = 1.59 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x\,\mathrm {acos}\left (x\right )}{{\left (1-x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*acos(x))/(1 - x^2)^(3/2),x)

[Out]

int((x*acos(x))/(1 - x^2)^(3/2), x)

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