3.7.61 \(\int \frac {x \sin ^{-1}(x)}{(1-x^2)^{3/2}} \, dx\) [661]

Optimal. Leaf size=19 \[ \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-\tanh ^{-1}(x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 19, 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 = {4767, 212} \begin {gather*} \frac {\text {ArcSin}(x)}{\sqrt {1-x^2}}-\tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*ArcSin[x])/(1 - x^2)^(3/2),x]

[Out]

ArcSin[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 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[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*ArcSin[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 \sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx &=\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-\int \frac {1}{1-x^2} \, dx\\ &=\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-\tanh ^{-1}(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-\tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[x]/Sqrt[1 - x^2] - ArcTanh[x]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(17)=34\).
time = 0.10, size = 46, normalized size = 2.42

method	result	size
default	\(-\frac {\sqrt {-x^{2}+1}\, \arcsin \left (x \right )}{x^{2}-1}-\ln \left (\frac {1}{\sqrt {-x^{2}+1}}+\frac {x}{\sqrt {-x^{2}+1}}\right )\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.94, size = 25, normalized size = 1.32 \begin {gather*} \frac {\arcsin \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*arcsin(x)/(-x^2+1)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
time = 0.54, size = 44, normalized size = 2.32 \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} \arcsin \left (x\right )}{2 \, {\left (x^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(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)*arcsin(x))/(x^2 - 1)

________________________________________________________________________________________

Sympy [A]
time = 4.51, size = 20, normalized size = 1.05 \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 {asin}{\left (x \right )}}{\sqrt {1 - x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.93, size = 27, normalized size = 1.42 \begin {gather*} \frac {\arcsin \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*arcsin(x)/(-x^2+1)^(3/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x\,\mathrm {asin}\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*asin(x))/(1 - x^2)^(3/2),x)

[Out]

int((x*asin(x))/(1 - x^2)^(3/2), x)

________________________________________________________________________________________