3.1 \(\int \sin ^{-1}(x) \log (x) \, dx\)

Optimal. Leaf size=51 \[ -2 \sqrt{1-x^2}+\sqrt{1-x^2} \log (x)+\tanh ^{-1}\left (\sqrt{1-x^2}\right )-x (1-\log (x)) \sin ^{-1}(x) \]

[Out]

-2*Sqrt[1 - x^2] + ArcTanh[Sqrt[1 - x^2]] - x*ArcSin[x]*(1 - Log[x]) + Sqrt[1 - x^2]*Log[x]

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Rubi [A]  time = 0.0291379, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.4, Rules used = {4619, 261, 2387, 266, 50, 63, 206} \[ -2 \sqrt{1-x^2}+\sqrt{1-x^2} \log (x)+\tanh ^{-1}\left (\sqrt{1-x^2}\right )-x \sin ^{-1}(x)+x \log (x) \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]*Log[x],x]

[Out]

-2*Sqrt[1 - x^2] - x*ArcSin[x] + ArcTanh[Sqrt[1 - x^2]] + Sqrt[1 - x^2]*Log[x] + x*ArcSin[x]*Log[x]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 2387

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)*(x_))]^(m_.), x_Symbol] :> With[{u
= IntHide[Px*F[d*(e + f*x)]^m, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; F
reeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCos
h}, F]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sin ^{-1}(x) \log (x) \, dx &=\sqrt{1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)-\int \left (\frac{\sqrt{1-x^2}}{x}+\sin ^{-1}(x)\right ) \, dx\\ &=\sqrt{1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)-\int \frac{\sqrt{1-x^2}}{x} \, dx-\int \sin ^{-1}(x) \, dx\\ &=-x \sin ^{-1}(x)+\sqrt{1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{1-x}}{x} \, dx,x,x^2\right )+\int \frac{x}{\sqrt{1-x^2}} \, dx\\ &=-2 \sqrt{1-x^2}-x \sin ^{-1}(x)+\sqrt{1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^2\right )\\ &=-2 \sqrt{1-x^2}-x \sin ^{-1}(x)+\sqrt{1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^2}\right )\\ &=-2 \sqrt{1-x^2}-x \sin ^{-1}(x)+\tanh ^{-1}\left (\sqrt{1-x^2}\right )+\sqrt{1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0205966, size = 52, normalized size = 1.02 \[ -2 \sqrt{1-x^2}+\left (\sqrt{1-x^2}-1\right ) \log (x)+\log \left (\sqrt{1-x^2}+1\right )+x (\log (x)-1) \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[x]*Log[x],x]

[Out]

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

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Maple [B]  time = 0.039, size = 93, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}+1} \left ( \arcsin \left ( x \right ) \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \ln \left ( 2\,{\frac{\tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) }{ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}+1}} \right ) - \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}\ln \left ( 2\,{\frac{\tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) }{ \left ( \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) \right ) ^{2}+1}} \right ) -\arcsin \left ( x \right ) \tan \left ( 1/2\,\arcsin \left ( x \right ) \right ) -2 \right ) }-\ln \left ( \left ( \tan \left ({\frac{\arcsin \left ( x \right ) }{2}} \right ) \right ) ^{2}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)*ln(x),x)

[Out]

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

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Maxima [A]  time = 1.42246, size = 78, normalized size = 1.53 \begin{align*}{\left (x \log \left (x\right ) - x\right )} \arcsin \left (x\right ) + \sqrt{-x^{2} + 1} \log \left (x\right ) - 2 \, \sqrt{-x^{2} + 1} + \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)*log(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.51791, size = 167, normalized size = 3.27 \begin{align*} x \arcsin \left (x\right ) \log \left (x\right ) - x \arcsin \left (x\right ) + \sqrt{-x^{2} + 1}{\left (\log \left (x\right ) - 2\right )} + \frac{1}{2} \, \log \left (\sqrt{-x^{2} + 1} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{-x^{2} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)*log(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.8642, size = 102, normalized size = 2. \begin{align*} x \log{\left (x \right )} \operatorname{asin}{\left (x \right )} - x \operatorname{asin}{\left (x \right )} + \sqrt{1 - x^{2}} \log{\left (x \right )} - \sqrt{1 - x^{2}} - \begin{cases} - \frac{x}{\sqrt{-1 + \frac{1}{x^{2}}}} - \operatorname{acosh}{\left (\frac{1}{x} \right )} + \frac{1}{x \sqrt{-1 + \frac{1}{x^{2}}}} & \text{for}\: \frac{1}{\left |{x^{2}}\right |} > 1 \\\frac{i x}{\sqrt{1 - \frac{1}{x^{2}}}} + i \operatorname{asin}{\left (\frac{1}{x} \right )} - \frac{i}{x \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x)*ln(x),x)

[Out]

x*log(x)*asin(x) - x*asin(x) + sqrt(1 - x**2)*log(x) - sqrt(1 - x**2) - Piecewise((-x/sqrt(-1 + x**(-2)) - aco
sh(1/x) + 1/(x*sqrt(-1 + x**(-2))), 1/Abs(x**2) > 1), (I*x/sqrt(1 - 1/x**2) + I*asin(1/x) - I/(x*sqrt(1 - 1/x*
*2)), True))

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Giac [B]  time = 1.1764, size = 367, normalized size = 7.2 \begin{align*} x \arcsin \left (x\right ) \log \left (x\right ) + \sqrt{-x^{2} + 1} \log \left (x\right ) - \frac{2 \, x \arcsin \left (x\right )}{{\left (\sqrt{-x^{2} + 1} + 1\right )}{\left (\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} + \frac{x^{2} \log \left (\sqrt{-x^{2} + 1} + 1\right )}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}{\left (\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} + \frac{\log \left (\sqrt{-x^{2} + 1} + 1\right )}{\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1} - \frac{x^{2} \log \left ({\left | x \right |}\right )}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}{\left (\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} - \frac{\log \left ({\left | x \right |}\right )}{\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1} + \frac{2 \, x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}{\left (\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} - \frac{2}{\frac{x^{2}}{{\left (\sqrt{-x^{2} + 1} + 1\right )}^{2}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)*log(x),x, algorithm="giac")

[Out]

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