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]

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

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, 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 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])

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 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 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 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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.46, size = 54, normalized size = 1.06 \[ x \arcsin \relax (x) \log \relax (x) - x \arcsin \relax (x) + \sqrt {-x^{2} + 1} {\left (\log \relax (x) - 2\right )} + \frac {1}{2} \, \log \left (\sqrt {-x^{2} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {-x^{2} + 1} - 1\right ) \]

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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giac [B]  time = 1.15, size = 272, normalized size = 5.33 \[ x \arcsin \relax (x) \log \relax (x) + \sqrt {-x^{2} + 1} \log \relax (x) - \frac {2 \, x \arcsin \relax (x)}{{\left (\sqrt {-x^{2} + 1} + 1\right )} {\left (\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} + \frac {x^{2} \log \left (\sqrt {-x^{2} + 1} + 1\right )}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2} {\left (\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} + \frac {\log \left (\sqrt {-x^{2} + 1} + 1\right )}{\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1} - \frac {x^{2} \log \left ({\left | x \right |}\right )}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2} {\left (\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} - \frac {\log \left ({\left | x \right |}\right )}{\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1} + \frac {2 \, x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2} {\left (\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} - \frac {2}{\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1} \]

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)

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maple [B]  time = 0.06, size = 93, normalized size = 1.82 \[ -\ln \left (\tan ^{2}\left (\frac {\arcsin \relax (x )}{2}\right )+1\right )+\frac {2 \arcsin \relax (x ) \ln \left (\frac {2 \tan \left (\frac {\arcsin \relax (x )}{2}\right )}{\tan ^{2}\left (\frac {\arcsin \relax (x )}{2}\right )+1}\right ) \tan \left (\frac {\arcsin \relax (x )}{2}\right )-2 \ln \left (\frac {2 \tan \left (\frac {\arcsin \relax (x )}{2}\right )}{\tan ^{2}\left (\frac {\arcsin \relax (x )}{2}\right )+1}\right ) \left (\tan ^{2}\left (\frac {\arcsin \relax (x )}{2}\right )\right )-2 \arcsin \relax (x ) \tan \left (\frac {\arcsin \relax (x )}{2}\right )-4}{\tan ^{2}\left (\frac {\arcsin \relax (x )}{2}\right )+1} \]

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 = 0.96, size = 58, normalized size = 1.14 \[ {\left (x \log \relax (x) - x\right )} \arcsin \relax (x) + \sqrt {-x^{2} + 1} \log \relax (x) - 2 \, \sqrt {-x^{2} + 1} + \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {asin}\relax (x)\,\ln \relax (x) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x)*log(x),x)

[Out]

int(asin(x)*log(x), x)

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sympy [A]  time = 7.76, size = 102, normalized size = 2.00 \[ x \log {\relax (x )} \operatorname {asin}{\relax (x )} - x \operatorname {asin}{\relax (x )} + \sqrt {1 - x^{2}} \log {\relax (x )} - \sqrt {1 - x^{2}} - \begin {cases} - \frac {x}{\sqrt {-1 + \frac {1}{x^{2}}}} - \operatorname {acosh}{\left (\frac {1}{x} \right )} + \frac {1}{x \sqrt {-1 + \frac {1}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {i x}{\sqrt {1 - \frac {1}{x^{2}}}} + i \operatorname {asin}{\left (\frac {1}{x} \right )} - \frac {i}{x \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases} \]

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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