∫ x log(1+√ ____ 1 − x2 )______________

Optimal. Leaf size=55 \[ \sqrt {1-x^2}-\log \left (1+\sqrt {1-x^2}\right )-\sqrt {1-x^2} \log \left (1+\sqrt {1-x^2}\right ) \]

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

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Rubi [A]
time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {267, 2634, 1605, 196, 45} \begin {gather*} \sqrt {1-x^2}-\sqrt {1-x^2} \log \left (\sqrt {1-x^2}+1\right )-\log \left (\sqrt {1-x^2}+1\right ) \end {gather*}

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

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d

*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]

&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /

; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

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]

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coef

f[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x,

r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]

/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

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

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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.75 \begin {gather*} \sqrt {1-x^2}-\left (1+\sqrt {1-x^2}\right ) \log \left (1+\sqrt {1-x^2}\right ) \end {gather*}

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

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

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Mathics [A]
time = 7.39, size = 36, normalized size = 0.65 \begin {gather*} 1+\sqrt {1-x^2}-\text {Log}\left [1+\sqrt {1-x^2}\right ] \left (1+\sqrt {1-x^2}\right ) \end {gather*}

mathics('Integrate[x*Log[1 + Sqrt[1 - x^2]]/Sqrt[1 - x^2],x]')

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

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method	result	size

derivativedivides	\(-\ln \left (1+\sqrt {-x^{2}+1}\right ) \left (1+\sqrt {-x^{2}+1}\right )+1+\sqrt {-x^{2}+1}\)	\(37\)
default	\(-\ln \left (1+\sqrt {-x^{2}+1}\right ) \left (1+\sqrt {-x^{2}+1}\right )+1+\sqrt {-x^{2}+1}\)	\(37\)

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

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

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Maxima [A]
time = 0.25, size = 36, normalized size = 0.65 \begin {gather*} -{\left (\sqrt {-x^{2} + 1} + 1\right )} \log \left (\sqrt {-x^{2} + 1} + 1\right ) + \sqrt {-x^{2} + 1} + 1 \end {gather*}

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

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

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Fricas [A]
time = 0.33, size = 35, normalized size = 0.64 \begin {gather*} -{\left (\sqrt {-x^{2} + 1} + 1\right )} \log \left (\sqrt {-x^{2} + 1} + 1\right ) + \sqrt {-x^{2} + 1} \end {gather*}

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

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

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Sympy [A]
time = 3.58, size = 31, normalized size = 0.56 \begin {gather*} \sqrt {1 - x^{2}} - \left (\sqrt {1 - x^{2}} + 1\right ) \log {\left (\sqrt {1 - x^{2}} + 1 \right )} + 1 \end {gather*}

integrate(x*ln(1+(-x**2+1)**(1/2))/(-x**2+1)**(1/2),x)

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

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Giac [A]
time = 0.00, size = 39, normalized size = 0.71 \begin {gather*} \sqrt {-x^{2}+1}+1-\left (\sqrt {-x^{2}+1}+1\right ) \ln \left (\sqrt {-x^{2}+1}+1\right ) \end {gather*}

integrate(x*log(1+(-x^2+1)^(1/2))/(-x^2+1)^(1/2),x)

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

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Mupad [B]
time = 0.34, size = 27, normalized size = 0.49 \begin {gather*} -\left (\ln \left (\sqrt {1-x^2}+1\right )-1\right )\,\left (\sqrt {1-x^2}+1\right ) \end {gather*}

int((x*log((1 - x^2)^(1/2) + 1))/(1 - x^2)^(1/2),x)

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

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3.1.24 \(\int \frac {x \log (1+\sqrt {1-x^2})}{\sqrt {1-x^2}} \, dx\) [24]