3.15 \(\int \frac{\log (x+\sqrt{1+x^2})}{(1-x^2)^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ \frac{x \log \left (\sqrt{x^2+1}+x\right )}{\sqrt{1-x^2}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0352362, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {191, 2554, 275, 216} \[ \frac{x \log \left (\sqrt{x^2+1}+x\right )}{\sqrt{1-x^2}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[x + Sqrt[1 + x^2]]/(1 - x^2)^(3/2),x]

[Out]

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 2554

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]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 216

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

Rubi steps

\begin{align*} \int \frac{\log \left (x+\sqrt{1+x^2}\right )}{\left (1-x^2\right )^{3/2}} \, dx &=\frac{x \log \left (x+\sqrt{1+x^2}\right )}{\sqrt{1-x^2}}-\int \frac{x}{\sqrt{1-x^4}} \, dx\\ &=\frac{x \log \left (x+\sqrt{1+x^2}\right )}{\sqrt{1-x^2}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{2} \sin ^{-1}\left (x^2\right )+\frac{x \log \left (x+\sqrt{1+x^2}\right )}{\sqrt{1-x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0951354, size = 64, normalized size = 1.88 \[ \frac{1}{2} \sqrt{1-x^2} \left (-\frac{2 x \log \left (\sqrt{x^2+1}+x\right )}{x^2-1}-\frac{\sqrt{x^2+1} \sin ^{-1}\left (x^2\right )}{\sqrt{1-x^4}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x + Sqrt[1 + x^2]]/(1 - x^2)^(3/2),x]

[Out]

(Sqrt[1 - x^2]*(-((Sqrt[1 + x^2]*ArcSin[x^2])/Sqrt[1 - x^4]) - (2*x*Log[x + Sqrt[1 + x^2]])/(-1 + x^2)))/2

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{\ln \left ( x+\sqrt{{x}^{2}+1} \right ) \left ( -{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x+(x^2+1)^(1/2))/(-x^2+1)^(3/2),x)

[Out]

int(ln(x+(x^2+1)^(1/2))/(-x^2+1)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x + \sqrt{x^{2} + 1}\right )}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x+(x^2+1)^(1/2))/(-x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(log(x + sqrt(x^2 + 1))/(-x^2 + 1)^(3/2), x)

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Fricas [B]  time = 2.55405, size = 153, normalized size = 4.5 \begin{align*} -\frac{\sqrt{-x^{2} + 1} x \log \left (x + \sqrt{x^{2} + 1}\right ) -{\left (x^{2} - 1\right )} \arctan \left (\frac{\sqrt{x^{2} + 1} \sqrt{-x^{2} + 1} - 1}{x^{2}}\right )}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x+(x^2+1)^(1/2))/(-x^2+1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.13983, size = 49, normalized size = 1.44 \begin{align*} -\frac{\sqrt{-x^{2} + 1} x \log \left (x + \sqrt{x^{2} + 1}\right )}{x^{2} - 1} - \frac{1}{2} \, \arcsin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x+(x^2+1)^(1/2))/(-x^2+1)^(3/2),x, algorithm="giac")

[Out]

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