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

Optimal. Leaf size=32 \[ \frac{x \log \left (\sqrt{x^2-1}+x\right )}{\sqrt{x^2+1}}-\frac{1}{2} \cosh ^{-1}\left (x^2\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0406656, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {191, 2554, 276, 52} \[ \frac{x \log \left (\sqrt{x^2-1}+x\right )}{\sqrt{x^2+1}}-\frac{1}{2} \cosh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcCosh[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 276

Int[(x_)^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m +
1, 2*n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a1 + b1*x^(n/k))^p*(a2 + b2*x^(n/k))^p, x], x, x^k], x] /; k
!= 1] /; FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && IntegerQ[m]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

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

Mathematica [B]  time = 0.0831153, size = 89, normalized size = 2.78 \[ \frac{4 x \log \left (\sqrt{x^2-1}+x\right )+\frac{\sqrt{x^2-1} \left (x^2+1\right ) \left (\log \left (1-\frac{x^2}{\sqrt{x^4-1}}\right )-\log \left (\frac{x^2}{\sqrt{x^4-1}}+1\right )\right )}{\sqrt{x^4-1}}}{4 \sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.012, 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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [B]  time = 1.98225, size = 149, normalized size = 4.66 \begin{align*} \frac{2 \, \sqrt{x^{2} + 1} x \log \left (x + \sqrt{x^{2} - 1}\right ) +{\left (x^{2} + 1\right )} \log \left (-x^{2} + \sqrt{x^{2} + 1} \sqrt{x^{2} - 1}\right )}{2 \,{\left (x^{2} + 1\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="fricas")

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Giac [A]  time = 1.11607, size = 49, normalized size = 1.53 \begin{align*} \frac{x \log \left (x + \sqrt{x^{2} - 1}\right )}{\sqrt{x^{2} + 1}} + \frac{1}{2} \, \log \left (x^{2} - \sqrt{x^{4} - 1}\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]

x*log(x + sqrt(x^2 - 1))/sqrt(x^2 + 1) + 1/2*log(x^2 - sqrt(x^4 - 1))