3.1.15 \(\int \frac {\log (x+\sqrt {1+x^2})}{(1-x^2)^{3/2}} \, dx\) [15]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {197, 2634, 281, 222} \begin {gather*} \frac {x \log \left (\sqrt {x^2+1}+x\right )}{\sqrt {1-x^2}}-\frac {\text {ArcSin}\left (x^2\right )}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

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 222

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]

Rule 281

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 2634

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]

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} \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(34)=68\).
time = 0.08, size = 76, normalized size = 2.24 \begin {gather*} \frac {1}{2} \sqrt {1-x^2} \left (-\frac {\sqrt {1+x^2} \tan ^{-1}\left (\frac {x^2}{\sqrt {1-x^4}}\right )}{\sqrt {1-x^4}}-\frac {2 x \log \left (x+\sqrt {1+x^2}\right )}{-1+x^2}\right ) \end {gather*}

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (x +\sqrt {x^{2}+1}\right )}{\left (-x^{2}+1\right )^{\frac {3}{2}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (28) = 56\).
time = 0.68, size = 62, normalized size = 1.82 \begin {gather*} -\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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.48, size = 36, normalized size = 1.06 \begin {gather*} -\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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (x+\sqrt {x^2+1}\right )}{{\left (1-x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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