3.382 \(\int \frac{\sqrt{\frac{a}{x^2}}}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=22 \[ x \left (-\sqrt{\frac{a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]

[Out]

-(Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^2]])

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Rubi [A]  time = 0.0085699, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {15, 266, 63, 207} \[ x \left (-\sqrt{\frac{a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x^2]/Sqrt[1 + x^2],x]

[Out]

-(Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^2]])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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 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 207

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

Rubi steps

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

Mathematica [A]  time = 0.003938, size = 22, normalized size = 1. \[ x \left (-\sqrt{\frac{a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x^2]/Sqrt[1 + x^2],x]

[Out]

-(Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^2]])

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Maple [A]  time = 0.005, size = 19, normalized size = 0.9 \begin{align*} -\sqrt{{\frac{a}{{x}^{2}}}}x{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{2}+1}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/x^2)^(1/2)/(x^2+1)^(1/2),x)

[Out]

-(a/x^2)^(1/2)*x*arctanh(1/(x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{2}}}}{\sqrt{x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a/x^2)^(1/2)/(x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a/x^2)/sqrt(x^2 + 1), x)

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Fricas [A]  time = 1.00657, size = 182, normalized size = 8.27 \begin{align*} \left [x \sqrt{\frac{a}{x^{2}}} \log \left (\frac{\sqrt{x^{2} + 1} - 1}{x}\right ), 2 \, \sqrt{-a} \arctan \left (-\frac{\sqrt{-a} x^{2} \sqrt{\frac{a}{x^{2}}} - \sqrt{x^{2} + 1} \sqrt{-a} x \sqrt{\frac{a}{x^{2}}}}{a}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a/x^2)^(1/2)/(x^2+1)^(1/2),x, algorithm="fricas")

[Out]

[x*sqrt(a/x^2)*log((sqrt(x^2 + 1) - 1)/x), 2*sqrt(-a)*arctan(-(sqrt(-a)*x^2*sqrt(a/x^2) - sqrt(x^2 + 1)*sqrt(-
a)*x*sqrt(a/x^2))/a)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{2}}}}{\sqrt{x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a/x**2)**(1/2)/(x**2+1)**(1/2),x)

[Out]

Integral(sqrt(a/x**2)/sqrt(x**2 + 1), x)

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Giac [A]  time = 1.12686, size = 41, normalized size = 1.86 \begin{align*} -\frac{1}{2} \, \sqrt{a}{\left (\log \left (\sqrt{x^{2} + 1} + 1\right ) - \log \left (\sqrt{x^{2} + 1} - 1\right )\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a/x^2)^(1/2)/(x^2+1)^(1/2),x, algorithm="giac")

[Out]

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