3.4.82 \(\int \frac {\sqrt {\frac {a}{x^2}}}{\sqrt {1+x^2}} \, dx\) [382]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {15, 272, 65, 213} \begin {gather*} x \left (-\sqrt {\frac {a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt {x^2+1}\right ) \end {gather*}

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 65

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 213

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

Rule 272

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]]

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 ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^2\right )\\ &=\left (\sqrt {\frac {a}{x^2}} x\right ) \text {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 = 0.02, size = 22, normalized size = 1.00 \begin {gather*} -\sqrt {\frac {a}{x^2}} x \tanh ^{-1}\left (\sqrt {1+x^2}\right ) \end {gather*}

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]])

________________________________________________________________________________________

Maple [A]
time = 0.20, size = 19, normalized size = 0.86

method	result	size
default	\(-\sqrt {\frac {a}{x^{2}}}\, x \arctanh \left (\frac {1}{\sqrt {x^{2}+1}}\right )\)	\(19\)
meijerg	\(\frac {\sqrt {\frac {a}{x^{2}}}\, x \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )\right ) \sqrt {\pi }\right )}{2 \sqrt {\pi }}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/x^2)^(1/2)/(x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A] Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (18) = 36\).
time = 0.33, size = 76, normalized size = 3.45 \begin {gather*} \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 {gather*}

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)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a}{x^{2}}}}{\sqrt {x^{2} + 1}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 3.67, size = 30, normalized size = 1.36 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {\frac {a}{x^2}}}{\sqrt {x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/x^2)^(1/2)/(x^2 + 1)^(1/2),x)

[Out]

int((a/x^2)^(1/2)/(x^2 + 1)^(1/2), x)

________________________________________________________________________________________