3.20.17 \(\int \frac {1}{\sqrt {a+\frac {b}{x^2}} x} \, dx\) [1917]

Optimal. Leaf size=24 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[Out]

arctanh((a+b/x^2)^(1/2)/a^(1/2))/a^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^2]*x),x]

[Out]

ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]]/Sqrt[a]

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 214

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(24)=48\).
time = 0.01, size = 50, normalized size = 2.08 \begin {gather*} \frac {\sqrt {b+a x^2} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b+a x^2}}\right )}{\sqrt {a} \sqrt {a+\frac {b}{x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^2]*x),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(18)=36\).
time = 0.02, size = 46, normalized size = 1.92

method	result	size
default	\(\frac {\sqrt {a \,x^{2}+b}\, \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right )}{\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {a}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((a*x^2+b)/x^2)^(1/2)/x*(a*x^2+b)^(1/2)*ln(x*a^(1/2)+(a*x^2+b)^(1/2))/a^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
time = 0.53, size = 37, normalized size = 1.54 \begin {gather*} -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{2 \, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (18) = 36\).
time = 0.41, size = 80, normalized size = 3.33 \begin {gather*} \left [\frac {\log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right )}{2 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.50, size = 17, normalized size = 0.71 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(sqrt(a)*x/sqrt(b))/sqrt(a)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).
time = 1.44, size = 38, normalized size = 1.58 \begin {gather*} \frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, \sqrt {a}} - \frac {\log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.31, size = 18, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b/x^2)^(1/2)),x)

[Out]

atanh((a + b/x^2)^(1/2)/a^(1/2))/a^(1/2)

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