\(\int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx\) [1925]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {342, 223, 212} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{\sqrt {b}} \]

[In]

Int[1/(Sqrt[a + b/x^2]*x^2),x]

[Out]

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

Rule 212

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\sqrt {b+a x^2} \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {a+\frac {b}{x^2}} x} \]

[In]

Integrate[1/(Sqrt[a + b/x^2]*x^2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86

method	result	size
default	\(-\frac {\sqrt {a \,x^{2}+b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right )}{\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {b}}\)	\(52\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.82 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=\left [\frac {\log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right )}{2 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{b}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=- \frac {\operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{\sqrt {b}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{2 \, \sqrt {b}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).

Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + \frac {\arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

-arctan(sqrt(b)/sqrt(-b))*sgn(x)/sqrt(-b) + arctan(sqrt(a*x^2 + b)/sqrt(-b))/(sqrt(-b)*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2} \, dx=-\frac {\ln \left (\sqrt {a+\frac {b}{x^2}}+\frac {\sqrt {b}}{x}\right )}{\sqrt {b}} \]

[In]

int(1/(x^2*(a + b/x^2)^(1/2)),x)

[Out]

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