\(\int \frac {1}{\sqrt {a+b x^2}} \, dx\) [77]

Optimal result

Integrand size = 11, antiderivative size = 25 \[ \int \frac {1}{\sqrt {a+b x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {223, 212} \[ \int \frac {1}{\sqrt {a+b x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}} \]

[In]

Int[1/Sqrt[a + b*x^2],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+b x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}} \]

[In]

Integrate[1/Sqrt[a + b*x^2],x]

[Out]

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

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\)	\(21\)
pseudoelliptic	\(\frac {\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{\sqrt {b}}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\sqrt {a+b x^2}} \, dx=\left [\frac {\log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{2 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{b}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {a+b x^2}} \, dx=\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{\sqrt {b}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt {a+b x^2}} \, dx=\frac {\operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt {a+b x^2}} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} x - \frac {a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, \sqrt {b}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt {a+b x^2}} \, dx=\frac {\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{\sqrt {b}} \]

[In]

int(1/(a + b*x^2)^(1/2),x)

[Out]

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