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

Optimal. Leaf size=25 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0052679, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {217, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + c*x^2],x]

[Out]

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

Rule 217

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 206

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

Rubi steps

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

Mathematica [A]  time = 0.0060221, size = 25, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + c*x^2],x]

[Out]

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

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Maple [A]  time = 0.046, size = 21, normalized size = 0.8 \begin{align*}{\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.13793, size = 153, normalized size = 6.12 \begin{align*} \left [\frac{\log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right )}{2 \, \sqrt{c}}, -\frac{\sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.59242, size = 17, normalized size = 0.68 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.3434, size = 31, normalized size = 1.24 \begin{align*} -\frac{\log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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