3.580 \(\int \frac{1}{\sqrt{\pi +b x^2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{\sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]

[Out]

ArcSinh[(Sqrt[b]*x)/Sqrt[Pi]]/Sqrt[b]

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Rubi [A]  time = 0.005851, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {215} \[ \frac{\sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Pi + b*x^2],x]

[Out]

ArcSinh[(Sqrt[b]*x)/Sqrt[Pi]]/Sqrt[b]

Rule 215

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{\pi +b x^2}} \, dx &=\frac{\sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0075118, size = 19, normalized size = 1. \[ \frac{\sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Pi + b*x^2],x]

[Out]

ArcSinh[(Sqrt[b]*x)/Sqrt[Pi]]/Sqrt[b]

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Maple [A]  time = 0.005, size = 21, normalized size = 1.1 \begin{align*}{\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+\pi } \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+Pi)^(1/2),x)

[Out]

ln(x*b^(1/2)+(b*x^2+Pi)^(1/2))/b^(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/(b*x^2+pi)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.35324, size = 157, normalized size = 8.26 \begin{align*} \left [\frac{\log \left (-\pi - 2 \, b x^{2} - 2 \, \sqrt{\pi + b x^{2}} \sqrt{b} x\right )}{2 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{\pi + b x^{2}}}\right )}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.93452, size = 17, normalized size = 0.89 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{\pi }} \right )}}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 3.00574, size = 30, normalized size = 1.58 \begin{align*} -\frac{\log \left (-\sqrt{b} x + \sqrt{\pi + b x^{2}}\right )}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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