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(x*b^(1/2)/Pi^(1/2))/b^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\pi }}\right )}{\sqrt {b}} \]

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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fricas [B]  time = 0.92, size = 59, normalized size = 3.11 \[ \left [\frac {\log \left (-\pi - 2 \, b x^{2} - 2 \, \sqrt {\pi + b x^{2}} \sqrt {b} x\right )}{2 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {\pi + b x^{2}}}\right )}{b}\right ] \]

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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giac [A]  time = 1.12, size = 22, normalized size = 1.16 \[ -\frac {\log \left (-\sqrt {b} x + \sqrt {\pi + b x^{2}}\right )}{\sqrt {b}} \]

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)

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maple [A]  time = 0.00, size = 21, normalized size = 1.11 \[ \frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+\pi }\right )}{\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 13, normalized size = 0.68 \[ \frac {\operatorname {arsinh}\left (\frac {b x}{\sqrt {\pi b}}\right )}{\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.12, size = 20, normalized size = 1.05 \[ \frac {\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+\Pi }\right )}{\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(Pi + b*x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.98, size = 17, normalized size = 0.89 \[ \frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {\pi }} \right )}}{\sqrt {b}} \]

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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