3.6.80 \(\int \frac {1}{\sqrt {\pi +b x^2}} \, dx\) [580]

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 = 0.00, 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 = {221} \begin {gather*} \frac {\sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\pi }}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Pi + b*x^2],x]

[Out]

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

Rule 221

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 = 0.02, size = 28, normalized size = 1.47 \begin {gather*} -\frac {\log \left (-\sqrt {b} x+\sqrt {\pi +b x^2}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Pi + b*x^2],x]

[Out]

-(Log[-(Sqrt[b]*x) + Sqrt[Pi + b*x^2]]/Sqrt[b])

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Maple [A]
time = 0.05, size = 21, normalized size = 1.11

method	result	size
meijerg	\(\frac {\arcsinh \left (\frac {x \sqrt {b}}{\sqrt {\pi }}\right )}{\sqrt {b}}\)	\(14\)
default	\(\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+\pi }\right )}{\sqrt {b}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 13, normalized size = 0.68 \begin {gather*} \frac {\operatorname {arsinh}\left (\frac {b x}{\sqrt {\pi b}}\right )}{\sqrt {b}} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (13) = 26\).
time = 0.85, size = 59, normalized size = 3.11 \begin {gather*} \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 {gather*}

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 = 0.42, size = 17, normalized size = 0.89 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {\pi }} \right )}}{\sqrt {b}} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (13) = 26\).
time = 0.61, size = 36, normalized size = 1.89 \begin {gather*} \frac {1}{2} \, \sqrt {\pi + b x^{2}} x - \frac {\pi \log \left (-\sqrt {b} x + \sqrt {\pi + b x^{2}}\right )}{2 \, \sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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