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 = 0.014828, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}} \]

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]

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Rubi in Sympy [A]  time = 2.43196, size = 22, normalized size = 0.88 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+a)**(1/2),x)

[Out]

atanh(sqrt(c)*x/sqrt(a + c*x**2))/sqrt(c)

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Mathematica [A]  time = 0.010187, 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 = 0.003, size = 21, normalized size = 0.8 \[{1\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226916, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{2 \, \sqrt{c}}, \frac{\arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{\sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.57301, size = 17, normalized size = 0.68 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{c}} \]

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/XCAS [A]  time = 0.215241, size = 31, normalized size = 1.24 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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