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

Optimal. Leaf size=24 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}} \]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[c])

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Rubi [A]  time = 0.0187577, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(-1),x]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[c])

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Rubi in Sympy [A]  time = 2.44701, size = 22, normalized size = 0.92 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.00783446, size = 24, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(-1),x]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[c])

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Maple [A]  time = 0.005, size = 16, normalized size = 0.7 \[{1\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221177, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right )}{2 \, \sqrt{-a c}}, \frac{\arctan \left (\frac{\sqrt{a c} x}{a}\right )}{\sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.289894, size = 53, normalized size = 2.21 \[ - \frac{\sqrt{- \frac{1}{a c}} \log{\left (- a \sqrt{- \frac{1}{a c}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a c}} \log{\left (a \sqrt{- \frac{1}{a c}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+a),x)

[Out]

-sqrt(-1/(a*c))*log(-a*sqrt(-1/(a*c)) + x)/2 + sqrt(-1/(a*c))*log(a*sqrt(-1/(a*c
)) + x)/2

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GIAC/XCAS [A]  time = 0.208717, size = 20, normalized size = 0.83 \[ \frac{\arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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