3.274 \(\int \frac{1}{a+\sqrt{-a} x} \, dx\)

Optimal. Leaf size=20 \[ \frac{\log \left (\sqrt{-a} x+a\right )}{\sqrt{-a}} \]

[Out]

Log[a + Sqrt[-a]*x]/Sqrt[-a]

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Rubi [A]  time = 0.0179456, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\log \left (\sqrt{-a} x+a\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]  Int[(a + Sqrt[-a]*x)^(-1),x]

[Out]

Log[a + Sqrt[-a]*x]/Sqrt[-a]

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Rubi in Sympy [A]  time = 1.67653, size = 17, normalized size = 0.85 \[ \frac{\log{\left (a + x \sqrt{- a} \right )}}{\sqrt{- a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + x*sqrt(-a))/sqrt(-a)

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Mathematica [A]  time = 0.00722106, size = 20, normalized size = 1. \[ \frac{\log \left (\sqrt{-a} x+a\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + Sqrt[-a]*x)^(-1),x]

[Out]

Log[a + Sqrt[-a]*x]/Sqrt[-a]

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Maple [A]  time = 0.002, size = 17, normalized size = 0.9 \[{1\ln \left ( a+x\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+x*(-a)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.34102, size = 22, normalized size = 1.1 \[ \frac{\log \left (\sqrt{-a} x + a\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(-a)*x + a)/sqrt(-a)

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Fricas [A]  time = 0.218946, size = 22, normalized size = 1.1 \[ \frac{\log \left (\sqrt{-a} x + a\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(-a)*x + a)/sqrt(-a)

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Sympy [A]  time = 0.089785, size = 17, normalized size = 0.85 \[ \frac{\log{\left (a + x \sqrt{- a} \right )}}{\sqrt{- a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+x*(-a)**(1/2)),x)

[Out]

log(a + x*sqrt(-a))/sqrt(-a)

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GIAC/XCAS [A]  time = 0.203512, size = 23, normalized size = 1.15 \[ \frac{{\rm ln}\left ({\left | \sqrt{-a} x + a \right |}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(sqrt(-a)*x + a))/sqrt(-a)