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 = 0.0074811, 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, Rules used = {31} \[ \frac{\log \left (\sqrt{-a} x+a\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]

Int[(a + Sqrt[-a]*x)^(-1),x]

[Out]

Log[a + Sqrt[-a]*x]/Sqrt[-a]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{a+\sqrt{-a} x} \, dx &=\frac{\log \left (a+\sqrt{-a} x\right )}{\sqrt{-a}}\\ \end{align*}

Mathematica [A]  time = 0.0061207, 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 = 0.001, size = 17, normalized size = 0.9 \begin{align*}{\ln \left ( a+x\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}}} \end{align*}

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 = 1.06086, size = 22, normalized size = 1.1 \begin{align*} \frac{\log \left (\sqrt{-a} x + a\right )}{\sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+x*(-a)^(1/2)),x, algorithm="maxima")

[Out]

log(sqrt(-a)*x + a)/sqrt(-a)

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Fricas [A]  time = 1.42402, size = 42, normalized size = 2.1 \begin{align*} -\frac{\sqrt{-a} \log \left (x - \sqrt{-a}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+x*(-a)^(1/2)),x, algorithm="fricas")

[Out]

-sqrt(-a)*log(x - sqrt(-a))/a

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Sympy [A]  time = 0.070181, size = 17, normalized size = 0.85 \begin{align*} \frac{\log{\left (a + x \sqrt{- a} \right )}}{\sqrt{- a}} \end{align*}

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 [A]  time = 1.17921, size = 23, normalized size = 1.15 \begin{align*} \frac{\log \left ({\left | \sqrt{-a} x + a \right |}\right )}{\sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+x*(-a)^(1/2)),x, algorithm="giac")

[Out]

log(abs(sqrt(-a)*x + a))/sqrt(-a)