3.275 \(\int \frac{1}{a^2+\sqrt{-a} x} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log \left (a^2+\sqrt{-a} x\right )}{\sqrt{-a}} \]

[Out]

Log[a^2 + Sqrt[-a]*x]/Sqrt[-a]

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Rubi [A]  time = 0.0023876, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {31} \[ \frac{\log \left (a^2+\sqrt{-a} x\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + Sqrt[-a]*x)^(-1),x]

[Out]

Log[a^2 + 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^2+\sqrt{-a} x} \, dx &=\frac{\log \left (a^2+\sqrt{-a} x\right )}{\sqrt{-a}}\\ \end{align*}

Mathematica [A]  time = 0.0036597, size = 22, normalized size = 1. \[ \frac{\log \left (a^2+\sqrt{-a} x\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + Sqrt[-a]*x)^(-1),x]

[Out]

Log[a^2 + Sqrt[-a]*x]/Sqrt[-a]

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Maple [A]  time = 0.001, size = 19, normalized size = 0.9 \begin{align*}{\ln \left ({a}^{2}+x\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2+x*(-a)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.02064, size = 24, normalized size = 1.09 \begin{align*} \frac{\log \left (a^{2} + \sqrt{-a} x\right )}{\sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a^2 + sqrt(-a)*x)/sqrt(-a)

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Fricas [A]  time = 1.47102, size = 46, normalized size = 2.09 \begin{align*} -\frac{\sqrt{-a} \log \left (-\sqrt{-a} a + x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a)*log(-sqrt(-a)*a + x)/a

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Sympy [A]  time = 0.09407, size = 19, normalized size = 0.86 \begin{align*} \frac{\log{\left (a^{2} + x \sqrt{- a} \right )}}{\sqrt{- a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a**2 + x*sqrt(-a))/sqrt(-a)

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Giac [A]  time = 1.20031, size = 26, normalized size = 1.18 \begin{align*} \frac{\log \left ({\left | a^{2} + \sqrt{-a} x \right |}\right )}{\sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a^2 + sqrt(-a)*x))/sqrt(-a)