3.2 \(\int \frac{-2 \log (-\sqrt{-1+a x})+\log (-1+a x)}{2 \pi \sqrt{-1+a x}} \, dx\)

Optimal. Leaf size=15 \[ -\frac{2 \sqrt{1-a x}}{a} \]

[Out]

(-2*Sqrt[1 - a*x])/a

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Rubi [C]  time = 0.0558809, antiderivative size = 52, normalized size of antiderivative = 3.47, number of steps used = 5, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {12, 2295} \[ \frac{\sqrt{a x-1} \log (a x-1)}{\pi a}-\frac{2 \sqrt{a x-1} \log \left (-\sqrt{a x-1}\right )}{\pi a} \]

Antiderivative was successfully verified.

[In]

Int[(-2*Log[-Sqrt[-1 + a*x]] + Log[-1 + a*x])/(2*Pi*Sqrt[-1 + a*x]),x]

[Out]

(-2*Sqrt[-1 + a*x]*Log[-Sqrt[-1 + a*x]])/(a*Pi) + (Sqrt[-1 + a*x]*Log[-1 + a*x])/(a*Pi)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int \frac{-2 \log \left (-\sqrt{-1+a x}\right )+\log (-1+a x)}{2 \pi \sqrt{-1+a x}} \, dx &=\frac{\int \frac{-2 \log \left (-\sqrt{-1+a x}\right )+\log (-1+a x)}{\sqrt{-1+a x}} \, dx}{2 \pi }\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 \log (-x)+\log \left (x^2\right )\right ) \, dx,x,\sqrt{-1+a x}\right )}{a \pi }\\ &=\frac{\operatorname{Subst}\left (\int \log \left (x^2\right ) \, dx,x,\sqrt{-1+a x}\right )}{a \pi }-\frac{2 \operatorname{Subst}\left (\int \log (-x) \, dx,x,\sqrt{-1+a x}\right )}{a \pi }\\ &=-\frac{2 \sqrt{-1+a x} \log \left (-\sqrt{-1+a x}\right )}{a \pi }+\frac{\sqrt{-1+a x} \log (-1+a x)}{a \pi }\\ \end{align*}

Mathematica [C]  time = 0.025668, size = 37, normalized size = 2.47 \[ \frac{\sqrt{a x-1} \left (\log (a x-1)-2 \log \left (-\sqrt{a x-1}\right )\right )}{\pi a} \]

Antiderivative was successfully verified.

[In]

Integrate[(-2*Log[-Sqrt[-1 + a*x]] + Log[-1 + a*x])/(2*Pi*Sqrt[-1 + a*x]),x]

[Out]

(Sqrt[-1 + a*x]*(-2*Log[-Sqrt[-1 + a*x]] + Log[-1 + a*x]))/(a*Pi)

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Maple [C]  time = 0.003, size = 34, normalized size = 2.3 \begin{align*}{\frac{1}{a\pi }\sqrt{ax-1} \left ( \ln \left ( ax-1 \right ) -2\,\ln \left ( -\sqrt{ax-1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.947855, size = 55, normalized size = 3.67 \begin{align*} \frac{\sqrt{a x - 1} \log \left (a x - 1\right ) - 2 \, \sqrt{a x - 1} \log \left (-\sqrt{a x - 1}\right )}{\pi a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(a*x - 1)*log(a*x - 1) - 2*sqrt(a*x - 1)*log(-sqrt(a*x - 1)))/(pi*a)

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Fricas [A]  time = 2.14124, size = 4, normalized size = 0.27 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [A]  time = 57.8506, size = 42, normalized size = 2.8 \begin{align*} \frac{\begin{cases} \frac{- 2 \sqrt{a x - 1} \log{\left (- \sqrt{a x - 1} \right )} + \sqrt{a x - 1} \log{\left (a x - 1 \right )}}{a} & \text{for}\: a \neq 0 \\\pi x & \text{otherwise} \end{cases}}{\pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(ln(a*x-1)-2*ln(-(a*x-1)**(1/2)))/pi/(a*x-1)**(1/2),x)

[Out]

Piecewise(((-2*sqrt(a*x - 1)*log(-sqrt(a*x - 1)) + sqrt(a*x - 1)*log(a*x - 1))/a, Ne(a, 0)), (pi*x, True))/pi

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Giac [B]  time = 1.07061, size = 55, normalized size = 3.67 \begin{align*} \frac{\sqrt{a x - 1} \log \left (a x - 1\right ) - 2 \, \sqrt{a x - 1} \log \left (-\sqrt{a x - 1}\right )}{\pi a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(a*x - 1)*log(a*x - 1) - 2*sqrt(a*x - 1)*log(-sqrt(a*x - 1)))/(pi*a)