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

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

[Out]

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

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Rubi [C] Result contains higher order function than in optimal. Order 3 vs. order 2 in optimal.
time = 0.04, 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, 2332} \begin {gather*} \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} \end {gather*}

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 2332

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

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Mathematica [C] Result contains higher order function than in optimal. Order 3 vs. order 2 in optimal.
time = 0.02, size = 37, normalized size = 2.47 \begin {gather*} \frac {\sqrt {-1+a x} \left (-2 \log \left (-\sqrt {-1+a x}\right )+\log (-1+a x)\right )}{a \pi } \end {gather*}

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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: } \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception:

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.10, size = 42, normalized size = 2.80

method	result	size
gosper	\(\frac {\sqrt {a x -1}\, \left (\ln \left (a x -1\right )-2 \ln \left (-\sqrt {a x -1}\right )\right )}{a \pi }\)	\(34\)
derivativedivides	\(\frac {-2 \ln \left (-\sqrt {a x -1}\right ) \sqrt {a x -1}+\sqrt {a x -1}\, \ln \left (a x -1\right )}{\pi a}\)	\(42\)
default	\(\frac {-2 \ln \left (-\sqrt {a x -1}\right ) \sqrt {a x -1}+\sqrt {a x -1}\, \ln \left (a x -1\right )}{\pi a}\)	\(42\)
meijerg	\(\frac {i \sqrt {-\mathrm {signum}\left (a x -1\right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a x +1}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (a x -1\right )}\, a}\)	\(47\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (13) = 26\).
time = 0.24, size = 41, normalized size = 2.73 \begin {gather*} \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 {gather*}

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 = 0.30, size = 1, normalized size = 0.07 \begin {gather*} 0 \end {gather*}

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 = 30.50, size = 42, normalized size = 2.80 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (13) = 26\).
time = 0.01, size = 93, normalized size = 6.20 \begin {gather*} \frac {-2 \left (-\sqrt {a x-1} \ln \left (a x-1\right )+\frac {2 \sqrt {a x-1} \sqrt {a x-1}}{\sqrt {a x-1}}\right )+4 \left (-\sqrt {a x-1} \ln \left (-\sqrt {a x-1}\right )+\sqrt {a x-1}\right )}{2 \pi a} \end {gather*}

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)

[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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Mupad [B]
time = 0.51, size = 43, normalized size = 2.87 \begin {gather*} -\frac {2\,\ln \left (-\sqrt {a\,x-1}\right )\,\sqrt {a\,x-1}-\ln \left (a\,x-1\right )\,\sqrt {a\,x-1}}{\Pi \,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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