3.2414 \(\int \frac{1}{x \sqrt{-a^2+2 a b x-b^2 x^2}} \, dx\)

Optimal. Leaf size=77 \[ \frac{\log (x) (a-b x)}{a \sqrt{-a^2+2 a b x-b^2 x^2}}-\frac{(a-b x) \log (a-b x)}{a \sqrt{-a^2+2 a b x-b^2 x^2}} \]

[Out]

((a - b*x)*Log[x])/(a*Sqrt[-a^2 + 2*a*b*x - b^2*x^2]) - ((a - b*x)*Log[a - b*x])
/(a*Sqrt[-a^2 + 2*a*b*x - b^2*x^2])

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Rubi [A]  time = 0.0847769, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\log (x) (a-b x)}{a \sqrt{-a^2+2 a b x-b^2 x^2}}-\frac{(a-b x) \log (a-b x)}{a \sqrt{-a^2+2 a b x-b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[-a^2 + 2*a*b*x - b^2*x^2]),x]

[Out]

((a - b*x)*Log[x])/(a*Sqrt[-a^2 + 2*a*b*x - b^2*x^2]) - ((a - b*x)*Log[a - b*x])
/(a*Sqrt[-a^2 + 2*a*b*x - b^2*x^2])

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Rubi in Sympy [A]  time = 17.2565, size = 63, normalized size = 0.82 \[ - \frac{\sqrt{- a^{2} + 2 a b x - b^{2} x^{2}} \log{\left (x \right )}}{a \left (a - b x\right )} + \frac{\sqrt{- a^{2} + 2 a b x - b^{2} x^{2}} \log{\left (a - b x \right )}}{a \left (a - b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-a**2 + 2*a*b*x - b**2*x**2)*log(x)/(a*(a - b*x)) + sqrt(-a**2 + 2*a*b*x -
b**2*x**2)*log(a - b*x)/(a*(a - b*x))

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Mathematica [A]  time = 0.0227044, size = 36, normalized size = 0.47 \[ \frac{(a-b x) (\log (x)-\log (a-b x))}{a \sqrt{-(a-b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[-a^2 + 2*a*b*x - b^2*x^2]),x]

[Out]

((a - b*x)*(Log[x] - Log[a - b*x]))/(a*Sqrt[-(a - b*x)^2])

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Maple [A]  time = 0.005, size = 38, normalized size = 0.5 \[{\frac{ \left ( bx-a \right ) \left ( \ln \left ( bx-a \right ) -\ln \left ( x \right ) \right ) }{a}{\frac{1}{\sqrt{- \left ( bx-a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-(b*x-a)^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-(b*x - a)^2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221138, size = 92, normalized size = 1.19 \[ \sqrt{-\frac{1}{a^{2}}} \log \left (-\frac{i \,{\left (a^{2} \sqrt{-\frac{1}{a^{2}}} + 2 i \, b x - i \, a\right )}}{2 \, b}\right ) - \sqrt{-\frac{1}{a^{2}}} \log \left (\frac{i \,{\left (a^{2} \sqrt{-\frac{1}{a^{2}}} - 2 i \, b x + i \, a\right )}}{2 \, b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-(b*x - a)^2)*x),x, algorithm="fricas")

[Out]

sqrt(-1/a^2)*log(-1/2*I*(a^2*sqrt(-1/a^2) + 2*I*b*x - I*a)/b) - sqrt(-1/a^2)*log
(1/2*I*(a^2*sqrt(-1/a^2) - 2*I*b*x + I*a)/b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{- \left (- a + b x\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(-(-a + b*x)**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-(b*x - a)^2)*x),x, algorithm="giac")

[Out]

undef