Optimal. Leaf size=77 \[ \frac {(a-b x) \log (x)}{a \sqrt {-a^2+2 a b x-b^2 x^2}}-\frac {(a-b x) \log (a-b x)}{a \sqrt {-a^2+2 a b x-b^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {660, 36, 29, 31}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-a^2+2 a b x-b^2 x^2}} \, dx &=\frac {\left (a b-b^2 x\right ) \int \frac {1}{x \left (a b-b^2 x\right )} \, dx}{\sqrt {-a^2+2 a b x-b^2 x^2}}\\ &=\frac {\left (a b-b^2 x\right ) \int \frac {1}{x} \, dx}{a b \sqrt {-a^2+2 a b x-b^2 x^2}}+\frac {\left (b \left (a b-b^2 x\right )\right ) \int \frac {1}{a b-b^2 x} \, dx}{a \sqrt {-a^2+2 a b x-b^2 x^2}}\\ &=\frac {(a-b x) \log (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}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.47 \begin {gather*} \frac {(a-b x) (\log (x)-\log (a-b x))}{a \sqrt {-(a-b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.77, size = 36, normalized size = 0.47
method | result | size |
default | \(-\frac {\left (-b x +a \right ) \left (\ln \left (-b x +a \right )-\ln \left (x \right )\right )}{\sqrt {-\left (-b x +a \right )^{2}}\, a}\) | \(36\) |
risch | \(-\frac {\left (-b x +a \right ) \ln \left (b x -a \right )}{\sqrt {-\left (-b x +a \right )^{2}}\, a}+\frac {\left (-b x +a \right ) \ln \left (-x \right )}{\sqrt {-\left (-b x +a \right )^{2}}\, a}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.29, size = 38, normalized size = 0.49 \begin {gather*} -\frac {i \, \left (-1\right )^{-2 \, a b x + 2 \, a^{2}} \log \left (-\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 4.51, size = 70, normalized size = 0.91 \begin {gather*} -\sqrt {-\frac {1}{a^{2}}} \log \left (\frac {i \, a^{2} \sqrt {-\frac {1}{a^{2}}} + 2 \, b x - a}{2 \, b}\right ) + \sqrt {-\frac {1}{a^{2}}} \log \left (\frac {-i \, a^{2} \sqrt {-\frac {1}{a^{2}}} + 2 \, b x - a}{2 \, b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- \left (- a + b x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.36, size = 41, normalized size = 0.53 \begin {gather*} \frac {i \, \log \left ({\left | b x - a \right |}\right )}{a \mathrm {sgn}\left (-b x + a\right )} - \frac {i \, \log \left ({\left | x \right |}\right )}{a \mathrm {sgn}\left (-b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 54, normalized size = 0.70 \begin {gather*} -\frac {\ln \left (a\,b-\frac {a^2}{x}+\frac {\sqrt {-a^2}\,\sqrt {-a^2+2\,a\,b\,x-b^2\,x^2}}{x}\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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