Optimal. Leaf size=32 \[ -\frac {\tanh ^{-1}\left (\frac {a-b x}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {618, 206} \[ -\frac {\tanh ^{-1}\left (\frac {a-b x}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{b+2 a x-b x^2} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 a-2 b x\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {a-b x}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 1.28 \[ -\frac {\tan ^{-1}\left (\frac {b x-a}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 67, normalized size = 2.09 \[ \frac {\log \left (\frac {b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} + b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b x - a\right )}}{b x^{2} - 2 \, a x - b}\right )}{2 \, \sqrt {a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 55, normalized size = 1.72 \[ -\frac {\log \left (\frac {{\left | 2 \, b x - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b x - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, \sqrt {a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 31, normalized size = 0.97 \[ \frac {\arctanh \left (\frac {2 b x -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 49, normalized size = 1.53 \[ -\frac {\log \left (\frac {b x - a - \sqrt {a^{2} + b^{2}}}{b x - a + \sqrt {a^{2} + b^{2}}}\right )}{2 \, \sqrt {a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 28, normalized size = 0.88 \[ -\frac {\mathrm {atanh}\left (\frac {a-b\,x}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.25, size = 102, normalized size = 3.19 \[ - \frac {\sqrt {\frac {1}{a^{2} + b^{2}}} \log {\left (x + \frac {- a^{2} \sqrt {\frac {1}{a^{2} + b^{2}}} - a - b^{2} \sqrt {\frac {1}{a^{2} + b^{2}}}}{b} \right )}}{2} + \frac {\sqrt {\frac {1}{a^{2} + b^{2}}} \log {\left (x + \frac {a^{2} \sqrt {\frac {1}{a^{2} + b^{2}}} - a + b^{2} \sqrt {\frac {1}{a^{2} + b^{2}}}}{b} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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