Optimal. Leaf size=17 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b c} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {35, 208} \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b c} \]
Antiderivative was successfully verified.
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Rule 35
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) (a c-b c x)} \, dx &=\int \frac {1}{a^2 c-b^2 c x^2} \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 28, normalized size = 1.65 \[ \frac {\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.20, size = 39, normalized size = 2.29 \[ \frac {\log \left ({\left | b x + a \right |}\right )}{2 \, a b c} - \frac {\log \left ({\left | b x - a \right |}\right )}{2 \, a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 38, normalized size = 2.24 \[ -\frac {\ln \left (b x -a \right )}{2 a b c}+\frac {\ln \left (b x +a \right )}{2 a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 37, normalized size = 2.18 \[ \frac {\log \left (b x + a\right )}{2 \, a b c} - \frac {\log \left (b x - a\right )}{2 \, a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 17, normalized size = 1.00 \[ \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{a\,b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 22, normalized size = 1.29 \[ - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{2} - \frac {\log {\left (\frac {a}{b} + x \right )}}{2}}{a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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