Optimal. Leaf size=42 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b c^2}+\frac{1}{2 a b c^2 (a-b x)} \]
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Rubi [A] time = 0.0292954, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {44, 208} \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b c^2}+\frac{1}{2 a b c^2 (a-b x)} \]
Antiderivative was successfully verified.
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Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) (a c-b c x)^2} \, dx &=\int \left (\frac{1}{2 a c^2 (a-b x)^2}+\frac{1}{2 a c^2 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{2 a b c^2 (a-b x)}+\frac{\int \frac{1}{a^2-b^2 x^2} \, dx}{2 a c^2}\\ &=\frac{1}{2 a b c^2 (a-b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b c^2}\\ \end{align*}
Mathematica [A] time = 0.0152711, size = 53, normalized size = 1.26 \[ \frac{(b x-a) \log (a-b x)+(a-b x) \log (a+b x)+2 a}{4 a^2 b c^2 (a-b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 58, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{4\,{c}^{2}{a}^{2}b}}-{\frac{\ln \left ( bx-a \right ) }{4\,{c}^{2}{a}^{2}b}}-{\frac{1}{2\,{c}^{2}ba \left ( bx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06501, size = 81, normalized size = 1.93 \begin{align*} -\frac{1}{2 \,{\left (a b^{2} c^{2} x - a^{2} b c^{2}\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{2} b c^{2}} - \frac{\log \left (b x - a\right )}{4 \, a^{2} b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53512, size = 120, normalized size = 2.86 \begin{align*} \frac{{\left (b x - a\right )} \log \left (b x + a\right ) -{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \,{\left (a^{2} b^{2} c^{2} x - a^{3} b c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.468911, size = 48, normalized size = 1.14 \begin{align*} - \frac{1}{- 2 a^{2} b c^{2} + 2 a b^{2} c^{2} x} + \frac{- \frac{\log{\left (- \frac{a}{b} + x \right )}}{4} + \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{2} b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07984, size = 72, normalized size = 1.71 \begin{align*} -\frac{1}{2 \,{\left (b c x - a c\right )} a b c} + \frac{\log \left ({\left | -\frac{2 \, a c}{b c x - a c} - 1 \right |}\right )}{4 \, a^{2} b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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