Optimal. Leaf size=36 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}+\frac{1}{2 a b (a-b x)} \]
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Rubi [A] time = 0.0261313, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {627, 44, 208} \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}+\frac{1}{2 a b (a-b x)} \]
Antiderivative was successfully verified.
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Rule 627
Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a-b x)^2 (a+b x)} \, dx\\ &=\int \left (\frac{1}{2 a (a-b x)^2}+\frac{1}{2 a \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{2 a b (a-b x)}+\frac{\int \frac{1}{a^2-b^2 x^2} \, dx}{2 a}\\ &=\frac{1}{2 a b (a-b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}\\ \end{align*}
Mathematica [A] time = 0.0099478, size = 50, normalized size = 1.39 \[ \frac{(b x-a) \log (a-b x)+(a-b x) \log (a+b x)+2 a}{4 a^2 b (a-b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 49, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{4\,b{a}^{2}}}-{\frac{\ln \left ( bx-a \right ) }{4\,b{a}^{2}}}-{\frac{1}{2\,ab \left ( bx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06567, size = 65, normalized size = 1.81 \begin{align*} -\frac{1}{2 \,{\left (a b^{2} x - a^{2} b\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{2} b} - \frac{\log \left (b x - a\right )}{4 \, a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8425, size = 109, normalized size = 3.03 \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} x - a^{3} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.391543, size = 37, normalized size = 1.03 \begin{align*} - \frac{1}{- 2 a^{2} b + 2 a b^{2} x} + \frac{- \frac{\log{\left (- \frac{a}{b} + x \right )}}{4} + \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23356, size = 68, normalized size = 1.89 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{4 \, a^{2} b} - \frac{\log \left ({\left | b x - a \right |}\right )}{4 \, a^{2} b} - \frac{1}{2 \,{\left (b x - a\right )} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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