Optimal. Leaf size=35 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}-\frac{1}{2 a b (a+b x)} \]
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Rubi [A] time = 0.0308144, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, 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{1}{(a+b x) \left (a^2-b^2 x^2\right )} \, dx &=\int \frac{1}{(a-b x) (a+b x)^2} \, 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 = 0.0114628, size = 47, normalized size = 1.34 \[ \frac{-(a+b x) \log (a-b x)+(a+b x) \log (a+b x)-2 a}{4 a^2 b (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 47, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{4\,b{a}^{2}}}-{\frac{1}{2\,ab \left ( bx+a \right ) }}-{\frac{\ln \left ( bx-a \right ) }{4\,b{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.34774, size = 63, normalized size = 1.8 \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 = 1.73136, size = 109, normalized size = 3.11 \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 = 0.395931, size = 39, normalized size = 1.11 \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 = 1.20726, size = 65, normalized size = 1.86 \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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