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 = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 44
Rule 208
Rule 627
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.84, size = 49, normalized size = 1.40 \[ \frac {{\left (b x + a\right )} \log \left (b x + a\right ) - {\left (b x + a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \, {\left (a^{2} b^{2} x + a^{3} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 48, normalized size = 1.37 \[ \frac {\log \left ({\left | b x + a \right |}\right )}{4 \, a^{2} b} - \frac {\log \left ({\left | b x - a \right |}\right )}{4 \, a^{2} b} - \frac {1}{2 \, {\left (b x + a\right )} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 47, normalized size = 1.34 \[ -\frac {1}{2 \left (b x +a \right ) a b}-\frac {\ln \left (b x -a \right )}{4 a^{2} b}+\frac {\ln \left (b x +a \right )}{4 a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 47, normalized size = 1.34 \[ -\frac {1}{2 \, {\left (a b^{2} x + a^{2} b\right )}} + \frac {\log \left (b x + a\right )}{4 \, a^{2} b} - \frac {\log \left (b x - a\right )}{4 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 31, normalized size = 0.89 \[ \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{2\,a^2\,b}-\frac {1}{2\,a\,b\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 39, normalized size = 1.11 \[ - \frac {1}{2 a^{2} b + 2 a b^{2} x} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{4} - \frac {\log {\left (\frac {a}{b} + x \right )}}{4}}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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