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 = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 {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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 1.27, size = 54, normalized size = 1.50 \[ \frac {{\left (b x - a\right )} \log \left (b x + a\right ) - {\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \, {\left (a^{2} b^{2} x - a^{3} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 50, normalized size = 1.39 \[ \frac {\log \left ({\left | b x + a \right |}\right )}{4 \, a^{2} b} - \frac {\log \left ({\left | b x - a \right |}\right )}{4 \, a^{2} b} - \frac {1}{2 \, {\left (b x - a\right )} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 49, normalized size = 1.36 \[ -\frac {1}{2 \left (b x -a \right ) a b}-\frac {\ln \left (b x -a \right )}{4 a^{2} b}+\frac {\ln \left (b x +a \right )}{4 a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 48, normalized size = 1.33 \[ -\frac {1}{2 \, {\left (a b^{2} x - a^{2} b\right )}} + \frac {\log \left (b x + a\right )}{4 \, a^{2} b} - \frac {\log \left (b x - a\right )}{4 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 32, normalized size = 0.89 \[ \frac {1}{2\,a\,b\,\left (a-b\,x\right )}+\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{2\,a^2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 37, normalized size = 1.03 \[ - \frac {1}{- 2 a^{2} b + 2 a b^{2} x} + \frac {- \frac {\log {\left (- \frac {a}{b} + x \right )}}{4} + \frac {\log {\left (\frac {a}{b} + x \right )}}{4}}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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