Optimal. Leaf size=48 \[ \frac{2 b \log (x)}{(1-a)^2}-\frac{2 b \log (-a-b x+1)}{(1-a)^2}-\frac{a+1}{(1-a) x} \]
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Rubi [A] time = 0.0408594, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6163, 77} \[ \frac{2 b \log (x)}{(1-a)^2}-\frac{2 b \log (-a-b x+1)}{(1-a)^2}-\frac{a+1}{(1-a) x} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac{1+a+b x}{x^2 (1-a-b x)} \, dx\\ &=\int \left (\frac{-1-a}{(-1+a) x^2}+\frac{2 b}{(-1+a)^2 x}-\frac{2 b^2}{(-1+a)^2 (-1+a+b x)}\right ) \, dx\\ &=-\frac{1+a}{(1-a) x}+\frac{2 b \log (x)}{(1-a)^2}-\frac{2 b \log (1-a-b x)}{(1-a)^2}\\ \end{align*}
Mathematica [A] time = 0.0221775, size = 34, normalized size = 0.71 \[ \frac{a^2-2 b x \log (-a-b x+1)+2 b x \log (x)-1}{(a-1)^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 46, normalized size = 1. \begin{align*}{\frac{1}{ \left ( a-1 \right ) x}}+{\frac{a}{ \left ( a-1 \right ) x}}+2\,{\frac{b\ln \left ( x \right ) }{ \left ( a-1 \right ) ^{2}}}-2\,{\frac{b\ln \left ( bx+a-1 \right ) }{ \left ( a-1 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961006, size = 65, normalized size = 1.35 \begin{align*} -\frac{2 \, b \log \left (b x + a - 1\right )}{a^{2} - 2 \, a + 1} + \frac{2 \, b \log \left (x\right )}{a^{2} - 2 \, a + 1} + \frac{a + 1}{{\left (a - 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83429, size = 97, normalized size = 2.02 \begin{align*} -\frac{2 \, b x \log \left (b x + a - 1\right ) - 2 \, b x \log \left (x\right ) - a^{2} + 1}{{\left (a^{2} - 2 \, a + 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.538894, size = 143, normalized size = 2.98 \begin{align*} \frac{2 b \log{\left (x + \frac{- \frac{2 a^{3} b}{\left (a - 1\right )^{2}} + \frac{6 a^{2} b}{\left (a - 1\right )^{2}} + 2 a b - \frac{6 a b}{\left (a - 1\right )^{2}} - 2 b + \frac{2 b}{\left (a - 1\right )^{2}}}{4 b^{2}} \right )}}{\left (a - 1\right )^{2}} - \frac{2 b \log{\left (x + \frac{\frac{2 a^{3} b}{\left (a - 1\right )^{2}} - \frac{6 a^{2} b}{\left (a - 1\right )^{2}} + 2 a b + \frac{6 a b}{\left (a - 1\right )^{2}} - 2 b - \frac{2 b}{\left (a - 1\right )^{2}}}{4 b^{2}} \right )}}{\left (a - 1\right )^{2}} + \frac{a + 1}{x \left (a - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16605, size = 77, normalized size = 1.6 \begin{align*} -\frac{2 \, b^{2} \log \left ({\left | b x + a - 1 \right |}\right )}{a^{2} b - 2 \, a b + b} + \frac{2 \, b \log \left ({\left | x \right |}\right )}{a^{2} - 2 \, a + 1} + \frac{a^{2} - 1}{{\left (a - 1\right )}^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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