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 = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 77
Rule 6163
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 1.18, size = 39, normalized size = 0.81 \[ -\frac {2 \, b x \log \left (b x + a - 1\right ) - 2 \, b x \log \relax (x) - a^{2} + 1}{{\left (a^{2} - 2 \, a + 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 57, normalized size = 1.19 \[ -\frac {2 \, b^{2} \log \left ({\left | b x + a - 1 \right |}\right )}{a^{2} b - 2 \, a b + b} + \frac {2 \, b \log \left ({\left | x \right |}\right )}{a^{2} - 2 \, a + 1} + \frac {a^{2} - 1}{{\left (a - 1\right )}^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 46, normalized size = 0.96 \[ \frac {1}{\left (a -1\right ) x}+\frac {a}{\left (a -1\right ) x}+\frac {2 b \ln \relax (x )}{\left (a -1\right )^{2}}-\frac {2 b \ln \left (b x +a -1\right )}{\left (a -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 48, normalized size = 1.00 \[ -\frac {2 \, b \log \left (b x + a - 1\right )}{a^{2} - 2 \, a + 1} + \frac {2 \, b \log \relax (x)}{a^{2} - 2 \, a + 1} + \frac {a + 1}{{\left (a - 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 47, normalized size = 0.98 \[ \frac {a+1}{x\,\left (a-1\right )}-\frac {4\,b\,\mathrm {atanh}\left (\frac {2\,b\,x+\frac {a^2-2\,a+1}{a-1}}{a-1}\right )}{{\left (a-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.35, size = 144, normalized size = 3.00 \[ \frac {2 b \log {\left (x + \frac {- \frac {2 a^{3} b}{\left (a - 1\right )^{2}} + \frac {6 a^{2} b}{\left (a - 1\right )^{2}} + 2 a b - \frac {6 a b}{\left (a - 1\right )^{2}} - 2 b + \frac {2 b}{\left (a - 1\right )^{2}}}{4 b^{2}} \right )}}{\left (a - 1\right )^{2}} - \frac {2 b \log {\left (x + \frac {\frac {2 a^{3} b}{\left (a - 1\right )^{2}} - \frac {6 a^{2} b}{\left (a - 1\right )^{2}} + 2 a b + \frac {6 a b}{\left (a - 1\right )^{2}} - 2 b - \frac {2 b}{\left (a - 1\right )^{2}}}{4 b^{2}} \right )}}{\left (a - 1\right )^{2}} - \frac {- a - 1}{x \left (a - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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