Optimal. Leaf size=67 \[ \frac {2 b^2 \log (x)}{(1-a)^3}-\frac {2 b^2 \log (-a-b x+1)}{(1-a)^3}-\frac {2 b}{(1-a)^2 x}-\frac {a+1}{2 (1-a) x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 67, 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} \[ \frac {2 b^2 \log (x)}{(1-a)^3}-\frac {2 b^2 \log (-a-b x+1)}{(1-a)^3}-\frac {2 b}{(1-a)^2 x}-\frac {a+1}{2 (1-a) x^2} \]
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^3} \, dx &=\int \frac {1+a+b x}{x^3 (1-a-b x)} \, dx\\ &=\int \left (\frac {-1-a}{(-1+a) x^3}+\frac {2 b}{(-1+a)^2 x^2}-\frac {2 b^2}{(-1+a)^3 x}+\frac {2 b^3}{(-1+a)^3 (-1+a+b x)}\right ) \, dx\\ &=-\frac {1+a}{2 (1-a) x^2}-\frac {2 b}{(1-a)^2 x}+\frac {2 b^2 \log (x)}{(1-a)^3}-\frac {2 b^2 \log (1-a-b x)}{(1-a)^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 0.81 \[ \frac {(a-1) \left (a^2-4 b x-1\right )+4 b^2 x^2 \log (-a-b x+1)-4 b^2 x^2 \log (x)}{2 (a-1)^3 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 65, normalized size = 0.97 \[ \frac {4 \, b^{2} x^{2} \log \left (b x + a - 1\right ) - 4 \, b^{2} x^{2} \log \relax (x) + a^{3} - 4 \, {\left (a - 1\right )} b x - a^{2} - a + 1}{2 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 91, normalized size = 1.36 \[ \frac {2 \, b^{3} \log \left ({\left | b x + a - 1 \right |}\right )}{a^{3} b - 3 \, a^{2} b + 3 \, a b - b} - \frac {2 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} + \frac {a^{3} - a^{2} - 4 \, {\left (a b - b\right )} x - a + 1}{2 \, {\left (a - 1\right )}^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 63, normalized size = 0.94 \[ \frac {1}{2 \left (a -1\right ) x^{2}}+\frac {a}{2 \left (a -1\right ) x^{2}}-\frac {2 b}{\left (a -1\right )^{2} x}-\frac {2 b^{2} \ln \relax (x )}{\left (a -1\right )^{3}}+\frac {2 b^{2} \ln \left (b x +a -1\right )}{\left (a -1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 74, normalized size = 1.10 \[ \frac {2 \, b^{2} \log \left (b x + a - 1\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} - \frac {2 \, b^{2} \log \relax (x)}{a^{3} - 3 \, a^{2} + 3 \, a - 1} + \frac {a^{2} - 4 \, b x - 1}{2 \, {\left (a^{2} - 2 \, a + 1\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 66, normalized size = 0.99 \[ \frac {\frac {a+1}{2\,\left (a-1\right )}-\frac {2\,b\,x}{{\left (a-1\right )}^2}}{x^2}+\frac {4\,b^2\,\mathrm {atanh}\left (\frac {a^3-3\,a^2+3\,a-1}{{\left (a-1\right )}^3}+\frac {2\,b\,x}{a-1}\right )}{{\left (a-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.45, size = 209, normalized size = 3.12 \[ - \frac {2 b^{2} \log {\left (x + \frac {- \frac {2 a^{4} b^{2}}{\left (a - 1\right )^{3}} + \frac {8 a^{3} b^{2}}{\left (a - 1\right )^{3}} - \frac {12 a^{2} b^{2}}{\left (a - 1\right )^{3}} + 2 a b^{2} + \frac {8 a b^{2}}{\left (a - 1\right )^{3}} - 2 b^{2} - \frac {2 b^{2}}{\left (a - 1\right )^{3}}}{4 b^{3}} \right )}}{\left (a - 1\right )^{3}} + \frac {2 b^{2} \log {\left (x + \frac {\frac {2 a^{4} b^{2}}{\left (a - 1\right )^{3}} - \frac {8 a^{3} b^{2}}{\left (a - 1\right )^{3}} + \frac {12 a^{2} b^{2}}{\left (a - 1\right )^{3}} + 2 a b^{2} - \frac {8 a b^{2}}{\left (a - 1\right )^{3}} - 2 b^{2} + \frac {2 b^{2}}{\left (a - 1\right )^{3}}}{4 b^{3}} \right )}}{\left (a - 1\right )^{3}} - \frac {- a^{2} + 4 b x + 1}{x^{2} \left (2 a^{2} - 4 a + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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