Optimal. Leaf size=67 \[ -\frac {1+a}{2 (1-a) x^2}-\frac {2 b}{(1-a)^2 x}+\frac {2 b^2 \log (x)}{(1-a)^3}-\frac {2 b^2 \log (1-a-b x)}{(1-a)^3} \]
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Rubi [A]
time = 0.04, 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 = {6298, 78}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6298
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 = 0.02, size = 54, normalized size = 0.81 \begin {gather*} \frac {(-1+a) \left (-1+a^2-4 b x\right )-4 b^2 x^2 \log (x)+4 b^2 x^2 \log (1-a-b x)}{2 (-1+a)^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 57, normalized size = 0.85
method | result | size |
default | \(-\frac {-1-a}{2 \left (-1+a \right ) x^{2}}-\frac {2 b}{\left (-1+a \right )^{2} x}-\frac {2 b^{2} \ln \left (x \right )}{\left (-1+a \right )^{3}}+\frac {2 b^{2} \ln \left (b x +a -1\right )}{\left (-1+a \right )^{3}}\) | \(57\) |
norman | \(\frac {\frac {1+a}{-2+2 a}-\frac {2 b x}{a^{2}-2 a +1}}{x^{2}}-\frac {2 b^{2} \ln \left (x \right )}{a^{3}-3 a^{2}+3 a -1}+\frac {2 b^{2} \ln \left (b x +a -1\right )}{a^{3}-3 a^{2}+3 a -1}\) | \(80\) |
risch | \(\frac {\frac {1+a}{-2+2 a}-\frac {2 b x}{a^{2}-2 a +1}}{x^{2}}+\frac {2 b^{2} \ln \left (-b x -a +1\right )}{a^{3}-3 a^{2}+3 a -1}-\frac {2 b^{2} \ln \left (x \right )}{a^{3}-3 a^{2}+3 a -1}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 74, normalized size = 1.10 \begin {gather*} \frac {2 \, b^{2} \log \left (b x + a - 1\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} - \frac {2 \, b^{2} \log \left (x\right )}{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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 65, normalized size = 0.97 \begin {gather*} \frac {4 \, b^{2} x^{2} \log \left (b x + a - 1\right ) - 4 \, b^{2} x^{2} \log \left (x\right ) + 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (56) = 112\).
time = 0.25, size = 209, normalized size = 3.12 \begin {gather*} - \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} \cdot \left (2 a^{2} - 4 a + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 91, normalized size = 1.36 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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