Optimal. Leaf size=41 \[ -\frac {1-a}{(1+a) x}-\frac {2 b \log (x)}{(1+a)^2}+\frac {2 b \log (1+a+b x)}{(1+a)^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 41, 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 \log (x)}{(a+1)^2}+\frac {2 b \log (a+b x+1)}{(a+1)^2}-\frac {1-a}{(a+1) x} \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^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 = 0.03, size = 31, normalized size = 0.76 \begin {gather*} \frac {-1+a^2-2 b x \log (x)+2 b x \log (1+a+b x)}{(1+a)^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 42, normalized size = 1.02
method | result | size |
default | \(\frac {2 b \ln \left (b x +a +1\right )}{\left (1+a \right )^{2}}-\frac {1-a}{\left (1+a \right ) x}-\frac {2 b \ln \left (x \right )}{\left (1+a \right )^{2}}\) | \(42\) |
risch | \(-\frac {1}{\left (1+a \right ) x}+\frac {a}{\left (1+a \right ) x}-\frac {2 b \ln \left (x \right )}{a^{2}+2 a +1}+\frac {2 b \ln \left (-b x -a -1\right )}{a^{2}+2 a +1}\) | \(60\) |
norman | \(\frac {\frac {\left (a \,b^{2}-b^{2}\right ) x}{b \left (1+a \right )}-1+a}{\left (b x +a +1\right ) x}-\frac {2 b \ln \left (x \right )}{a^{2}+2 a +1}+\frac {2 b \ln \left (b x +a +1\right )}{a^{2}+2 a +1}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 48, normalized size = 1.17 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 36, normalized size = 0.88 \begin {gather*} \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 {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. 143 vs.
\(2 (36) = 72\).
time = 0.21, size = 143, normalized size = 3.49 \begin {gather*} - \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 {1 - a}{x \left (a + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (38) = 76\).
time = 0.45, size = 80, normalized size = 1.95 \begin {gather*} -\frac {2 \, b^{2} \log \left ({\left | -\frac {a}{b x + a + 1} - \frac {1}{b x + a + 1} + 1 \right |}\right )}{a^{2} b + 2 \, a b + b} - \frac {a b - b}{{\left (a + 1\right )}^{2} {\left (\frac {a}{b x + a + 1} + \frac {1}{b x + a + 1} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.92, size = 47, normalized size = 1.15 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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