Optimal. Leaf size=15 \[ -x+\frac {2 \log (1+a x)}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6260, 45}
\begin {gather*} \frac {2 \log (a x+1)}{a}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6260
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \, dx &=\int \frac {1-a x}{1+a x} \, dx\\ &=\int \left (-1+\frac {2}{1+a x}\right ) \, dx\\ &=-x+\frac {2 \log (1+a x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -x+\frac {2 \log (1+a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 16, normalized size = 1.07
method | result | size |
default | \(-x +\frac {2 \ln \left (a x +1\right )}{a}\) | \(16\) |
risch | \(-x +\frac {2 \ln \left (a x +1\right )}{a}\) | \(16\) |
norman | \(\frac {-a \,x^{2}-x}{a x +1}+\frac {2 \ln \left (a x +1\right )}{a}\) | \(31\) |
meijerg | \(-\frac {\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )}{a}+\frac {x}{a x +1}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 15, normalized size = 1.00 \begin {gather*} -x + \frac {2 \, \log \left (a x + 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 17, normalized size = 1.13 \begin {gather*} -\frac {a x - 2 \, \log \left (a x + 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 10, normalized size = 0.67 \begin {gather*} - x + \frac {2 \log {\left (a x + 1 \right )}}{a} \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. 64 vs.
\(2 (15) = 30\).
time = 0.41, size = 64, normalized size = 4.27 \begin {gather*} -a^{2} {\left (\frac {a x + 1}{a^{3}} + \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a^{3}} - \frac {1}{{\left (a x + 1\right )} a^{3}}\right )} - \frac {1}{{\left (a x + 1\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 15, normalized size = 1.00 \begin {gather*} \frac {2\,\ln \left (a\,x+1\right )}{a}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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