Optimal. Leaf size=16 \[ -x+\frac {2 \log (1+a+b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6296, 45}
\begin {gather*} \frac {2 \log (a+b x+1)}{b}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6296
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a+b x)} \, dx &=\int \frac {1-a-b x}{1+a+b x} \, dx\\ &=\int \left (-1+\frac {2}{1+a+b x}\right ) \, dx\\ &=-x+\frac {2 \log (1+a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -x+\frac {2 \log (1+a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 17, normalized size = 1.06
method | result | size |
default | \(-x +\frac {2 \ln \left (b x +a +1\right )}{b}\) | \(17\) |
risch | \(-x +\frac {2 \ln \left (b x +a +1\right )}{b}\) | \(17\) |
norman | \(\frac {\frac {a^{2}+2 a +1}{b}-b \,x^{2}}{b x +a +1}+\frac {2 \ln \left (b x +a +1\right )}{b}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 16, normalized size = 1.00 \begin {gather*} -x + \frac {2 \, \log \left (b x + a + 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 18, normalized size = 1.12 \begin {gather*} -\frac {b x - 2 \, \log \left (b x + a + 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 12, normalized size = 0.75 \begin {gather*} - x + \frac {2 \log {\left (a + b x + 1 \right )}}{b} \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. 38 vs.
\(2 (16) = 32\).
time = 0.42, size = 38, normalized size = 2.38 \begin {gather*} -\frac {b x + a + 1}{b} - \frac {2 \, \log \left (\frac {{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2} {\left | b \right |}}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 16, normalized size = 1.00 \begin {gather*} \frac {2\,\ln \left (a+b\,x+1\right )}{b}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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