Optimal. Leaf size=20 \[ -\frac {x}{c}+\frac {\log (1+a x)}{a c} \]
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Rubi [A]
time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6266, 6264, 45}
\begin {gather*} \frac {\log (a x+1)}{a c}-\frac {x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6264
Rule 6266
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\frac {a \int \frac {e^{-2 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac {a \int \frac {x}{1+a x} \, dx}{c}\\ &=-\frac {a \int \left (\frac {1}{a}-\frac {1}{a (1+a x)}\right ) \, dx}{c}\\ &=-\frac {x}{c}+\frac {\log (1+a x)}{a c}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} \frac {-a x+\log (1+a x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 23, normalized size = 1.15
method | result | size |
risch | \(-\frac {x}{c}+\frac {\ln \left (a x +1\right )}{a c}\) | \(21\) |
default | \(\frac {a \left (-\frac {x}{a}+\frac {\ln \left (a x +1\right )}{a^{2}}\right )}{c}\) | \(23\) |
norman | \(\frac {-\frac {x}{c}-\frac {a \,x^{2}}{c}}{a x +1}+\frac {\ln \left (a x +1\right )}{a c}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 20, normalized size = 1.00 \begin {gather*} -\frac {x}{c} + \frac {\log \left (a x + 1\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 20, normalized size = 1.00 \begin {gather*} -\frac {a x - \log \left (a x + 1\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.95 \begin {gather*} - a \left (\frac {x}{a c} - \frac {\log {\left (a x + 1 \right )}}{a^{2} c}\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. 41 vs.
\(2 (20) = 40\).
time = 0.41, size = 41, normalized size = 2.05 \begin {gather*} -\frac {a x + 1}{a c} - \frac {\log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 18, normalized size = 0.90 \begin {gather*} \frac {\ln \left (a\,x+1\right )-a\,x}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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