Optimal. Leaf size=35 \[ -\frac {x}{c}+\frac {1}{a c (1+a x)}+\frac {2 \log (1+a x)}{a c} \]
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Rubi [A]
time = 0.09, antiderivative size = 35, 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 = {6292, 6285, 45}
\begin {gather*} \frac {1}{a c (a x+1)}+\frac {2 \log (a x+1)}{a c}-\frac {x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rule 6292
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {a^2 \int \frac {e^{-2 \tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \int \frac {x^2}{(1+a x)^2} \, dx}{c}\\ &=-\frac {a^2 \int \left (\frac {1}{a^2}+\frac {1}{a^2 (1+a x)^2}-\frac {2}{a^2 (1+a x)}\right ) \, dx}{c}\\ &=-\frac {x}{c}+\frac {1}{a c (1+a x)}+\frac {2 \log (1+a x)}{a c}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.80 \begin {gather*} \frac {-x+\frac {1}{a+a^2 x}+\frac {2 \log (1+a x)}{a}}{c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 37, normalized size = 1.06
method | result | size |
risch | \(-\frac {x}{c}+\frac {1}{\left (a x +1\right ) a c}+\frac {2 \ln \left (a x +1\right )}{a c}\) | \(36\) |
default | \(\frac {a^{2} \left (-\frac {x}{a^{2}}+\frac {2 \ln \left (a x +1\right )}{a^{3}}+\frac {1}{a^{3} \left (a x +1\right )}\right )}{c}\) | \(37\) |
norman | \(\frac {-\frac {3 a \,x^{2}}{c}-\frac {a^{2} x^{3}}{c}-\frac {2 x}{c}}{\left (a x +1\right )^{2}}+\frac {2 \ln \left (a x +1\right )}{a c}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 33, normalized size = 0.94 \begin {gather*} -\frac {x}{c} + \frac {1}{a^{2} c x + a c} + \frac {2 \, \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 = 0.40, size = 39, normalized size = 1.11 \begin {gather*} -\frac {a^{2} x^{2} + a x - 2 \, {\left (a x + 1\right )} \log \left (a x + 1\right ) - 1}{a^{2} c x + a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 37, normalized size = 1.06 \begin {gather*} - a^{2} \left (- \frac {1}{a^{4} c x + a^{3} c} + \frac {x}{a^{2} c} - \frac {2 \log {\left (a x + 1 \right )}}{a^{3} c}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 55, normalized size = 1.57 \begin {gather*} -\frac {a x + 1}{a c} - \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c} + \frac {1}{{\left (a x + 1\right )} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 33, normalized size = 0.94 \begin {gather*} \frac {1}{a\,\left (c+a\,c\,x\right )}-\frac {x}{c}+\frac {2\,\ln \left (a\,x+1\right )}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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