Optimal. Leaf size=35 \[ -\frac {1}{a c (a x+1)}-\frac {2 \log (a x+1)}{a c}+\frac {x}{c} \]
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Rubi [A] time = 0.16, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 43} \[ -\frac {1}{a c (a x+1)}-\frac {2 \log (a x+1)}{a c}+\frac {x}{c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rule 6157
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\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 = 0.04, size = 28, normalized size = 0.80 \[ \frac {-\frac {1}{a^2 x+a}-\frac {2 \log (a x+1)}{a}+x}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 38, normalized size = 1.09 \[ \frac {a^{2} x^{2} + a x - 2 \, {\left (a x + 1\right )} \log \left (a x + 1\right ) - 1}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 36, normalized size = 1.03 \[ \frac {x}{c} - \frac {2 \, \log \left ({\left | a x + 1 \right |}\right )}{a c} - \frac {1}{{\left (a x + 1\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 1.03 \[ \frac {x}{c}-\frac {1}{a c \left (a x +1\right )}-\frac {2 \ln \left (a x +1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 34, normalized size = 0.97 \[ \frac {x}{c} - \frac {1}{a^{2} c x + a c} - \frac {2 \, \log \left (a x + 1\right )}{a c} \]
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 \[ \frac {x}{c}-\frac {1}{a\,\left (c+a\,c\,x\right )}-\frac {2\,\ln \left (a\,x+1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 36, normalized size = 1.03 \[ a^{2} \left (- \frac {1}{a^{4} c x + a^{3} c} + \frac {x}{a^{2} c} - \frac {2 \log {\left (a x + 1 \right )}}{a^{3} c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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