Optimal. Leaf size=35 \[ \frac {1}{a c (a x+1)}+\frac {2 \log (a x+1)}{a c}-\frac {x}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, 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 = {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.
[In]
[Out]
Rule 43
Rule 6150
Rule 6157
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 39, normalized size = 1.11 \[ -\frac {a^{2} x^{2} + a x - 2 \, {\left (a x + 1\right )} \log \left (a x + 1\right ) - 1}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 55, normalized size = 1.57 \[ -\frac {a x + 1}{a c} - \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c} + \frac {1}{{\left (a x + 1\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 33, normalized size = 0.94 \[ -\frac {x}{c} + \frac {1}{a^{2} c x + a c} + \frac {2 \, \log \left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 33, normalized size = 0.94 \[ \frac {1}{a\,\left (c+a\,c\,x\right )}-\frac {x}{c}+\frac {2\,\ln \left (a\,x+1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 37, normalized size = 1.06 \[ - a^{2} \left (- \frac {1}{a^{4} c x + a^{3} c} + \frac {x}{a^{2} c} - \frac {2 \log {\left (a x + 1 \right )}}{a^{3} c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________