Optimal. Leaf size=21 \[ \frac {c}{a^2 x}-\frac {2 c \log (x)}{a}+c (-x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6157, 6150, 43} \[ \frac {c}{a^2 x}-\frac {2 c \log (x)}{a}+c (-x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\frac {c \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \frac {(1+a x)^2}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \left (a^2+\frac {1}{x^2}+\frac {2 a}{x}\right ) \, dx}{a^2}\\ &=\frac {c}{a^2 x}-c x-\frac {2 c \log (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {c}{a^2 x}-\frac {2 c \log (x)}{a}+c (-x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 27, normalized size = 1.29 \[ -\frac {a^{2} c x^{2} + 2 \, a c x \log \relax (x) - c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 22, normalized size = 1.05 \[ -c x - \frac {2 \, c \log \left ({\left | x \right |}\right )}{a} + \frac {c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 22, normalized size = 1.05 \[ \frac {c}{a^{2} x}-c x -\frac {2 c \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 21, normalized size = 1.00 \[ -c x - \frac {2 \, c \log \relax (x)}{a} + \frac {c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.84, size = 24, normalized size = 1.14 \[ -\frac {c\,\left (a^2\,x^2+2\,a\,x\,\ln \relax (x)-1\right )}{a^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 20, normalized size = 0.95 \[ \frac {- a^{2} c x - 2 a c \log {\relax (x )} + \frac {c}{x}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________