Optimal. Leaf size=13 \[ \frac {x}{c (1-a x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6140, 34} \[ \frac {x}{c (1-a x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 34
Rule 6140
Rubi steps
\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac {\int \frac {1+a x}{(1-a x)^3} \, dx}{c}\\ &=\frac {x}{c (1-a x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 25, normalized size = 1.92 \[ \frac {(a x+1)^2}{4 a c (1-a x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 19, normalized size = 1.46 \[ \frac {x}{a^{2} c x^{2} - 2 \, a c x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 12, normalized size = 0.92 \[ \frac {x}{{\left (a x - 1\right )}^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 28, normalized size = 2.15 \[ \frac {\frac {1}{a \left (a x -1\right )}+\frac {1}{a \left (a x -1\right )^{2}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 19, normalized size = 1.46 \[ \frac {x}{a^{2} c x^{2} - 2 \, a c x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.92, size = 12, normalized size = 0.92 \[ \frac {x}{c\,{\left (a\,x-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.24, size = 17, normalized size = 1.31 \[ \frac {x}{a^{2} c x^{2} - 2 a c x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________