Optimal. Leaf size=14 \[ -\frac{x}{c^2 (1-a x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0510269, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 34} \[ -\frac{x}{c^2 (1-a x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6129
Rule 34
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\\ &=-\frac{\int \frac{1+a x}{(1-a x)^3} \, dx}{c^2}\\ &=-\frac{x}{c^2 (1-a x)^2}\\ \end{align*}
Mathematica [A] time = 0.0081405, size = 25, normalized size = 1.79 \[ -\frac{(a x+1)^2}{4 a c^2 (1-a x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.047, size = 30, normalized size = 2.1 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{1}{a \left ( ax-1 \right ) ^{2}}}-{\frac{1}{a \left ( ax-1 \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00542, size = 35, normalized size = 2.5 \begin{align*} -\frac{x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.35504, size = 49, normalized size = 3.5 \begin{align*} -\frac{x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.370919, size = 24, normalized size = 1.71 \begin{align*} - \frac{x}{a^{2} c^{2} x^{2} - 2 a c^{2} x + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.11755, size = 46, normalized size = 3.29 \begin{align*} -\frac{1}{{\left (a c x - c\right )}^{2} a} - \frac{1}{{\left (a c x - c\right )} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]