Optimal. Leaf size=29 \[ \frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6127, 651} \[ \frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 651
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac {\int \frac {1}{(c-a c x) \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 26, normalized size = 0.90 \[ \frac {\sqrt {a x+1}}{a c^2 \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 37, normalized size = 1.28 \[ \frac {a x - \sqrt {-a^{2} x^{2} + 1} - 1}{a^{2} c^{2} x - a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.24, size = 70, normalized size = 2.41 \[ -c^{2} {\left (\frac {\sqrt {-\frac {2 \, c}{a c x - c} - 1} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}{a^{2} c^{4}} - \frac {i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}{a^{2} c^{4}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 28, normalized size = 0.97 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{\left (a x -1\right ) a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a c x - c\right )}^{2} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 47, normalized size = 1.62 \[ \frac {\sqrt {1-a^2\,x^2}}{c^2\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________