Optimal. Leaf size=32 \[ \frac {\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 32, 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} \[ \frac {\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 651
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a c^2 (1-a x)^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 29, normalized size = 0.91 \[ \frac {(a x+1)^{5/2}}{5 a c^2 (1-a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 89, normalized size = 2.78 \[ \frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-a^{2} x^{2} + 1} - 1}{5 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.31, size = 83, normalized size = 2.59 \[ -\frac {-i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c) + \frac {{\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}}{5 \, c^{2} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 35, normalized size = 1.09 \[ -\frac {\left (a x +1\right )^{4}}{5 \left (a x -1\right ) c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 146, normalized size = 4.56 \[ \frac {8}{5 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{2} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a c^{2}\right )}} + \frac {12}{5 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a c^{2}\right )}} + \frac {x}{5 \, \sqrt {-a^{2} x^{2} + 1} c^{2}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 34, normalized size = 1.06 \[ -\frac {\sqrt {1-a^2\,x^2}\,{\left (a\,x+1\right )}^2}{5\,a\,c^2\,{\left (a\,x-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {3 a x}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________