Optimal. Leaf size=28 \[ -\frac {1-a x}{a c^2 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 28, 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, 637} \[ -\frac {1-a x}{a c^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 637
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac {\int \frac {c-a c x}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {1-a x}{a c^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.96 \[ -\frac {\sqrt {1-a x}}{a c^2 \sqrt {a x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 35, normalized size = 1.25 \[ -\frac {a x + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{2} c^{2} x + a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.29, size = 69, normalized size = 2.46 \[ c^{2} {\left (\frac {i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}{a^{2} c^{4}} + \frac {\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}{a^{2} c^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 34, normalized size = 1.21 \[ \frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a x -1\right ) c^{2} a \left (a x +1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )}^{2} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 46, normalized size = 1.64 \[ \frac {\sqrt {1-a^2\,x^2}}{c^2\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\right )\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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