Optimal. Leaf size=19 \[ \frac {x}{c^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 19, 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, 191} \[ \frac {x}{c^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx &=\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {x}{c^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ \frac {x}{c^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 33, normalized size = 1.74 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2} c^{3} x^{2} - c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 29, normalized size = 1.53 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x}{{\left (a^{2} x^{2} - 1\right )} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 32, normalized size = 1.68 \[ \frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}{\left (a x -1\right )^{2} c^{3} \left (a x +1\right )^{2}} \]
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 \[ -\int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )}^{3} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 17, normalized size = 0.89 \[ \frac {x}{c^3\,\sqrt {1-a^2\,x^2}} \]
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 \[ - \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\right )\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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