Optimal. Leaf size=65 \[ \frac {\sqrt {1-a^2 x^2}}{3 a c^3 (1-a x)}+\frac {\sqrt {1-a^2 x^2}}{3 a c^3 (1-a x)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 = {6127, 659, 651} \[ \frac {\sqrt {1-a^2 x^2}}{3 a c^3 (1-a x)}+\frac {\sqrt {1-a^2 x^2}}{3 a c^3 (1-a x)^2} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx &=\frac {\int \frac {1}{(c-a c x)^2 \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {\sqrt {1-a^2 x^2}}{3 a c^3 (1-a x)^2}+\frac {\int \frac {1}{(c-a c x) \sqrt {1-a^2 x^2}} \, dx}{3 c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{3 a c^3 (1-a x)^2}+\frac {\sqrt {1-a^2 x^2}}{3 a c^3 (1-a x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.52 \[ -\frac {(a x-2) \sqrt {a x+1}}{3 a c^3 (1-a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 62, normalized size = 0.95 \[ \frac {2 \, a^{2} x^{2} - 4 \, a x - \sqrt {-a^{2} x^{2} + 1} {\left (a x - 2\right )} + 2}{3 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 91, normalized size = 1.40 \[ -\frac {2 \, {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 2\right )}}{3 \, c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 0.51 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a x -2\right )}{3 \left (a x -1\right )^{2} c^{3} a} \]
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 \[ -\int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a c x - c\right )}^{3} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 32, normalized size = 0.49 \[ -\frac {\sqrt {1-a^2\,x^2}\,\left (a\,x-2\right )}{3\,a\,c^3\,{\left (a\,x-1\right )}^2} \]
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 \[ - \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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