Optimal. Leaf size=32 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6127, 651} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 651
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.91 \[ \frac {(a x+1)^{3/2}}{3 a c^2 (1-a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.42, size = 60, normalized size = 1.88 \[ \frac {a^{2} x^{2} - 2 \, a x + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 1\right )} + 1}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.35, size = 66, normalized size = 2.06 \[ -\frac {i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c) + \frac {{\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}}{3 \, c^{2} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 35, normalized size = 1.09 \[ -\frac {\left (a x +1\right )^{2}}{3 \left (a x -1\right ) c^{2} \sqrt {-a^{2} x^{2}+1}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 73, normalized size = 2.28 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 32, normalized size = 1.00 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (a\,x+1\right )}{3\,a\,c^2\,{\left (a\,x-1\right )}^2} \]
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 \[ \frac {\int \frac {a x}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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