Optimal. Leaf size=65 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^2 c^3 (1-a x)^4}-\frac {4 \left (1-a^2 x^2\right )^{3/2}}{15 a^2 c^3 (1-a x)^3} \]
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Rubi [A] time = 0.08, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6128, 793, 651} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^2 c^3 (1-a x)^4}-\frac {4 \left (1-a^2 x^2\right )^{3/2}}{15 a^2 c^3 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 651
Rule 793
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x}{(c-a c x)^3} \, dx &=c \int \frac {x \sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^2 c^3 (1-a x)^4}-\frac {4 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx}{5 a}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^2 c^3 (1-a x)^4}-\frac {4 \left (1-a^2 x^2\right )^{3/2}}{15 a^2 c^3 (1-a x)^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.54 \[ \frac {(a x+1)^{3/2} (4 a x-1)}{15 a^2 c^3 (1-a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.62, size = 91, normalized size = 1.40 \[ -\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x + {\left (4 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} - 1}{15 \, {\left (a^{5} c^{3} x^{3} - 3 \, a^{4} c^{3} x^{2} + 3 \, a^{3} c^{3} x - a^{2} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 121, normalized size = 1.86 \[ \frac {2 \, {\left (\frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} + \frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - 1\right )}}{15 \, a c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 41, normalized size = 0.63 \[ \frac {\left (4 a x -1\right ) \left (a x +1\right )^{2}}{15 \left (a x -1\right )^{2} c^{3} \sqrt {-a^{2} x^{2}+1}\, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 132, normalized size = 2.03 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{5} c^{3} x^{3} - 3 \, a^{4} c^{3} x^{2} + 3 \, a^{3} c^{3} x - a^{2} c^{3}\right )}} - \frac {11 \, \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{4} c^{3} x^{2} - 2 \, a^{3} c^{3} x + a^{2} c^{3}\right )}} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{3} c^{3} x - a^{2} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 143, normalized size = 2.20 \[ -\frac {360\,a^6\,c^3\,\sqrt {1-a^2\,x^2}-600\,a^6\,c^3\,{\left (1-a^2\,x^2\right )}^{3/2}+225\,a^6\,c^3\,{\left (1-a^2\,x^2\right )}^{5/2}+360\,a^7\,c^3\,x\,\sqrt {1-a^2\,x^2}-420\,a^7\,c^3\,x\,{\left (1-a^2\,x^2\right )}^{3/2}+60\,a^7\,c^3\,x\,{\left (1-a^2\,x^2\right )}^{5/2}}{225\,a^8\,c^6\,{\left (a^2\,x^2-1\right )}^3} \]
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 \[ - \frac {\int \frac {x}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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