Optimal. Leaf size=55 \[ \frac {a x+1}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x}{3 a c^2 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6148, 778, 191} \[ \frac {a x+1}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x}{3 a c^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 778
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {1+a x}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a c^2}\\ &=\frac {1+a x}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x}{3 a c^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.80 \[ \frac {-a^2 x^2+a x-1}{3 a^2 c^2 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 89, normalized size = 1.62 \[ \frac {a^{3} x^{3} - a^{2} x^{2} - a x + {\left (a^{2} x^{2} - a x + 1\right )} \sqrt {-a^{2} x^{2} + 1} + 1}{3 \, {\left (a^{5} c^{2} x^{3} - a^{4} c^{2} x^{2} - a^{3} c^{2} x + a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} x}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 41, normalized size = 0.75 \[ -\frac {a^{2} x^{2}-a x +1}{3 \left (a x -1\right ) c^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2}} \]
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 \[ a \int \frac {x^{2}}{{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} + \frac {1}{3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 63, normalized size = 1.15 \[ \frac {1}{3\,a^2\,c^2\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {x}{3\,a\,c^2\,\sqrt {1-a^2\,x^2}}+\frac {x}{3\,a\,c^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \]
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 \[ \frac {\int \frac {x}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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