Optimal. Leaf size=52 \[ \frac {2 x}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {a x+1}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6138, 639, 191} \[ \frac {2 x}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {a x+1}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 639
Rule 6138
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1+a x}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {1+a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac {1+a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x}{3 c^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 45, normalized size = 0.87 \[ \frac {2 a^2 x^2-2 a x-1}{3 a c^2 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 89, normalized size = 1.71 \[ \frac {a^{3} x^{3} - a^{2} x^{2} - a x - {\left (2 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} + 1}{3 \, {\left (a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 42, normalized size = 0.81 \[ \frac {2 a^{2} x^{2}-2 a x -1}{3 \left (a x -1\right ) c^{2} \sqrt {-a^{2} x^{2}+1}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 48, normalized size = 0.92 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (-2\,a^2\,x^2+2\,a\,x+1\right )}{3\,a\,c^2\,{\left (a\,x-1\right )}^2\,\left (a\,x+1\right )} \]
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 \[ \frac {\int \frac {a x}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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