Optimal. Leaf size=51 \[ \frac {2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac {(1-2 a x) e^{\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6185, 6183} \[ \frac {2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac {(1-2 a x) e^{\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6183
Rule 6185
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac {e^{\coth ^{-1}(a x)} (1-2 a x)}{3 a c^2 \left (1-a^2 x^2\right )}+\frac {2 \int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{3 c}\\ &=\frac {2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac {e^{\coth ^{-1}(a x)} (1-2 a x)}{3 a c^2 \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 50, normalized size = 0.98 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^2 x^2-2 a x-1\right )}{3 c^2 (a x-1)^2 (a x+1)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 58, normalized size = 1.14 \[ \frac {{\left (2 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 70, normalized size = 1.37 \[ \frac {\frac {{\left (a x + 1\right )} {\left (\frac {6 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{{\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}} + 3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{12 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 49, normalized size = 0.96 \[ \frac {2 a^{2} x^{2}-2 a x -1}{3 \left (a^{2} x^{2}-1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 65, normalized size = 1.27 \[ \frac {1}{12} \, a {\left (\frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac {\frac {6 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 50, normalized size = 0.98 \[ \frac {-2\,a^2\,x^2+2\,a\,x+1}{\left (3\,a\,c^2-3\,a^3\,c^2\,x^2\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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 \[ \frac {\int \frac {1}{a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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