Optimal. Leaf size=55 \[ \frac {(2 a x+1) e^{-\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )}-\frac {2 e^{-\coth ^{-1}(a x)}}{3 a c^2} \]
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Rubi [A] time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ \frac {(2 a x+1) e^{-\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )}-\frac {2 e^{-\coth ^{-1}(a x)}}{3 a c^2} \]
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 = 48, normalized size = 0.87 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a^2 x^2+2 a x-1\right )}{3 (a x-1) (a c x+c)^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.64, size = 50, normalized size = 0.91 \[ -\frac {{\left (2 \, a^{2} x^{2} + 2 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 49, normalized size = 0.89 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (2 a^{2} x^{2}+2 a x -1\right )}{3 \left (a^{2} x^{2}-1\right ) a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 67, normalized size = 1.22 \[ \frac {1}{12} \, a {\left (\frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}} - \frac {3}{a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 55, normalized size = 1.00 \[ -\frac {\frac {6\,\left (a\,x-1\right )}{a\,x+1}-\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+3}{12\,a\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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 \[ \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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