Optimal. Leaf size=12 \[ -\frac {x e^{\text {sech}^{-1}(a x)}}{a} \]
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Rubi [B] time = 1.05, antiderivative size = 26, normalized size of antiderivative = 2.17, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6725, 260, 6341, 1956, 74} \[ -\frac {\sqrt {1-a x}}{a^2 \sqrt {\frac {1}{a x+1}}} \]
Warning: Unable to verify antiderivative.
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Rule 74
Rule 260
Rule 1956
Rule 6341
Rule 6725
Rubi steps
\begin {align*} \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx &=\int \left (\frac {x}{-1+a^2 x^2}-\frac {a e^{\text {sech}^{-1}(a x)} x^2}{-1+a^2 x^2}\right ) \, dx\\ &=-\left (a \int \frac {e^{\text {sech}^{-1}(a x)} x^2}{-1+a^2 x^2} \, dx\right )+\int \frac {x}{-1+a^2 x^2} \, dx\\ &=\frac {\log \left (1-a^2 x^2\right )}{2 a^2}+\int \frac {x \sqrt {\frac {1}{1+a x}}}{\sqrt {1-a x}} \, dx-\int \frac {x}{-1+a^2 x^2} \, dx\\ &=\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {x}{\sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=-\frac {\sqrt {1-a x}}{a^2 \sqrt {\frac {1}{1+a x}}}\\ \end {align*}
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Mathematica [B] time = 0.28, size = 28, normalized size = 2.33 \[ -\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 35, normalized size = 2.92 \[ -\frac {x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{a} \]
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 \[ \int -\frac {{\left (a x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} - 1\right )} x}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 3.00 \[ -\frac {x \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{a} \]
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 \[ -\int \frac {{\left (a x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} - 1\right )} x}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.96, size = 76, normalized size = 6.33 \[ \frac {\ln \left (\frac {1}{x}\right )}{a^2}-\frac {\ln \left (a+\frac {1}{x}\right )}{2\,a^2}-\frac {\ln \left (\frac {1}{x}-a\right )}{2\,a^2}+\frac {\ln \left (a^2\,x^2-1\right )}{2\,a^2}-\frac {x\,\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}}{a} \]
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 \[ - a \int \frac {x^{2} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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