Optimal. Leaf size=45 \[ -\frac {\sqrt {1-c x}}{c^3 \sqrt {\frac {1}{c x+1}}}-\frac {\log \left (1-c^2 x^2\right )}{2 c^3} \]
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Rubi [A] time = 0.13, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6341, 1956, 74, 260} \[ -\frac {\log \left (1-c^2 x^2\right )}{2 c^3}-\frac {\sqrt {1-c x}}{c^3 \sqrt {\frac {1}{c x+1}}} \]
Antiderivative was successfully verified.
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Rule 74
Rule 260
Rule 1956
Rule 6341
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(c x)} x^2}{1-c^2 x^2} \, dx &=\frac {\int \frac {x \sqrt {\frac {1}{1+c x}}}{\sqrt {1-c x}} \, dx}{c}+\frac {\int \frac {x}{1-c^2 x^2} \, dx}{c}\\ &=-\frac {\log \left (1-c^2 x^2\right )}{2 c^3}+\frac {\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{c}\\ &=-\frac {\sqrt {1-c x}}{c^3 \sqrt {\frac {1}{1+c x}}}-\frac {\log \left (1-c^2 x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 44, normalized size = 0.98 \[ -\frac {\log \left (1-c^2 x^2\right )+2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 49, normalized size = 1.09 \[ -\frac {2 \, c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} + \log \left (c^{2} x^{2} - 1\right )}{2 \, c^{3}} \]
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 \[ \int -\frac {x^{2} {\left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 52, normalized size = 1.16 \[ -\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}}{c^{2}}-\frac {\ln \left (c^{2} x^{2}-1\right )}{2 c^{3}} \]
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 \[ -\frac {\log \left (c x + 1\right )}{2 \, c^{3}} - \frac {\log \left (c x - 1\right )}{2 \, c^{3}} - \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x}{c^{3} x^{2} - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.70, size = 44, normalized size = 0.98 \[ -\frac {\ln \left (c^2\,x^2-1\right )}{2\,c^3}-\frac {x\,\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}}{c^2} \]
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 \[ - \frac {\int \frac {x}{c^{2} x^{2} - 1}\, dx + \int \frac {c x^{2} \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{2} - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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