Optimal. Leaf size=42 \[ -\frac {\sqrt {1-c x}}{c x \sqrt {\frac {1}{c x+1}}}-\frac {1}{c x}+\tanh ^{-1}(c x) \]
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Rubi [A] time = 0.14, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6341, 1956, 95, 325, 206} \[ -\frac {\sqrt {1-c x}}{c x \sqrt {\frac {1}{c x+1}}}-\frac {1}{c x}+\tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 95
Rule 206
Rule 325
Rule 1956
Rule 6341
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(c x)}}{x \left (1-c^2 x^2\right )} \, dx &=\frac {\int \frac {\sqrt {\frac {1}{1+c x}}}{x^2 \sqrt {1-c x}} \, dx}{c}+\frac {\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx}{c}\\ &=-\frac {1}{c x}+c \int \frac {1}{1-c^2 x^2} \, dx+\frac {\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \sqrt {1-c x} \sqrt {1+c x}} \, dx}{c}\\ &=-\frac {1}{c x}-\frac {\sqrt {1-c x}}{c x \sqrt {\frac {1}{1+c x}}}+\tanh ^{-1}(c x)\\ \end {align*}
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Mathematica [A] time = 0.17, size = 59, normalized size = 1.40 \[ -\sqrt {\frac {1-c x}{c x+1}} \left (\frac {1}{c x}+1\right )-\frac {1}{c x}-\frac {1}{2} \log (1-c x)+\frac {1}{2} \log (c x+1) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.92, size = 62, normalized size = 1.48 \[ -\frac {2 \, c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} - c x \log \left (c x + 1\right ) + c x \log \left (c x - 1\right ) + 2}{2 \, c x} \]
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 \[ \int -\frac {\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}}{{\left (c^{2} x^{2} - 1\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 61, normalized size = 1.45 \[ -\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \mathrm {csgn}\relax (c )^{2}-\frac {1}{c x}+\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2} \]
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 \[ \frac {-\frac {1}{x}}{c} - \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c^{3} x^{4} - c x^{2}}\,{d x} + \frac {1}{2} \, \log \left (c x + 1\right ) - \frac {1}{2} \, \log \left (c x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.53, size = 37, normalized size = 0.88 \[ \mathrm {atanh}\left (c\,x\right )-\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}-\frac {1}{c\,x} \]
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 \[ - \frac {\int \frac {c x \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{4} - x^{2}}\, dx + \int \frac {1}{c^{2} x^{4} - x^{2}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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