Optimal. Leaf size=71 \[ -\frac {\log \left (1-c^2 x^2\right )}{2 c}+\frac {\log (x)}{c}-\frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {c x+1}\right )}{c} \]
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Rubi [A] time = 0.12, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6339, 1956, 92, 208, 266, 36, 29, 31} \[ -\frac {\log \left (1-c^2 x^2\right )}{2 c}+\frac {\log (x)}{c}-\frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {c x+1}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 92
Rule 208
Rule 266
Rule 1956
Rule 6339
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx &=\frac {\int \frac {\sqrt {\frac {1}{1+c x}}}{x \sqrt {1-c x}} \, dx}{c}+\frac {\int \frac {1}{x \left (1-c^2 x^2\right )} \, dx}{c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 c}+\frac {\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx}{c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c}+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )-\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{c-c x^2} \, dx,x,\sqrt {1-c x} \sqrt {1+c x}\right )\\ &=-\frac {\sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c x} \sqrt {1+c x}\right )}{c}+\frac {\log (x)}{c}-\frac {\log \left (1-c^2 x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 1.03 \[ -\frac {\log \left (1-c^2 x^2\right )}{2 c}+\frac {2 \log (x)}{c}-\frac {\log \left (c x \sqrt {\frac {1-c x}{c x+1}}+\sqrt {\frac {1-c x}{c x+1}}+1\right )}{c} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.91, size = 92, normalized size = 1.30 \[ -\frac {\log \left (c^{2} x^{2} - 1\right ) + \log \left (c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} + 1\right ) - \log \left (c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} - 1\right ) - 2 \, \log \relax (x)}{2 \, c} \]
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 \[ \int -\frac {\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}}{c^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 87, normalized size = 1.23 \[ -\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{\sqrt {-c^{2} x^{2}+1}}+\frac {\ln \relax (x )}{c}-\frac {\ln \left (c x +1\right )}{2 c}-\frac {\ln \left (c x -1\right )}{2 c} \]
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 \[ \frac {\log \relax (x)}{c} - \frac {\log \left (c x + 1\right )}{2 \, c} - \frac {\log \left (c x - 1\right )}{2 \, c} - \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c^{3} x^{3} - c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 59, normalized size = 0.83 \[ \frac {\ln \relax (x)}{c}-\frac {4\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{c\,x}+1}-1}\right )}{c}-\frac {\ln \left (3\,c^2\,x^2-3\right )}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.43, size = 48, normalized size = 0.68 \[ - \frac {\log {\left (-1 + \frac {1}{c x} \right )}}{2 c} - \frac {\log {\left (\sqrt {1 + \frac {1}{c x}} \right )}}{c} - \frac {2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {1 + \frac {1}{c x}}}{2} \right )}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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