Integrand size = 19, antiderivative size = 71 \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx=-\frac {\sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c x} \sqrt {1+c x}\right )}{c}+\frac {\log (x)}{c}-\frac {\log \left (1-c^2 x^2\right )}{2 c} \]
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Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6474, 1972, 94, 214, 272, 36, 29, 31} \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx=-\frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {arctanh}\left (\sqrt {1-c x} \sqrt {c x+1}\right )}{c}-\frac {\log \left (1-c^2 x^2\right )}{2 c}+\frac {\log (x)}{c} \]
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Rule 29
Rule 31
Rule 36
Rule 94
Rule 214
Rule 272
Rule 1972
Rule 6474
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c}+\frac {1}{2} c \text {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 ) \text {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} \text {arctanh}\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*}
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx=\frac {2 \log (x)}{c}-\frac {\log \left (1-c^2 x^2\right )}{2 c}-\frac {\log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{c} \]
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Time = 0.70 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.15
method | result | size |
default | \(-\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{\sqrt {-c^{2} x^{2}+1}}+\frac {-\frac {\ln \left (c x +1\right )}{2}+\ln \left (x \right )-\frac {\ln \left (c x -1\right )}{2}}{c}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (45) = 90\).
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.30 \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx=-\frac {\log \left (c^{2} x^{2} - 1\right ) + \log \left (c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} + 1\right ) - \log \left (c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} - 1\right ) - 2 \, \log \left (x\right )}{2 \, c} \]
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\[ \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx=- \frac {\int \frac {c x \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{3} - x}\, dx + \int \frac {1}{c^{2} x^{3} - x}\, dx}{c} \]
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\[ \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx=\int { -\frac {\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}}{c^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx=\int { -\frac {\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}}{c^{2} x^{2} - 1} \,d x } \]
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Time = 5.95 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{1-c^2 x^2} \, dx=\frac {\ln \left (x\right )}{c}-\frac {4\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{c\,x}+1}-1}\right )}{c}-\frac {\ln \left (3\,c^2\,x^2-3\right )}{2\,c} \]
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