Integrand size = 17, antiderivative size = 91 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]
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Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3861, 3859, 209, 3880} \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]
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Rule 209
Rule 3859
Rule 3861
Rule 3880
Rubi steps \begin{align*} \text {integral}& = -\left (i \int \frac {\text {csch}(c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx\right )+\frac {\int \sqrt {a+i a \text {csch}(c+d x)} \, dx}{a} \\ & = -\frac {(2 i) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {i a \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,\frac {i a \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 1.78 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\frac {\sqrt {a} \left (2 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right )\right ) \coth (c+d x)}{d \sqrt {i a (i+\text {csch}(c+d x))} \sqrt {a+i a \text {csch}(c+d x)}} \]
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\[\int \frac {1}{\sqrt {a +i a \,\operatorname {csch}\left (d x +c \right )}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (72) = 144\).
Time = 0.29 (sec) , antiderivative size = 551, normalized size of antiderivative = 6.05 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (2 \, {\left (\sqrt {2} {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} + a e^{\left (d x + c\right )} - i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-2 \, {\left (\sqrt {2} {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} - a e^{\left (d x + c\right )} + i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {2 \, {\left ({\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} + e^{\left (d x + c\right )} + i\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {2 \, {\left ({\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} - e^{\left (d x + c\right )} - i\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )} + 2 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} - \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )} + 2 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]
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\[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\int \frac {1}{\sqrt {i a \operatorname {csch}{\left (c + d x \right )} + a}}\, dx \]
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\[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \operatorname {csch}\left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \operatorname {csch}\left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}}} \,d x \]
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