Integrand size = 17, antiderivative size = 72 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3860, 21, 3859, 209} \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}} \]
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Rule 21
Rule 209
Rule 3859
Rule 3860
Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}+(2 a) \int \frac {\frac {a}{2}+\frac {1}{2} i a \text {csch}(c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx \\ & = \frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}+a \int \sqrt {a+i a \text {csch}(c+d x)} \, dx \\ & = \frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}-\frac {\left (2 i a^2\right ) \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 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}} \\ \end{align*}
Time = 1.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.39 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=-\frac {2 i a \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)} \left (-\sqrt [4]{-1} \text {arctanh}\left ((-1)^{3/4} \sqrt {i+\text {csch}(c+d x)}\right )+\sqrt {i+\text {csch}(c+d x)}\right )}{d (-i+\text {csch}(c+d x)) \sqrt {i+\text {csch}(c+d x)}} \]
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\[\int \left (a +i a \,\operatorname {csch}\left (d x +c \right )\right )^{\frac {3}{2}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (60) = 120\).
Time = 0.27 (sec) , antiderivative size = 461, normalized size of antiderivative = 6.40 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\frac {\sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left (a^{2} e^{\left (d x + c\right )} + i \, a^{2} + {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-d x - c\right )}}{d}\right ) - \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left (a^{2} e^{\left (d x + c\right )} + i \, a^{2} - {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-d x - c\right )}}{d}\right ) + \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a^{3}}{d^{2}}} + {\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} e^{\left (d x + c\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (-\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a^{3}}{d^{2}}} - {\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} e^{\left (d x + c\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) + 4 \, {\left (a e^{\left (d x + c\right )} - i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}}{2 \, d} \]
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\[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\int \left (i a \operatorname {csch}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
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\[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\int { {\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\int { {\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\int {\left (a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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