Integrand size = 14, antiderivative size = 30 \[ \int \sqrt {i \sinh (c+d x)} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{d} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2719} \[ \int \sqrt {i \sinh (c+d x)} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d} \]
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Rule 2719
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \sqrt {i \sinh (c+d x)} \, dx=\frac {2 i E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i (c+d x)\right )\right |2\right )}{d} \]
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Time = 1.48 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.03
method | result | size |
default | \(\frac {i \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (d x +c \right ) d}\) | \(91\) |
risch | \(\frac {\sqrt {2}\, \sqrt {i \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{-d x -c}}}{d}-\frac {\left (-\frac {2 i \left (-i+i {\mathrm e}^{2 d x +2 c}\right )}{\sqrt {{\mathrm e}^{d x +c} \left (-i+i {\mathrm e}^{2 d x +2 c}\right )}}-\frac {\sqrt {{\mathrm e}^{d x +c}+1}\, \sqrt {-2 \,{\mathrm e}^{d x +c}+2}\, \sqrt {-{\mathrm e}^{d x +c}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i {\mathrm e}^{3 d x +3 c}-i {\mathrm e}^{d x +c}}}\right ) \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{-d x -c}}\, \sqrt {i \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{d x +c}}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(229\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \sqrt {i \sinh (c+d x)} \, dx=-\frac {2 \, {\left (\sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + \sqrt {2} \sqrt {i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )\right )}}{d} \]
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\[ \int \sqrt {i \sinh (c+d x)} \, dx=\int \sqrt {i \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt {i \sinh (c+d x)} \, dx=\int { \sqrt {i \, \sinh \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {i \sinh (c+d x)} \, dx=\int { \sqrt {i \, \sinh \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {i \sinh (c+d x)} \, dx=\int \sqrt {\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
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