Integrand size = 17, antiderivative size = 31 \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {2 i a \cosh (c+d x)}{d \sqrt {a+i a \sinh (c+d x)}} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2725} \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {2 i a \cosh (c+d x)}{d \sqrt {a+i a \sinh (c+d x)}} \]
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Rule 2725
Rubi steps \begin{align*} \text {integral}& = \frac {2 i a \cosh (c+d x)}{d \sqrt {a+i a \sinh (c+d x)}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(31)=62\).
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.39 \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {2 \left (i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {a+i a \sinh (c+d x)}}{d \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27 ) = 54\).
Time = 2.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.87
method | result | size |
risch | \(\frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{d x +c}-i\right ) {\mathrm e}^{-d x -c}}\, \left ({\mathrm e}^{d x +c}+i\right ) \left ({\mathrm e}^{d x +c}-i\right )}{\left (i {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{d x +c}-i\right ) d}\) | \(89\) |
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none
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} {\left (e^{\left (d x + c\right )} + i\right )}}{d} \]
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\[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\int \sqrt {i a \sinh {\left (c + d x \right )} + a}\, dx \]
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\[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\int { \sqrt {i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
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\[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\int { \sqrt {i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
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Time = 1.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \sqrt {a+i a \sinh (c+d x)} \, dx=\frac {\sqrt {2}\,\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-c-d\,x}\,{\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}}{d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \]
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