Integrand size = 17, antiderivative size = 52 \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d} \]
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Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2728, 212} \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d} \]
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Rule 212
Rule 2728
Rubi steps \begin{align*} \text {integral}& = \frac {(2 i) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cosh (c+d x)}{\sqrt {a+i a \sinh (c+d x)}}\right )}{d} \\ & = \frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\frac {(2+2 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1-i \tanh \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (-i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{d \sqrt {a+i a \sinh (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (41 ) = 82\).
Time = 6.70 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.08
method | result | size |
risch | \(-\frac {2 \left ({\mathrm e}^{d x +c}-i\right ) \sqrt {2}\, {\mathrm e}^{-d x -c}}{d \sqrt {a \left (i {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{d x +c}-i\right ) {\mathrm e}^{-d x -c}}}-\frac {2 \left (-{\mathrm e}^{d x +c}+i\right ) \left (a^{\frac {3}{2}}+\arctan \left (\frac {\sqrt {i {\mathrm e}^{d x +c} a}}{\sqrt {a}}\right ) a \sqrt {i {\mathrm e}^{d x +c} a}\right ) \sqrt {2}\, {\mathrm e}^{-d x -c}}{d \,a^{\frac {3}{2}} \sqrt {a \left (i {\mathrm e}^{2 d x +2 c}+2 \,{\mathrm e}^{d x +c}-i\right ) {\mathrm e}^{-d x -c}}}\) | \(160\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (39) = 78\).
Time = 0.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=i \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {1}{2} \, \sqrt {2} a d \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) - i \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {1}{2} \, \sqrt {2} a d \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) \]
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\[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\int \frac {1}{\sqrt {i a \sinh {\left (c + d x \right )} + a}}\, dx \]
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\[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \sinh \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \sinh \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx=\int \frac {1}{\sqrt {a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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