Integrand size = 11, antiderivative size = 13 \[ \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx=-i \log (i-\sinh (x)) \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3964, 31} \[ \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx=-i \log (-\sinh (x)+i) \]
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Rule 31
Rule 3964
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{i+i x} \, dx,x,i \sinh (x)\right ) \\ & = -i \log (i-\sinh (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx=-i \log (i-\sinh (x)) \]
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Time = 1.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15
method | result | size |
risch | \(i x -2 i \ln \left ({\mathrm e}^{x}-i\right )\) | \(15\) |
derivativedivides | \(i \ln \left (\operatorname {csch}\left (x \right )\right )-\frac {i \ln \left (1+\operatorname {csch}\left (x \right )^{2}\right )}{2}-\arctan \left (\operatorname {csch}\left (x \right )\right )\) | \(23\) |
default | \(i \ln \left (\operatorname {csch}\left (x \right )\right )-\frac {i \ln \left (1+\operatorname {csch}\left (x \right )^{2}\right )}{2}-\arctan \left (\operatorname {csch}\left (x \right )\right )\) | \(23\) |
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx=i \, x - 2 i \, \log \left (e^{x} - i\right ) \]
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\[ \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx=\int \frac {\coth {\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx=-i \, x - 2 i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx=i \, x - 2 i \, \log \left (e^{x} - i\right ) \]
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Time = 2.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\coth (x)}{i+\text {csch}(x)} \, dx=x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,2{}\mathrm {i} \]
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