Integrand size = 15, antiderivative size = 32 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {x}{a}-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3862, 8} \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {x}{a}-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))} \]
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Rule 8
Rule 3862
Rubi steps \begin{align*} \text {integral}& = -\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))}+\frac {\int a \, dx}{a^2} \\ & = \frac {x}{a}-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {1}{b}+\frac {x}{a}-\frac {2 \sinh \left (\frac {1}{2} (a+b x)\right )}{a b \left (\cosh \left (\frac {1}{2} (a+b x)\right )-i \sinh \left (\frac {1}{2} (a+b x)\right )\right )} \]
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Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {x}{a}+\frac {2 i}{b a \left ({\mathrm e}^{b x +a}+i\right )}\) | \(27\) |
parallelrisch | \(\frac {i x b +\tanh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (b x -2 i\right )}{b a \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i\right )}\) | \(44\) |
derivativedivides | \(\frac {\ln \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {2}{\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i}-\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{a b}\) | \(54\) |
default | \(\frac {\ln \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {2}{\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i}-\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{a b}\) | \(54\) |
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {b x e^{\left (b x + a\right )} + i \, b x + 2 i}{a b e^{\left (b x + a\right )} + i \, a b} \]
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\[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=- \frac {i \int \frac {1}{\operatorname {csch}{\left (a + b x \right )} - i}\, dx}{a} \]
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Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {b x + a}{a b} + \frac {2 i}{{\left (a e^{\left (-b x - a\right )} - i \, a\right )} b} \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {\frac {b x + a}{a} + \frac {2 i}{a {\left (e^{\left (b x + a\right )} + i\right )}}}{b} \]
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Time = 2.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {x}{a}+\frac {2{}\mathrm {i}}{a\,b\,\left ({\mathrm {e}}^{a+b\,x}+1{}\mathrm {i}\right )} \]
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