Optimal. Leaf size=41 \[ -\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))} \]
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Rubi [A] time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2747, 3770, 2648} \[ -\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2747
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\left (i \int \frac {1}{a+i a \sinh (c+d x)} \, dx\right )+\frac {\int \text {csch}(c+d x) \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\cosh (c+d x)}{d (a+i a \sinh (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 52, normalized size = 1.27 \[ -\frac {\text {sech}(c+d x) \left (i \sinh (c+d x)+\sqrt {\cosh ^2(c+d x)} \tanh ^{-1}\left (\sqrt {\cosh ^2(c+d x)}\right )-1\right )}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.60, size = 57, normalized size = 1.39 \[ -\frac {{\left (e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - {\left (e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 2}{a d e^{\left (d x + c\right )} - i \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 48, normalized size = 1.17 \[ -\frac {\frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac {\log \left (e^{\left (d x + c\right )} - 1\right )}{a} - \frac {2}{a {\left (e^{\left (d x + c\right )} - i\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 42, normalized size = 1.02 \[ -\frac {2 i}{d a \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 62, normalized size = 1.51 \[ -\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} + \frac {2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 56, normalized size = 1.37 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^2\,d^2}}{a\,d}\right )}{\sqrt {-a^2\,d^2}}+\frac {2}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {\operatorname {csch}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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