Optimal. Leaf size=27 \[ -\frac {i \cosh (c+d x)}{d (1-i \sinh (c+d x))} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2648} \[ -\frac {i \cosh (c+d x)}{d (1-i \sinh (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin {align*} \int \frac {1}{1-i \sinh (c+d x)} \, dx &=-\frac {i \cosh (c+d x)}{d (1-i \sinh (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 42, normalized size = 1.56 \[ \frac {2 \sinh \left (\frac {1}{2} (c+d x)\right )}{d \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 16, normalized size = 0.59 \[ -\frac {2 i}{d e^{\left (d x + c\right )} + i \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 15, normalized size = 0.56 \[ -\frac {2 i}{d {\left (e^{\left (d x + c\right )} + i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.74 \[ \frac {2}{d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 20, normalized size = 0.74 \[ \frac {2}{d {\left (i \, e^{\left (-d x - c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 17, normalized size = 0.63 \[ -\frac {2{}\mathrm {i}}{d\,\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 0.63 \[ \frac {2 e^{c}}{d e^{c} + i d e^{- d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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