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 = {2727}
\begin {gather*} \frac {i \cosh (c+d x)}{d (1+i \sinh (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
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 = 0.05, size = 42, normalized size = 1.56 \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.81, size = 20, normalized size = 0.74
method | result | size |
risch | \(\frac {2 i}{d \left ({\mathrm e}^{d x +c}-i\right )}\) | \(18\) |
derivativedivides | \(\frac {2}{d \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(20\) |
default | \(\frac {2}{d \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 20, normalized size = 0.74 \begin {gather*} -\frac {2}{d {\left (i \, e^{\left (-d x - c\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 16, normalized size = 0.59 \begin {gather*} \frac {2 i}{d e^{\left (d x + c\right )} - i \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 0.56 \begin {gather*} \frac {2 i}{d e^{c} e^{d x} - i d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 15, normalized size = 0.56 \begin {gather*} \frac {2 i}{d {\left (e^{\left (d x + c\right )} - i\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 17, normalized size = 0.63 \begin {gather*} \frac {2{}\mathrm {i}}{d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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