Optimal. Leaf size=23 \[ -\frac {i \log (i-\sinh (c+d x))}{a d} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2746, 31}
\begin {gather*} -\frac {i \log (-\sinh (c+d x)+i)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,i a \sinh (c+d x)\right )}{a d}\\ &=-\frac {i \log (i-\sinh (c+d x))}{a d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} -\frac {i \log (i-\sinh (c+d x))}{a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 23, normalized size = 1.00
method | result | size |
derivativedivides | \(-\frac {i \ln \left (a +i a \sinh \left (d x +c \right )\right )}{d a}\) | \(23\) |
default | \(-\frac {i \ln \left (a +i a \sinh \left (d x +c \right )\right )}{d a}\) | \(23\) |
risch | \(\frac {i x}{a}+\frac {2 i c}{a d}-\frac {2 i \ln \left ({\mathrm e}^{d x +c}-i\right )}{a d}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 20, normalized size = 0.87 \begin {gather*} -\frac {i \, \log \left (i \, a \sinh \left (d x + c\right ) + a\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 23, normalized size = 1.00 \begin {gather*} \frac {i \, d x - 2 i \, \log \left (e^{\left (d x + c\right )} - i\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 22, normalized size = 0.96 \begin {gather*} \frac {i x}{a} - \frac {2 i \log {\left (e^{d x} - i e^{- c} \right )}}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 32, normalized size = 1.39 \begin {gather*} -\frac {-\frac {i \, {\left (d x + c\right )}}{a} + \frac {2 i \, \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 19, normalized size = 0.83 \begin {gather*} -\frac {\ln \left (\mathrm {sinh}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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