Optimal. Leaf size=42 \[ \frac{\tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac{i}{2 d (a+i a \sinh (c+d x))} \]
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Rubi [A] time = 0.0566407, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2667, 44, 206} \[ \frac{\tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac{i}{2 d (a+i a \sinh (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^2} \, dx,x,i a \sinh (c+d x)\right )}{d}\\ &=-\frac{(i a) \operatorname{Subst}\left (\int \left (\frac{1}{2 a (a+x)^2}+\frac{1}{2 a \left (a^2-x^2\right )}\right ) \, dx,x,i a \sinh (c+d x)\right )}{d}\\ &=\frac{i}{2 d (a+i a \sinh (c+d x))}-\frac{i \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \sinh (c+d x)\right )}{2 d}\\ &=\frac{\tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac{i}{2 d (a+i a \sinh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0480522, size = 30, normalized size = 0.71 \[ \frac{\tan ^{-1}(\sinh (c+d x))+\frac{1}{\sinh (c+d x)-i}}{2 a d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 91, normalized size = 2.2 \begin{align*}{\frac{{\frac{i}{2}}}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +i \right ) }-{\frac{i}{da} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{{\frac{i}{2}}}{da}\ln \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{da} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16804, size = 117, normalized size = 2.79 \begin{align*} -\frac{2 \, e^{\left (-d x - c\right )}}{{\left (4 i \, a e^{\left (-d x - c\right )} + 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a\right )} d} - \frac{i \, \log \left (e^{\left (-d x - c\right )} + i\right )}{2 \, a d} + \frac{i \, \log \left (i \, e^{\left (-d x - c\right )} + 1\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12009, size = 267, normalized size = 6.36 \begin{align*} \frac{{\left (i \, e^{\left (2 \, d x + 2 \, c\right )} + 2 \, e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} + i\right ) +{\left (-i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + i\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 2 \, e^{\left (d x + c\right )}}{2 \, a d e^{\left (2 \, d x + 2 \, c\right )} - 4 i \, a d e^{\left (d x + c\right )} - 2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{sech}{\left (c + d x \right )}}{i \sinh{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15309, size = 147, normalized size = 3.5 \begin{align*} -\frac{i \, \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} - 2 i\right )}{4 \, a d} + \frac{i \, \log \left (i \, e^{\left (d x + c\right )} - i \, e^{\left (-d x - c\right )} - 2\right )}{4 \, a d} - \frac{-i \, e^{\left (d x + c\right )} + i \, e^{\left (-d x - c\right )} - 6}{4 \, a d{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} - 2 i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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