Optimal. Leaf size=47 \[ \frac{2 \tanh (c+d x)}{3 a d}+\frac{i \text{sech}(c+d x)}{3 d (a+i a \sinh (c+d x))} \]
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Rubi [A] time = 0.054996, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2672, 3767, 8} \[ \frac{2 \tanh (c+d x)}{3 a d}+\frac{i \text{sech}(c+d x)}{3 d (a+i a \sinh (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=\frac{i \text{sech}(c+d x)}{3 d (a+i a \sinh (c+d x))}+\frac{2 \int \text{sech}^2(c+d x) \, dx}{3 a}\\ &=\frac{i \text{sech}(c+d x)}{3 d (a+i a \sinh (c+d x))}+\frac{(2 i) \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{3 a d}\\ &=\frac{i \text{sech}(c+d x)}{3 d (a+i a \sinh (c+d x))}+\frac{2 \tanh (c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.0461272, size = 47, normalized size = 1. \[ \frac{\text{sech}(c+d x) (\cosh (2 (c+d x))-2 i \sinh (c+d x))}{3 a d (\sinh (c+d x)-i)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 75, normalized size = 1.6 \begin{align*} 2\,{\frac{1}{da} \left ( 1/4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-1}-1/3\, \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{-3}+{\frac{i/2}{ \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}+3/4\, \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18232, size = 140, normalized size = 2.98 \begin{align*} \frac{8 \, e^{\left (-d x - c\right )}}{{\left (6 \, a e^{\left (-d x - c\right )} + 6 \, a e^{\left (-3 \, d x - 3 \, c\right )} - 3 i \, a e^{\left (-4 \, d x - 4 \, c\right )} + 3 i \, a\right )} d} + \frac{4 i}{{\left (6 \, a e^{\left (-d x - c\right )} + 6 \, a e^{\left (-3 \, d x - 3 \, c\right )} - 3 i \, a e^{\left (-4 \, d x - 4 \, c\right )} + 3 i \, a\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07062, size = 144, normalized size = 3.06 \begin{align*} -\frac{4 \,{\left (-2 i \, e^{\left (d x + c\right )} - 1\right )}}{3 \, a d e^{\left (4 \, d x + 4 \, c\right )} - 6 i \, a d e^{\left (3 \, d x + 3 \, c\right )} - 6 i \, a d e^{\left (d x + c\right )} - 3 \, 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}^{2}{\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 [A] time = 1.15949, size = 81, normalized size = 1.72 \begin{align*} \frac{1}{2 \, a d{\left (i \, e^{\left (d x + c\right )} - 1\right )}} - \frac{-3 i \, e^{\left (2 \, d x + 2 \, c\right )} - 12 \, e^{\left (d x + c\right )} + 5 i}{6 \, a d{\left (e^{\left (d x + c\right )} - i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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