Optimal. Leaf size=22 \[ \frac{x}{a}-\frac{i \cosh (c+d x)}{a d} \]
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Rubi [A] time = 0.0431233, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2682, 8} \[ \frac{x}{a}-\frac{i \cosh (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 2682
Rule 8
Rubi steps
\begin{align*} \int \frac{\cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \cosh (c+d x)}{a d}+\frac{\int 1 \, dx}{a}\\ &=\frac{x}{a}-\frac{i \cosh (c+d x)}{a d}\\ \end{align*}
Mathematica [B] time = 0.1447, size = 139, normalized size = 6.32 \[ \frac{\cosh ^3(c+d x) \left (-i \sqrt{1+i \sinh (c+d x)} \sinh (c+d x)+\sqrt{1+i \sinh (c+d x)}-2 \sqrt{1-i \sinh (c+d x)} \sin ^{-1}\left (\frac{\sqrt{1-i \sinh (c+d x)}}{\sqrt{2}}\right )\right )}{a d \sqrt{1+i \sinh (c+d x)} (\sinh (c+d x)-i) (\sinh (c+d x)+i)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 85, normalized size = 3.9 \begin{align*}{\frac{-i}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{i}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06201, size = 59, normalized size = 2.68 \begin{align*} \frac{d x + c}{a d} - \frac{i \, e^{\left (d x + c\right )}}{2 \, a d} - \frac{i \, 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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Fricas [A] time = 2.11257, size = 92, normalized size = 4.18 \begin{align*} \frac{{\left (2 \, d x e^{\left (d x + c\right )} - i \, e^{\left (2 \, d x + 2 \, c\right )} - i\right )} 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 [A] time = 0.433829, size = 82, normalized size = 3.73 \begin{align*} \begin{cases} \frac{\left (- 2 i a d e^{2 c} e^{d x} - 2 i a d e^{- d x}\right ) e^{- c}}{4 a^{2} d^{2}} & \text{for}\: 4 a^{2} d^{2} e^{c} \neq 0 \\x \left (- \frac{\left (i e^{2 c} - 2 e^{c} - i\right ) e^{- c}}{2 a} - \frac{1}{a}\right ) & \text{otherwise} \end{cases} + \frac{x}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22428, size = 47, normalized size = 2.14 \begin{align*} \frac{2 \, d x + 2 \, c - i \, e^{\left (d x + c\right )} - i \, 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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