Optimal. Leaf size=25 \[ B \log (\sinh (x)+i)-\frac{A \cosh (x)}{1-i \sinh (x)} \]
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Rubi [A] time = 0.0830601, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4401, 2648, 2667, 31} \[ B \log (\sinh (x)+i)-\frac{A \cosh (x)}{1-i \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2648
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{i+\sinh (x)} \, dx &=\int \left (\frac{i A}{-1+i \sinh (x)}+\frac{i B \cosh (x)}{-1+i \sinh (x)}\right ) \, dx\\ &=(i A) \int \frac{1}{-1+i \sinh (x)} \, dx+(i B) \int \frac{\cosh (x)}{-1+i \sinh (x)} \, dx\\ &=-\frac{A \cosh (x)}{1-i \sinh (x)}+B \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,i \sinh (x)\right )\\ &=B \log (i+\sinh (x))-\frac{A \cosh (x)}{1-i \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0658703, size = 48, normalized size = 1.92 \[ -\frac{2 i A \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )-i \sinh \left (\frac{x}{2}\right )}-2 i B \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+B \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 46, normalized size = 1.8 \begin{align*} -B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +2\,B\ln \left ( \tanh \left ( x/2 \right ) +i \right ) -{2\,iA \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03481, size = 26, normalized size = 1.04 \begin{align*} B \log \left (\sinh \left (x\right ) + i\right ) - \frac{2 \, A}{e^{\left (-x\right )} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04706, size = 90, normalized size = 3.6 \begin{align*} -\frac{B x e^{x} + i \, B x - 2 \,{\left (B e^{x} + i \, B\right )} \log \left (e^{x} + i\right ) + 2 \, A}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.3108, size = 20, normalized size = 0.8 \begin{align*} - \frac{2 A}{e^{x} + i} - B x + 2 B \log{\left (e^{x} + i \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13634, size = 30, normalized size = 1.2 \begin{align*} -B x + 2 \, B \log \left (e^{x} + i\right ) - \frac{2 \, A}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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