Optimal. Leaf size=15 \[ \frac{\log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{c} \]
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Rubi [A] time = 0.0187025, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3124, 31} \[ \frac{\log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{c} \]
Antiderivative was successfully verified.
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Rule 3124
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+a \cosh (x)+c \sinh (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{2 a+2 c x} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{\log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{c}\\ \end{align*}
Mathematica [B] time = 0.0366816, size = 35, normalized size = 2.33 \[ \frac{\log \left (a \cosh \left (\frac{x}{2}\right )+c \sinh \left (\frac{x}{2}\right )\right )}{c}-\frac{\log \left (\cosh \left (\frac{x}{2}\right )\right )}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 14, normalized size = 0.9 \begin{align*}{\frac{1}{c}\ln \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02476, size = 49, normalized size = 3.27 \begin{align*} \frac{\log \left (-{\left (a - c\right )} e^{\left (-x\right )} - a - c\right )}{c} - \frac{\log \left (e^{\left (-x\right )} + 1\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.29608, size = 109, normalized size = 7.27 \begin{align*} \frac{\log \left ({\left (a + c\right )} \cosh \left (x\right ) +{\left (a + c\right )} \sinh \left (x\right ) + a - c\right ) - \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.65989, size = 17, normalized size = 1.13 \begin{align*} \begin{cases} \frac{\log{\left (\frac{a}{c} + \tanh{\left (\frac{x}{2} \right )} \right )}}{c} & \text{for}\: c \neq 0 \\\frac{\tanh{\left (\frac{x}{2} \right )}}{a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15186, size = 53, normalized size = 3.53 \begin{align*} \frac{{\left (a + c\right )} \log \left ({\left | a e^{x} + c e^{x} + a - c \right |}\right )}{a c + c^{2}} - \frac{\log \left (e^{x} + 1\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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