Optimal. Leaf size=53 \[ \frac{x (b B-c C)}{b^2-c^2}-\frac{(B c-b C) \log (b \cosh (x)+c \sinh (x))}{b^2-c^2} \]
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Rubi [A] time = 0.0464645, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3133} \[ \frac{x (b B-c C)}{b^2-c^2}-\frac{(B c-b C) \log (b \cosh (x)+c \sinh (x))}{b^2-c^2} \]
Antiderivative was successfully verified.
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Rule 3133
Rubi steps
\begin{align*} \int \frac{B \cosh (x)+C \sinh (x)}{b \cosh (x)+c \sinh (x)} \, dx &=\frac{(b B-c C) x}{b^2-c^2}-\frac{(B c-b C) \log (b \cosh (x)+c \sinh (x))}{b^2-c^2}\\ \end{align*}
Mathematica [A] time = 0.114514, size = 43, normalized size = 0.81 \[ \frac{x (b B-c C)+(b C-B c) \log (b \cosh (x)+c \sinh (x))}{(b-c) (b+c)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 145, normalized size = 2.7 \begin{align*} 2\,{\frac{B\ln \left ( \tanh \left ( x/2 \right ) +1 \right ) }{2\,b-2\,c}}-2\,{\frac{C\ln \left ( \tanh \left ( x/2 \right ) +1 \right ) }{2\,b-2\,c}}-{\frac{Bc}{ \left ( b-c \right ) \left ( b+c \right ) }\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+2\,c\tanh \left ( x/2 \right ) +b \right ) }+{\frac{bC}{ \left ( b-c \right ) \left ( b+c \right ) }\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+2\,c\tanh \left ( x/2 \right ) +b \right ) }-2\,{\frac{B\ln \left ( \tanh \left ( x/2 \right ) -1 \right ) }{2\,b+2\,c}}-2\,{\frac{C\ln \left ( \tanh \left ( x/2 \right ) -1 \right ) }{2\,b+2\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11514, size = 117, normalized size = 2.21 \begin{align*} C{\left (\frac{b \log \left (-{\left (b - c\right )} e^{\left (-2 \, x\right )} - b - c\right )}{b^{2} - c^{2}} + \frac{x}{b + c}\right )} - B{\left (\frac{c \log \left (-{\left (b - c\right )} e^{\left (-2 \, x\right )} - b - c\right )}{b^{2} - c^{2}} - \frac{x}{b + c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24471, size = 143, normalized size = 2.7 \begin{align*} \frac{{\left ({\left (B - C\right )} b +{\left (B - C\right )} c\right )} x +{\left (C b - B c\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + c \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{b^{2} - c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.883475, size = 326, normalized size = 6.15 \begin{align*} \begin{cases} \tilde{\infty } \left (B \log{\left (\sinh{\left (x \right )} \right )} + C x\right ) & \text{for}\: b = 0 \wedge c = 0 \\\frac{B x + C \log{\left (\cosh{\left (x \right )} \right )}}{b} & \text{for}\: c = 0 \\\frac{B x \sinh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} - \frac{B x \cosh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} - \frac{B \cosh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} - \frac{C x \sinh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} + \frac{C x \cosh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} - \frac{C \cosh{\left (x \right )}}{- 2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} & \text{for}\: b = - c \\\frac{B x \sinh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} + \frac{B x \cosh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} - \frac{B \cosh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} + \frac{C x \sinh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} + \frac{C x \cosh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} + \frac{C \cosh{\left (x \right )}}{2 c \sinh{\left (x \right )} + 2 c \cosh{\left (x \right )}} & \text{for}\: b = c \\\frac{B b x}{b^{2} - c^{2}} - \frac{B c \log{\left (\frac{b \cosh{\left (x \right )}}{c} + \sinh{\left (x \right )} \right )}}{b^{2} - c^{2}} + \frac{C b \log{\left (\frac{b \cosh{\left (x \right )}}{c} + \sinh{\left (x \right )} \right )}}{b^{2} - c^{2}} - \frac{C c x}{b^{2} - c^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13779, size = 73, normalized size = 1.38 \begin{align*} \frac{{\left (B - C\right )} x}{b - c} + \frac{{\left (C b - B c\right )} \log \left ({\left | b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c \right |}\right )}{b^{2} - c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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