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 = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.12, 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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fricas [A] time = 0.53, size = 60, normalized size = 1.13 \[ \frac {{\left ({\left (B - C\right )} b + {\left (B - C\right )} c\right )} x + {\left (C b - B c\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + c \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{b^{2} - c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 54, normalized size = 1.02 \[ \frac {{\left (B - C\right )} x}{b - c} + \frac {{\left (C b - B c\right )} \log \left ({\left | b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + b - c \right |}\right )}{b^{2} - c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 145, normalized size = 2.74 \[ -\frac {2 B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b +2 c}-\frac {2 C \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b +2 c}+\frac {2 B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b -2 c}-\frac {2 C \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b -2 c}-\frac {B c \ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +2 c \tanh \left (\frac {x}{2}\right )+b \right )}{\left (b -c \right ) \left (b +c \right )}+\frac {b C \ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +2 c \tanh \left (\frac {x}{2}\right )+b \right )}{\left (b -c \right ) \left (b +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 87, normalized size = 1.64 \[ C {\left (\frac {b \log \left (-{\left (b - c\right )} e^{\left (-2 \, x\right )} - b - c\right )}{b^{2} - c^{2}} + \frac {x}{b + c}\right )} - B {\left (\frac {c \log \left (-{\left (b - c\right )} e^{\left (-2 \, x\right )} - b - c\right )}{b^{2} - c^{2}} - \frac {x}{b + c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 53, normalized size = 1.00 \[ \frac {x\,\left (B\,b-C\,c\right )}{b^2-c^2}-\frac {\ln \left (b\,\mathrm {cosh}\relax (x)+c\,\mathrm {sinh}\relax (x)\right )\,\left (B\,c-C\,b\right )}{b^2-c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 326, normalized size = 6.15 \[ \begin {cases} \tilde {\infty } \left (B \log {\left (\sinh {\relax (x )} \right )} + C x\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {B x \sinh {\relax (x )}}{- 2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} - \frac {B x \cosh {\relax (x )}}{- 2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} - \frac {B \cosh {\relax (x )}}{- 2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} - \frac {C x \sinh {\relax (x )}}{- 2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} + \frac {C x \cosh {\relax (x )}}{- 2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} - \frac {C \cosh {\relax (x )}}{- 2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} & \text {for}\: b = - c \\\frac {B x \sinh {\relax (x )}}{2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} + \frac {B x \cosh {\relax (x )}}{2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} - \frac {B \cosh {\relax (x )}}{2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} + \frac {C x \sinh {\relax (x )}}{2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} + \frac {C x \cosh {\relax (x )}}{2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} + \frac {C \cosh {\relax (x )}}{2 c \sinh {\relax (x )} + 2 c \cosh {\relax (x )}} & \text {for}\: b = c \\\frac {B \log {\left (\sinh {\relax (x )} \right )} + C x}{c} & \text {for}\: b = 0 \\\frac {B b x}{b^{2} - c^{2}} - \frac {B c \log {\left (\cosh {\relax (x )} + \frac {c \sinh {\relax (x )}}{b} \right )}}{b^{2} - c^{2}} + \frac {C b \log {\left (\cosh {\relax (x )} + \frac {c \sinh {\relax (x )}}{b} \right )}}{b^{2} - c^{2}} - \frac {C c x}{b^{2} - c^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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