Integrand size = 32, antiderivative size = 22 \[ \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx=\frac {c \cosh (x)+b \sinh (x)}{a+b \cosh (x)+c \sinh (x)} \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {3229} \[ \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx=\frac {b \sinh (x)+c \cosh (x)}{a+b \cosh (x)+c \sinh (x)} \]
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Rule 3229
Rubi steps \begin{align*} \text {integral}& = \frac {c \cosh (x)+b \sinh (x)}{a+b \cosh (x)+c \sinh (x)} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx=\frac {-a c+b^2 \sinh (x)-c^2 \sinh (x)}{b (a+b \cosh (x)+c \sinh (x))} \]
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Time = 1.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64
method | result | size |
risch | \(-\frac {2 \left (a \,{\mathrm e}^{x}+b -c \right )}{{\mathrm e}^{2 x} b +{\mathrm e}^{2 x} c +2 a \,{\mathrm e}^{x}+b -c}\) | \(36\) |
default | \(\frac {-\frac {2 \left (a b -b^{2}+c^{2}\right ) \tanh \left (\frac {x}{2}\right )}{a -b}-\frac {2 a c}{a -b}}{a \tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right )^{2} b -2 c \tanh \left (\frac {x}{2}\right )-a -b}\) | \(73\) |
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.50 \[ \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx=-\frac {2 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b - c\right )}}{{\left (b + c\right )} \cosh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left ({\left (b + c\right )} \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b - c} \]
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Timed out. \[ \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx=-\frac {2 \, {\left (a e^{x} + b - c\right )}}{b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c} \]
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Timed out. \[ \int \frac {b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx=\int \frac {b^2+a\,\mathrm {cosh}\left (x\right )\,b-c^2+a\,\mathrm {sinh}\left (x\right )\,c}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^2} \,d x \]
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