Integrand size = 11, antiderivative size = 17 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3154} \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]
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Rule 3154
Rubi steps \begin{align*} \text {integral}& = \frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]
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Time = 5.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59
method | result | size |
risch | \(-\frac {2}{\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right ) \left (a +b \right )}\) | \(27\) |
default | \(\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (17) = 34\).
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.65 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (14) = 28\).
Time = 136.94 (sec) , antiderivative size = 604, normalized size of antiderivative = 35.53 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {\tanh {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tanh {\left (\frac {x}{2} \right )}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {\tanh {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tanh {\left (\frac {x}{2} \right )}}}{b^{2}} & \text {for}\: a = 0 \\\frac {x \tanh ^{4}{\left (\frac {x}{2} \right )}}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} - \frac {2 x \tanh ^{2}{\left (\frac {x}{2} \right )}}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} + \frac {x}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} + \frac {2 \tanh ^{3}{\left (\frac {x}{2} \right )}}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} + \frac {2 \tanh {\left (\frac {x}{2} \right )}}{2 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{4}{\left (\frac {x}{2} \right )} + 4 b^{2} \sinh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sinh ^{2}{\left (x \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} \tanh {\left (\frac {x}{2} \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}} & \text {for}\: a = - \frac {2 b \tanh {\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} + 1} \\\frac {2 \tanh {\left (\frac {x}{2} \right )}}{a^{2} \tanh ^{2}{\left (\frac {x}{2} \right )} + a^{2} + 2 a b \tanh {\left (\frac {x}{2} \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {2}{a^{2} - b^{2} + {\left (a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-2 \, x\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2}{{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )} {\left (a + b\right )}} \]
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Time = 2.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx=-\frac {2}{\left (a+b\right )\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
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