Integrand size = 27, antiderivative size = 11 \[ \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx=\frac {\sinh (x)}{b+a \cosh (x)} \]
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Time = 0.06 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3370, 2833, 8} \[ \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx=\frac {\sinh (x)}{a \cosh (x)+b} \]
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Rule 8
Rule 2833
Rule 3370
Rubi steps \begin{align*} \text {integral}& = \left (4 a^2\right ) \int \frac {a+b \cosh (x)}{\left (2 a b+2 a^2 \cosh (x)\right )^2} \, dx \\ & = \frac {\sinh (x)}{b+a \cosh (x)}+\frac {\int 0 \, dx}{a^2-b^2} \\ & = \frac {\sinh (x)}{b+a \cosh (x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx=\frac {\sinh (x)}{b+a \cosh (x)} \]
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Time = 0.16 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
parallelrisch | \(\frac {\sinh \left (x \right )}{b +a \cosh \left (x \right )}\) | \(12\) |
risch | \(-\frac {2 \left ({\mathrm e}^{x} b +a \right )}{a \left ({\mathrm e}^{2 x} a +2 \,{\mathrm e}^{x} b +a \right )}\) | \(27\) |
default | \(\frac {2 \tanh \left (\frac {x}{2}\right )}{a \tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right )^{2} b +a +b}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (11) = 22\).
Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 4.91 \[ \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx=-\frac {2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \]
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Timed out. \[ \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (11) = 22\).
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.36 \[ \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx=-\frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )} a} \]
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Time = 2.67 (sec) , antiderivative size = 51, normalized size of antiderivative = 4.64 \[ \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx=-\frac {\frac {2\,{\mathrm {e}}^x\,\left (a\,b^3-a^3\,b\right )}{a\,\left (a\,b^2-a^3\right )}+2}{a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}} \]
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