Integrand size = 25, antiderivative size = 34 \[ \int \frac {\cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a \log (a+b \sinh (c+d x))}{b^2 d}+\frac {\sinh (c+d x)}{b d} \]
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Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 45} \[ \int \frac {\cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\sinh (c+d x)}{b d}-\frac {a \log (a+b \sinh (c+d x))}{b^2 d} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x}{b (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b d} \\ & = \frac {\text {Subst}\left (\int \frac {x}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b^2 d} \\ & = \frac {\text {Subst}\left (\int \left (1-\frac {a}{a+x}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^2 d} \\ & = -\frac {a \log (a+b \sinh (c+d x))}{b^2 d}+\frac {\sinh (c+d x)}{b d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {\cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {a \log (a+b \sinh (c+d x))}{b^2}-\frac {\sinh (c+d x)}{b}}{d} \]
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Time = 1.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {\sinh \left (d x +c \right )}{b}-\frac {a \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{2}}}{d}\) | \(33\) |
default | \(\frac {\frac {\sinh \left (d x +c \right )}{b}-\frac {a \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{2}}}{d}\) | \(33\) |
risch | \(\frac {a x}{b^{2}}+\frac {{\mathrm e}^{d x +c}}{2 b d}-\frac {{\mathrm e}^{-d x -c}}{2 b d}+\frac {2 a c}{b^{2} d}-\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{2} d}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (34) = 68\).
Time = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.88 \[ \int \frac {\cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \, a d x \cosh \left (d x + c\right ) + b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} - 2 \, {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (a d x + b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - b}{2 \, {\left (b^{2} d \cosh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (27) = 54\).
Time = 0.87 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {\cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\begin {cases} \frac {x \sinh {\left (c \right )} \cosh {\left (c \right )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\cosh ^{2}{\left (c + d x \right )}}{2 a d} & \text {for}\: b = 0 \\\frac {x \sinh {\left (c \right )} \cosh {\left (c \right )}}{a + b \sinh {\left (c \right )}} & \text {for}\: d = 0 \\- \frac {a \log {\left (\frac {a}{b} + \sinh {\left (c + d x \right )} \right )}}{b^{2} d} + \frac {\sinh {\left (c + d x \right )}}{b d} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.44 \[ \int \frac {\cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (d x + c\right )} a}{b^{2} d} + \frac {e^{\left (d x + c\right )}}{2 \, b d} - \frac {e^{\left (-d x - c\right )}}{2 \, b d} - \frac {a \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{2} d} \]
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.76 \[ \int \frac {\cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}}{b} - \frac {2 \, a \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{2}}}{2 \, d} \]
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Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a\,\ln \left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )-b\,\mathrm {sinh}\left (c+d\,x\right )}{b^2\,d} \]
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