Integrand size = 12, antiderivative size = 28 \[ \int (c+d x) \cosh (a+b x) \, dx=-\frac {d \cosh (a+b x)}{b^2}+\frac {(c+d x) \sinh (a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3377, 2718} \[ \int (c+d x) \cosh (a+b x) \, dx=\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2} \]
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Rule 2718
Rule 3377
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int \sinh (a+b x) \, dx}{b} \\ & = -\frac {d \cosh (a+b x)}{b^2}+\frac {(c+d x) \sinh (a+b x)}{b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int (c+d x) \cosh (a+b x) \, dx=\frac {-d \cosh (a+b x)+b (c+d x) \sinh (a+b x)}{b^2} \]
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Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32
method | result | size |
parts | \(\frac {\sinh \left (b x +a \right ) d x}{b}+\frac {\sinh \left (b x +a \right ) c}{b}-\frac {d \cosh \left (b x +a \right )}{b^{2}}\) | \(37\) |
parallelrisch | \(\frac {-2 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right ) b \left (d x +c \right )+2 d}{b^{2} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\) | \(40\) |
risch | \(\frac {\left (d x b +c b -d \right ) {\mathrm e}^{b x +a}}{2 b^{2}}-\frac {\left (d x b +c b +d \right ) {\mathrm e}^{-b x -a}}{2 b^{2}}\) | \(47\) |
derivativedivides | \(\frac {\frac {d \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {d a \sinh \left (b x +a \right )}{b}+c \sinh \left (b x +a \right )}{b}\) | \(53\) |
default | \(\frac {\frac {d \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {d a \sinh \left (b x +a \right )}{b}+c \sinh \left (b x +a \right )}{b}\) | \(53\) |
meijerg | \(-\frac {2 d \cosh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {d \sinh \left (a \right ) \left (\cosh \left (b x \right ) x b -\sinh \left (b x \right )\right )}{b^{2}}+\frac {c \cosh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(95\) |
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (c+d x) \cosh (a+b x) \, dx=-\frac {d \cosh \left (b x + a\right ) - {\left (b d x + b c\right )} \sinh \left (b x + a\right )}{b^{2}} \]
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Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int (c+d x) \cosh (a+b x) \, dx=\begin {cases} \frac {c \sinh {\left (a + b x \right )}}{b} + \frac {d x \sinh {\left (a + b x \right )}}{b} - \frac {d \cosh {\left (a + b x \right )}}{b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cosh {\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).
Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43 \[ \int (c+d x) \cosh (a+b x) \, dx=\frac {c e^{\left (b x + a\right )}}{2 \, b} + \frac {{\left (b x e^{a} - e^{a}\right )} d e^{\left (b x\right )}}{2 \, b^{2}} - \frac {c e^{\left (-b x - a\right )}}{2 \, b} - \frac {{\left (b x + 1\right )} d e^{\left (-b x - a\right )}}{2 \, b^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int (c+d x) \cosh (a+b x) \, dx=\frac {{\left (b d x + b c - d\right )} e^{\left (b x + a\right )}}{2 \, b^{2}} - \frac {{\left (b d x + b c + d\right )} e^{\left (-b x - a\right )}}{2 \, b^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int (c+d x) \cosh (a+b x) \, dx=\frac {c\,\mathrm {sinh}\left (a+b\,x\right )+d\,x\,\mathrm {sinh}\left (a+b\,x\right )}{b}-\frac {d\,\mathrm {cosh}\left (a+b\,x\right )}{b^2} \]
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