Integrand size = 19, antiderivative size = 50 \[ \int (c+d x) (a+i a \sinh (e+f x)) \, dx=\frac {a (c+d x)^2}{2 d}+\frac {i a (c+d x) \cosh (e+f x)}{f}-\frac {i a d \sinh (e+f x)}{f^2} \]
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Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3398, 3377, 2717} \[ \int (c+d x) (a+i a \sinh (e+f x)) \, dx=\frac {i a (c+d x) \cosh (e+f x)}{f}+\frac {a (c+d x)^2}{2 d}-\frac {i a d \sinh (e+f x)}{f^2} \]
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Rule 2717
Rule 3377
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int (a (c+d x)+i a (c+d x) \sinh (e+f x)) \, dx \\ & = \frac {a (c+d x)^2}{2 d}+(i a) \int (c+d x) \sinh (e+f x) \, dx \\ & = \frac {a (c+d x)^2}{2 d}+\frac {i a (c+d x) \cosh (e+f x)}{f}-\frac {(i a d) \int \cosh (e+f x) \, dx}{f} \\ & = \frac {a (c+d x)^2}{2 d}+\frac {i a (c+d x) \cosh (e+f x)}{f}-\frac {i a d \sinh (e+f x)}{f^2} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int (c+d x) (a+i a \sinh (e+f x)) \, dx=\frac {a \left (f^2 x (2 c+d x)+2 i f (c+d x) \cosh (e+f x)-2 i d \sinh (e+f x)\right )}{2 f^2} \]
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Time = 0.75 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\frac {a \left (i f \left (d x +c \right ) \cosh \left (f x +e \right )-i \sinh \left (f x +e \right ) d +f \left (x \left (\frac {d x}{2}+c \right ) f +i c \right )\right )}{f^{2}}\) | \(48\) |
risch | \(\frac {a d \,x^{2}}{2}+a c x +\frac {i a \left (d f x +c f -d \right ) {\mathrm e}^{f x +e}}{2 f^{2}}+\frac {i a \left (d f x +c f +d \right ) {\mathrm e}^{-f x -e}}{2 f^{2}}\) | \(62\) |
parts | \(a \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {i a \left (\frac {d \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d e \cosh \left (f x +e \right )}{f}+c \cosh \left (f x +e \right )\right )}{f}\) | \(69\) |
derivativedivides | \(\frac {\frac {d a \left (f x +e \right )^{2}}{2 f}+\frac {i d a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d e a \left (f x +e \right )}{f}-\frac {i d e a \cosh \left (f x +e \right )}{f}+a c \left (f x +e \right )+i a c \cosh \left (f x +e \right )}{f}\) | \(96\) |
default | \(\frac {\frac {d a \left (f x +e \right )^{2}}{2 f}+\frac {i d a \left (\left (f x +e \right ) \cosh \left (f x +e \right )-\sinh \left (f x +e \right )\right )}{f}-\frac {d e a \left (f x +e \right )}{f}-\frac {i d e a \cosh \left (f x +e \right )}{f}+a c \left (f x +e \right )+i a c \cosh \left (f x +e \right )}{f}\) | \(96\) |
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Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.62 \[ \int (c+d x) (a+i a \sinh (e+f x)) \, dx=\frac {{\left (i \, a d f x + i \, a c f + i \, a d + {\left (i \, a d f x + i \, a c f - i \, a d\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (a d f^{2} x^{2} + 2 \, a c f^{2} x\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{2 \, f^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 3.24 \[ \int (c+d x) (a+i a \sinh (e+f x)) \, dx=a c x + \frac {a d x^{2}}{2} + \begin {cases} \frac {\left (\left (2 i a c f^{3} + 2 i a d f^{3} x + 2 i a d f^{2}\right ) e^{- f x} + \left (2 i a c f^{3} e^{2 e} + 2 i a d f^{3} x e^{2 e} - 2 i a d f^{2} e^{2 e}\right ) e^{f x}\right ) e^{- e}}{4 f^{4}} & \text {for}\: f^{4} e^{e} \neq 0 \\\frac {x^{2} \left (i a d e^{2 e} - i a d\right ) e^{- e}}{4} + \frac {x \left (i a c e^{2 e} - i a c\right ) e^{- e}}{2} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.32 \[ \int (c+d x) (a+i a \sinh (e+f x)) \, dx=\frac {1}{2} \, a d x^{2} + a c x + \frac {1}{2} i \, a d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {i \, a c \cosh \left (f x + e\right )}{f} \]
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Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.38 \[ \int (c+d x) (a+i a \sinh (e+f x)) \, dx=\frac {1}{2} \, a d x^{2} + a c x - \frac {{\left (-i \, a d f x - i \, a c f + i \, a d\right )} e^{\left (f x + e\right )}}{2 \, f^{2}} - \frac {{\left (-i \, a d f x - i \, a c f - i \, a d\right )} e^{\left (-f x - e\right )}}{2 \, f^{2}} \]
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Time = 0.91 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.12 \[ \int (c+d x) (a+i a \sinh (e+f x)) \, dx=\frac {\frac {a\,f\,\left (c\,\mathrm {cosh}\left (e+f\,x\right )\,2{}\mathrm {i}+d\,x\,\mathrm {cosh}\left (e+f\,x\right )\,2{}\mathrm {i}\right )}{2}-a\,d\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}{f^2}+\frac {a\,\left (d\,x^2+2\,c\,x\right )}{2} \]
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