Optimal. Leaf size=50 \[ \frac{i a (c+d x) \cosh (e+f x)}{f}+\frac{a (c+d x)^2}{2 d}-\frac{i a d \sinh (e+f x)}{f^2} \]
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Rubi [A] time = 0.0533599, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3317, 3296, 2637} \[ \frac{i a (c+d x) \cosh (e+f x)}{f}+\frac{a (c+d x)^2}{2 d}-\frac{i a d \sinh (e+f x)}{f^2} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int (c+d x) (a+i a \sinh (e+f x)) \, dx &=\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*}
Mathematica [A] time = 0.0842476, size = 48, normalized size = 0.96 \[ \frac{a \left (2 i f (c+d x) \cosh (e+f x)+f^2 x (2 c+d x)-2 i d \sinh (e+f x)\right )}{2 f^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 96, normalized size = 1.9 \begin{align*}{\frac{1}{f} \left ({\frac{da \left ( fx+e \right ) ^{2}}{2\,f}}+{\frac{ida \left ( \left ( fx+e \right ) \cosh \left ( fx+e \right ) -\sinh \left ( fx+e \right ) \right ) }{f}}-{\frac{dea \left ( fx+e \right ) }{f}}-{\frac{idea\cosh \left ( fx+e \right ) }{f}}+ca \left ( fx+e \right ) +ica\cosh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04286, size = 89, normalized size = 1.78 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.63207, size = 192, normalized size = 3.84 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.943545, size = 177, normalized size = 3.54 \begin{align*} a c x + \frac{a d x^{2}}{2} + \begin{cases} \frac{\left (\left (2 i a c f^{5} e^{e} + 2 i a d f^{5} x e^{e} + 2 i a d f^{4} e^{e}\right ) e^{- f x} + \left (2 i a c f^{5} e^{3 e} + 2 i a d f^{5} x e^{3 e} - 2 i a d f^{4} e^{3 e}\right ) e^{f x}\right ) e^{- 2 e}}{4 f^{6}} & \text{for}\: 4 f^{6} e^{2 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29728, size = 96, normalized size = 1.92 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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