Optimal. Leaf size=122 \[ \frac{2 i a^2 (c+d x) \cosh (e+f x)}{f}-\frac{a^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac{a^2 (c+d x)^2}{2 d}+\frac{1}{2} a^2 c x+\frac{a^2 d \sinh ^2(e+f x)}{4 f^2}-\frac{2 i a^2 d \sinh (e+f x)}{f^2}+\frac{1}{4} a^2 d x^2 \]
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Rubi [A] time = 0.105015, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3317, 3296, 2637, 3310} \[ \frac{2 i a^2 (c+d x) \cosh (e+f x)}{f}-\frac{a^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac{a^2 (c+d x)^2}{2 d}+\frac{1}{2} a^2 c x+\frac{a^2 d \sinh ^2(e+f x)}{4 f^2}-\frac{2 i a^2 d \sinh (e+f x)}{f^2}+\frac{1}{4} a^2 d x^2 \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2637
Rule 3310
Rubi steps
\begin{align*} \int (c+d x) (a+i a \sinh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 i a^2 (c+d x) \sinh (e+f x)-a^2 (c+d x) \sinh ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^2}{2 d}+\left (2 i a^2\right ) \int (c+d x) \sinh (e+f x) \, dx-a^2 \int (c+d x) \sinh ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^2}{2 d}+\frac{2 i a^2 (c+d x) \cosh (e+f x)}{f}-\frac{a^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac{a^2 d \sinh ^2(e+f x)}{4 f^2}+\frac{1}{2} a^2 \int (c+d x) \, dx-\frac{\left (2 i a^2 d\right ) \int \cosh (e+f x) \, dx}{f}\\ &=\frac{1}{2} a^2 c x+\frac{1}{4} a^2 d x^2+\frac{a^2 (c+d x)^2}{2 d}+\frac{2 i a^2 (c+d x) \cosh (e+f x)}{f}-\frac{2 i a^2 d \sinh (e+f x)}{f^2}-\frac{a^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac{a^2 d \sinh ^2(e+f x)}{4 f^2}\\ \end{align*}
Mathematica [A] time = 1.20878, size = 86, normalized size = 0.7 \[ \frac{a^2 (-2 (3 (e+f x) (-2 c f+d e-d f x)+f (c+d x) \sinh (2 (e+f x))+8 i d \sinh (e+f x))+16 i f (c+d x) \cosh (e+f x)+d \cosh (2 (e+f x)))}{8 f^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 215, normalized size = 1.8 \begin{align*}{\frac{1}{f} \left ({\frac{d{a}^{2} \left ( fx+e \right ) ^{2}}{2\,f}}+{\frac{2\,id{a}^{2} \left ( \left ( fx+e \right ) \cosh \left ( fx+e \right ) -\sinh \left ( fx+e \right ) \right ) }{f}}-{\frac{d{a}^{2}}{f} \left ({\frac{ \left ( fx+e \right ) \cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}-{\frac{ \left ( fx+e \right ) ^{2}}{4}}-{\frac{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{4}} \right ) }-{\frac{de{a}^{2} \left ( fx+e \right ) }{f}}-{\frac{2\,ide{a}^{2}\cosh \left ( fx+e \right ) }{f}}+{\frac{de{a}^{2}}{f} \left ({\frac{\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}-{\frac{fx}{2}}-{\frac{e}{2}} \right ) }+c{a}^{2} \left ( fx+e \right ) +2\,ic{a}^{2}\cosh \left ( fx+e \right ) -c{a}^{2} \left ({\frac{\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}-{\frac{fx}{2}}-{\frac{e}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18573, size = 225, normalized size = 1.84 \begin{align*} \frac{1}{2} \, a^{2} d x^{2} + \frac{1}{16} \,{\left (4 \, x^{2} - \frac{{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} + \frac{{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} a^{2} d + \frac{1}{8} \, a^{2} c{\left (4 \, x - \frac{e^{\left (2 \, f x + 2 \, e\right )}}{f} + \frac{e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c x + i \, a^{2} 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{2 i \, a^{2} 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 = 2.9555, size = 389, normalized size = 3.19 \begin{align*} \frac{{\left (2 \, a^{2} d f x + 2 \, a^{2} c f + a^{2} d -{\left (2 \, a^{2} d f x + 2 \, a^{2} c f - a^{2} d\right )} e^{\left (4 \, f x + 4 \, e\right )} +{\left (16 i \, a^{2} d f x + 16 i \, a^{2} c f - 16 i \, a^{2} d\right )} e^{\left (3 \, f x + 3 \, e\right )} + 12 \,{\left (a^{2} d f^{2} x^{2} + 2 \, a^{2} c f^{2} x\right )} e^{\left (2 \, f x + 2 \, e\right )} +{\left (16 i \, a^{2} d f x + 16 i \, a^{2} c f + 16 i \, a^{2} d\right )} e^{\left (f x + e\right )}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.06403, size = 364, normalized size = 2.98 \begin{align*} \frac{3 a^{2} c x}{2} + \frac{3 a^{2} d x^{2}}{4} + \begin{cases} \frac{\left (\left (32 a^{2} c f^{9} e^{2 e} + 32 a^{2} d f^{9} x e^{2 e} + 16 a^{2} d f^{8} e^{2 e}\right ) e^{- 2 f x} + \left (- 32 a^{2} c f^{9} e^{6 e} - 32 a^{2} d f^{9} x e^{6 e} + 16 a^{2} d f^{8} e^{6 e}\right ) e^{2 f x} + \left (256 i a^{2} c f^{9} e^{3 e} + 256 i a^{2} d f^{9} x e^{3 e} + 256 i a^{2} d f^{8} e^{3 e}\right ) e^{- f x} + \left (256 i a^{2} c f^{9} e^{5 e} + 256 i a^{2} d f^{9} x e^{5 e} - 256 i a^{2} d f^{8} e^{5 e}\right ) e^{f x}\right ) e^{- 4 e}}{256 f^{10}} & \text{for}\: 256 f^{10} e^{4 e} \neq 0 \\\frac{x^{2} \left (- a^{2} d e^{4 e} + 4 i a^{2} d e^{3 e} - 4 i a^{2} d e^{e} - a^{2} d\right ) e^{- 2 e}}{8} + \frac{x \left (- a^{2} c e^{4 e} + 4 i a^{2} c e^{3 e} - 4 i a^{2} c e^{e} - a^{2} c\right ) e^{- 2 e}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24014, size = 215, normalized size = 1.76 \begin{align*} \frac{3}{4} \, a^{2} d x^{2} + \frac{3}{2} \, a^{2} c x - \frac{{\left (2 \, a^{2} d f x + 2 \, a^{2} c f - a^{2} d\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{2}} + \frac{{\left (i \, a^{2} d f x + i \, a^{2} c f - i \, a^{2} d\right )} e^{\left (f x + e\right )}}{f^{2}} + \frac{{\left (i \, a^{2} d f x + i \, a^{2} c f + i \, a^{2} d\right )} e^{\left (-f x - e\right )}}{f^{2}} + \frac{{\left (2 \, a^{2} d f x + 2 \, a^{2} c f + a^{2} d\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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