Optimal. Leaf size=116 \[ \frac{a^2 (c+d x)^2}{2 d}+\frac{2 a b (c+d x) \cosh (e+f x)}{f}-\frac{2 a b d \sinh (e+f x)}{f^2}+\frac{b^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{2 f}-\frac{1}{2} b^2 c x-\frac{b^2 d \sinh ^2(e+f x)}{4 f^2}-\frac{1}{4} b^2 d x^2 \]
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Rubi [A] time = 0.105548, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3317, 3296, 2637, 3310} \[ \frac{a^2 (c+d x)^2}{2 d}+\frac{2 a b (c+d x) \cosh (e+f x)}{f}-\frac{2 a b d \sinh (e+f x)}{f^2}+\frac{b^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{2 f}-\frac{1}{2} b^2 c x-\frac{b^2 d \sinh ^2(e+f x)}{4 f^2}-\frac{1}{4} b^2 d x^2 \]
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+b \sinh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 a b (c+d x) \sinh (e+f x)+b^2 (c+d x) \sinh ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^2}{2 d}+(2 a b) \int (c+d x) \sinh (e+f x) \, dx+b^2 \int (c+d x) \sinh ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^2}{2 d}+\frac{2 a b (c+d x) \cosh (e+f x)}{f}+\frac{b^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac{b^2 d \sinh ^2(e+f x)}{4 f^2}-\frac{1}{2} b^2 \int (c+d x) \, dx-\frac{(2 a b d) \int \cosh (e+f x) \, dx}{f}\\ &=-\frac{1}{2} b^2 c x-\frac{1}{4} b^2 d x^2+\frac{a^2 (c+d x)^2}{2 d}+\frac{2 a b (c+d x) \cosh (e+f x)}{f}-\frac{2 a b d \sinh (e+f x)}{f^2}+\frac{b^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac{b^2 d \sinh ^2(e+f x)}{4 f^2}\\ \end{align*}
Mathematica [A] time = 0.745308, size = 98, normalized size = 0.84 \[ -\frac{2 \left (2 a^2-b^2\right ) (e+f x) (d (e-f x)-2 c f)-16 a b f (c+d x) \cosh (e+f x)+16 a b d \sinh (e+f x)-2 b^2 f (c+d x) \sinh (2 (e+f x))+b^2 d \cosh (2 (e+f x))}{8 f^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 208, normalized size = 1.8 \begin{align*}{\frac{1}{f} \left ({\frac{d{a}^{2} \left ( fx+e \right ) ^{2}}{2\,f}}+2\,{\frac{dab \left ( \left ( fx+e \right ) \cosh \left ( fx+e \right ) -\sinh \left ( fx+e \right ) \right ) }{f}}+{\frac{d{b}^{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}}-2\,{\frac{deab\cosh \left ( fx+e \right ) }{f}}-{\frac{de{b}^{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\,cab\cosh \left ( fx+e \right ) +c{b}^{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 = 1.25498, size = 221, normalized size = 1.91 \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 )} b^{2} d - \frac{1}{8} \, b^{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 + a b 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 \, a b 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 = 2.29991, size = 294, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (2 \, a^{2} - b^{2}\right )} d f^{2} x^{2} + 4 \,{\left (2 \, a^{2} - b^{2}\right )} c f^{2} x - b^{2} d \cosh \left (f x + e\right )^{2} - b^{2} d \sinh \left (f x + e\right )^{2} + 16 \,{\left (a b d f x + a b c f\right )} \cosh \left (f x + e\right ) - 4 \,{\left (4 \, a b d -{\left (b^{2} d f x + b^{2} c f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{8 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.935067, size = 219, normalized size = 1.89 \begin{align*} \begin{cases} a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{2 a b c \cosh{\left (e + f x \right )}}{f} + \frac{2 a b d x \cosh{\left (e + f x \right )}}{f} - \frac{2 a b d \sinh{\left (e + f x \right )}}{f^{2}} + \frac{b^{2} c x \sinh ^{2}{\left (e + f x \right )}}{2} - \frac{b^{2} c x \cosh ^{2}{\left (e + f x \right )}}{2} + \frac{b^{2} c \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{2 f} + \frac{b^{2} d x^{2} \sinh ^{2}{\left (e + f x \right )}}{4} - \frac{b^{2} d x^{2} \cosh ^{2}{\left (e + f x \right )}}{4} + \frac{b^{2} d x \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{2 f} - \frac{b^{2} d \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} & \text{for}\: f \neq 0 \\\left (a + b \sinh{\left (e \right )}\right )^{2} \left (c x + \frac{d x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23683, size = 220, normalized size = 1.9 \begin{align*} \frac{1}{2} \, a^{2} d x^{2} - \frac{1}{4} \, b^{2} d x^{2} + a^{2} c x - \frac{1}{2} \, b^{2} c x + \frac{{\left (2 \, b^{2} d f x + 2 \, b^{2} c f - b^{2} d\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{2}} + \frac{{\left (a b d f x + a b c f - a b d\right )} e^{\left (f x + e\right )}}{f^{2}} + \frac{{\left (a b d f x + a b c f + a b d\right )} e^{\left (-f x - e\right )}}{f^{2}} - \frac{{\left (2 \, b^{2} d f x + 2 \, b^{2} c f + b^{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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