Optimal. Leaf size=55 \[ -\frac{d \cosh ^2(a+b x)}{4 b^2}+\frac{(c+d x) \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac{c x}{2}+\frac{d x^2}{4} \]
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Rubi [A] time = 0.0249827, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3310} \[ -\frac{d \cosh ^2(a+b x)}{4 b^2}+\frac{(c+d x) \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac{c x}{2}+\frac{d x^2}{4} \]
Antiderivative was successfully verified.
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Rule 3310
Rubi steps
\begin{align*} \int (c+d x) \cosh ^2(a+b x) \, dx &=-\frac{d \cosh ^2(a+b x)}{4 b^2}+\frac{(c+d x) \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac{1}{2} \int (c+d x) \, dx\\ &=\frac{c x}{2}+\frac{d x^2}{4}-\frac{d \cosh ^2(a+b x)}{4 b^2}+\frac{(c+d x) \cosh (a+b x) \sinh (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.15492, size = 51, normalized size = 0.93 \[ \frac{2 b ((c+d x) \sinh (2 (a+b x))+2 a c+b x (2 c+d x))-d \cosh (2 (a+b x))}{8 b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 103, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ({\frac{ \left ( bx+a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2}}+{\frac{ \left ( bx+a \right ) ^{2}}{4}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4}} \right ) }-{\frac{da}{b} \left ({\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) }+c \left ({\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06472, size = 119, normalized size = 2.16 \begin{align*} \frac{1}{16} \,{\left (4 \, x^{2} + \frac{{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} - \frac{{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} d + \frac{1}{8} \, c{\left (4 \, x + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{b} - \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82963, size = 163, normalized size = 2.96 \begin{align*} \frac{2 \, b^{2} d x^{2} + 4 \, b^{2} c x - d \cosh \left (b x + a\right )^{2} + 4 \,{\left (b d x + b c\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - d \sinh \left (b x + a\right )^{2}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.673685, size = 126, normalized size = 2.29 \begin{align*} \begin{cases} - \frac{c x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac{c x \cosh ^{2}{\left (a + b x \right )}}{2} - \frac{d x^{2} \sinh ^{2}{\left (a + b x \right )}}{4} + \frac{d x^{2} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac{c \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b} + \frac{d x \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b} - \frac{d \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \cosh ^{2}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25421, size = 85, normalized size = 1.55 \begin{align*} \frac{1}{4} \, d x^{2} + \frac{1}{2} \, c x + \frac{{\left (2 \, b d x + 2 \, b c - d\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{2}} - \frac{{\left (2 \, b d x + 2 \, b c + d\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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