Optimal. Leaf size=44 \[ -\frac{\sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac{x \sinh ^2(a+b x)}{2 b}+\frac{x}{4 b} \]
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Rubi [A] time = 0.0234004, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5372, 2635, 8} \[ -\frac{\sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac{x \sinh ^2(a+b x)}{2 b}+\frac{x}{4 b} \]
Antiderivative was successfully verified.
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Rule 5372
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x \cosh (a+b x) \sinh (a+b x) \, dx &=\frac{x \sinh ^2(a+b x)}{2 b}-\frac{\int \sinh ^2(a+b x) \, dx}{2 b}\\ &=-\frac{\cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{x \sinh ^2(a+b x)}{2 b}+\frac{\int 1 \, dx}{4 b}\\ &=\frac{x}{4 b}-\frac{\cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac{x \sinh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0549291, size = 28, normalized size = 0.64 \[ -\frac{\sinh (2 (a+b x))-2 b x \cosh (2 (a+b x))}{8 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 53, normalized size = 1.2 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2}}-{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{4}}-{\frac{bx}{4}}-{\frac{a}{4}}-{\frac{a \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19771, size = 62, normalized size = 1.41 \begin{align*} \frac{{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{16 \, b^{2}} + \frac{{\left (2 \, b x + 1\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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Fricas [A] time = 1.99045, size = 112, normalized size = 2.55 \begin{align*} \frac{b x \cosh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{2} - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.610406, size = 56, normalized size = 1.27 \begin{align*} \begin{cases} \frac{x \sinh ^{2}{\left (a + b x \right )}}{4 b} + \frac{x \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac{\sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh{\left (a \right )} \cosh{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16262, size = 55, normalized size = 1.25 \begin{align*} \frac{{\left (2 \, b x - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{2}} + \frac{{\left (2 \, b x + 1\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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