Optimal. Leaf size=25 \[ \frac{\sinh (a+b x) \cosh (a+b x)}{2 b}-\frac{x}{2} \]
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Rubi [A] time = 0.0091, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ \frac{\sinh (a+b x) \cosh (a+b x)}{2 b}-\frac{x}{2} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sinh ^2(a+b x) \, dx &=\frac{\cosh (a+b x) \sinh (a+b x)}{2 b}-\frac{\int 1 \, dx}{2}\\ &=-\frac{x}{2}+\frac{\cosh (a+b x) \sinh (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.024184, size = 23, normalized size = 0.92 \[ \frac{\sinh (2 (a+b x))-2 (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 27, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ({\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2}}-{\frac{bx}{2}}-{\frac{a}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07968, size = 43, normalized size = 1.72 \begin{align*} -\frac{1}{2} \, x + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05974, size = 59, normalized size = 2.36 \begin{align*} -\frac{b x - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.26181, size = 46, normalized size = 1.84 \begin{align*} \begin{cases} \frac{x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac{x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac{\sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b} & \text{for}\: b \neq 0 \\x \sinh ^{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 [B] time = 1.32558, size = 65, normalized size = 2.6 \begin{align*} -\frac{4 \, b x -{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, a - e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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