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 = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 8
Rule 2635
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.47, size = 23, normalized size = 0.92 \[ -\frac {b x - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 32, normalized size = 1.28 \[ -\frac {1}{2} \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 1.08 \[ \frac {\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.61, size = 32, normalized size = 1.28 \[ -\frac {1}{2} \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 18, normalized size = 0.72 \[ \frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}-\frac {x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 46, normalized size = 1.84 \[ \begin {cases} \frac {x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \sinh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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