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 = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 8
Rule 2635
Rule 5372
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.56, size = 42, normalized size = 0.95 \[ \frac {b x \cosh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{2} - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 41, normalized size = 0.93 \[ \frac {{\left (2 \, b x - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{2}} + \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 53, normalized size = 1.20 \[ \frac {\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{2}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}-\frac {b x}{4}-\frac {a}{4}-\frac {a \left (\cosh ^{2}\left (b x +a \right )\right )}{2}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 46, normalized size = 1.05 \[ \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{16 \, b^{2}} + \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 28, normalized size = 0.64 \[ -\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )-2\,b\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{8\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 56, normalized size = 1.27 \[ \begin {cases} \frac {x \sinh ^{2}{\left (a + b x \right )}}{4 b} + \frac {x \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \sinh {\relax (a )} \cosh {\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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