Optimal. Leaf size=22 \[ -\frac{x}{2}+\frac{\sinh ^2(x)}{2}+\frac{1}{2} \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.0342234, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3089, 2564, 30, 2635, 8} \[ -\frac{x}{2}+\frac{\sinh ^2(x)}{2}+\frac{1}{2} \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 3089
Rule 2564
Rule 30
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sinh (x) (\cosh (x)+\sinh (x)) \, dx &=-\left (i \int \left (i \cosh (x) \sinh (x)+i \sinh ^2(x)\right ) \, dx\right )\\ &=\int \cosh (x) \sinh (x) \, dx+\int \sinh ^2(x) \, dx\\ &=\frac{1}{2} \cosh (x) \sinh (x)-\frac{\int 1 \, dx}{2}-\operatorname{Subst}(\int x \, dx,x,i \sinh (x))\\ &=-\frac{x}{2}+\frac{1}{2} \cosh (x) \sinh (x)+\frac{\sinh ^2(x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0041325, size = 22, normalized size = 1. \[ -\frac{x}{2}+\frac{1}{4} \sinh (2 x)+\frac{\cosh ^2(x)}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 17, normalized size = 0.8 \begin{align*}{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}}-{\frac{x}{2}}+{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01952, size = 30, normalized size = 1.36 \begin{align*} \frac{1}{2} \, \cosh \left (x\right )^{2} - \frac{1}{2} \, x + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03138, size = 89, normalized size = 4.05 \begin{align*} -\frac{{\left (2 \, x - 1\right )} \cosh \left (x\right ) -{\left (2 \, x + 1\right )} \sinh \left (x\right )}{4 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.21644, size = 31, normalized size = 1.41 \begin{align*} \frac{x \sinh ^{2}{\left (x \right )}}{2} - \frac{x \cosh ^{2}{\left (x \right )}}{2} + \frac{\sinh{\left (x \right )} \cosh{\left (x \right )}}{2} + \frac{\cosh ^{2}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14004, size = 14, normalized size = 0.64 \begin{align*} -\frac{1}{2} \, x + \frac{1}{4} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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