Optimal. Leaf size=53 \[ \frac{2 x \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d}-\frac{4 \sqrt{a \cosh (c+d x)+a}}{d^2} \]
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Rubi [A] time = 0.0621659, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3319, 3296, 2638} \[ \frac{2 x \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d}-\frac{4 \sqrt{a \cosh (c+d x)+a}}{d^2} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x \sqrt{a+a \cosh (c+d x)} \, dx &=\left (\sqrt{a+a \cosh (c+d x)} \csc \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int x \sin \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right ) \, dx\\ &=\frac{2 x \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}-\frac{\left (2 \sqrt{a+a \cosh (c+d x)} \csc \left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{i d x}{2}\right )\right ) \int \sinh \left (\frac{c}{2}+\frac{d x}{2}\right ) \, dx}{d}\\ &=-\frac{4 \sqrt{a+a \cosh (c+d x)}}{d^2}+\frac{2 x \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.119905, size = 34, normalized size = 0.64 \[ \frac{2 \left (d x \tanh \left (\frac{1}{2} (c+d x)\right )-2\right ) \sqrt{a (\cosh (c+d x)+1)}}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 64, normalized size = 1.2 \begin{align*}{\frac{\sqrt{2} \left ( dx{{\rm e}^{dx+c}}-dx-2\,{{\rm e}^{dx+c}}-2 \right ) }{ \left ({{\rm e}^{dx+c}}+1 \right ){d}^{2}}\sqrt{a \left ({{\rm e}^{dx+c}}+1 \right ) ^{2}{{\rm e}^{-dx-c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76643, size = 81, normalized size = 1.53 \begin{align*} -\frac{{\left (\sqrt{2} \sqrt{a} d x -{\left (\sqrt{2} \sqrt{a} d x e^{c} - 2 \, \sqrt{2} \sqrt{a} e^{c}\right )} e^{\left (d x\right )} + 2 \, \sqrt{2} \sqrt{a}\right )} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{a \left (\cosh{\left (c + d x \right )} + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29389, size = 90, normalized size = 1.7 \begin{align*} \frac{\sqrt{2}{\left (\sqrt{a} d x e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \sqrt{a} d x e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} - 2 \, \sqrt{a} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - 2 \, \sqrt{a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}\right )}}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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