Optimal. Leaf size=88 \[ -\frac{8 x \sqrt{a \cosh (c+d x)+a}}{d^2}+\frac{16 \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d^3}+\frac{2 x^2 \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d} \]
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Rubi [A] time = 0.113328, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3319, 3296, 2637} \[ -\frac{8 x \sqrt{a \cosh (c+d x)+a}}{d^2}+\frac{16 \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d^3}+\frac{2 x^2 \tanh \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \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^2 \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^2 \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}-\frac{\left (4 \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 \sinh \left (\frac{c}{2}+\frac{d x}{2}\right ) \, dx}{d}\\ &=-\frac{8 x \sqrt{a+a \cosh (c+d x)}}{d^2}+\frac{2 x^2 \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}+\frac{\left (8 \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 \cosh \left (\frac{c}{2}+\frac{d x}{2}\right ) \, dx}{d^2}\\ &=-\frac{8 x \sqrt{a+a \cosh (c+d x)}}{d^2}+\frac{16 \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d^3}+\frac{2 x^2 \sqrt{a+a \cosh (c+d x)} \tanh \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.149346, size = 44, normalized size = 0.5 \[ \frac{2 \left (\left (d^2 x^2+8\right ) \tanh \left (\frac{1}{2} (c+d x)\right )-4 d x\right ) \sqrt{a (\cosh (c+d x)+1)}}{d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 86, normalized size = 1. \begin{align*}{\frac{\sqrt{2} \left ({d}^{2}{x}^{2}{{\rm e}^{dx+c}}-{d}^{2}{x}^{2}-4\,dx{{\rm e}^{dx+c}}-4\,dx+8\,{{\rm e}^{dx+c}}-8 \right ) }{ \left ({{\rm e}^{dx+c}}+1 \right ){d}^{3}}\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 = 1.65905, size = 122, normalized size = 1.39 \begin{align*} -\frac{{\left (\sqrt{2} \sqrt{a} d^{2} x^{2} + 4 \, \sqrt{2} \sqrt{a} d x -{\left (\sqrt{2} \sqrt{a} d^{2} x^{2} e^{c} - 4 \, \sqrt{2} \sqrt{a} d x e^{c} + 8 \, \sqrt{2} \sqrt{a} e^{c}\right )} e^{\left (d x\right )} + 8 \, \sqrt{2} \sqrt{a}\right )} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d^{3}} \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^{2} \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 = 1.33322, size = 144, normalized size = 1.64 \begin{align*} \frac{\sqrt{2}{\left (\sqrt{a} d^{2} x^{2} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \sqrt{a} d^{2} x^{2} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} - 4 \, \sqrt{a} d x e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - 4 \, \sqrt{a} d x e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} + 8 \, \sqrt{a} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - 8 \, \sqrt{a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}\right )}}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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