Optimal. Leaf size=68 \[ 12 x^2 \sqrt{a \cos (x)+a}+2 x^3 \tan \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-96 \sqrt{a \cos (x)+a}-48 x \tan \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a} \]
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Rubi [A] time = 0.110838, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3319, 3296, 2638} \[ 12 x^2 \sqrt{a \cos (x)+a}+2 x^3 \tan \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a}-96 \sqrt{a \cos (x)+a}-48 x \tan \left (\frac{x}{2}\right ) \sqrt{a \cos (x)+a} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^3 \sqrt{a+a \cos (x)} \, dx &=\left (\sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int x^3 \cos \left (\frac{x}{2}\right ) \, dx\\ &=2 x^3 \sqrt{a+a \cos (x)} \tan \left (\frac{x}{2}\right )-\left (6 \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int x^2 \sin \left (\frac{x}{2}\right ) \, dx\\ &=12 x^2 \sqrt{a+a \cos (x)}+2 x^3 \sqrt{a+a \cos (x)} \tan \left (\frac{x}{2}\right )-\left (24 \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int x \cos \left (\frac{x}{2}\right ) \, dx\\ &=12 x^2 \sqrt{a+a \cos (x)}-48 x \sqrt{a+a \cos (x)} \tan \left (\frac{x}{2}\right )+2 x^3 \sqrt{a+a \cos (x)} \tan \left (\frac{x}{2}\right )+\left (48 \sqrt{a+a \cos (x)} \sec \left (\frac{x}{2}\right )\right ) \int \sin \left (\frac{x}{2}\right ) \, dx\\ &=-96 \sqrt{a+a \cos (x)}+12 x^2 \sqrt{a+a \cos (x)}-48 x \sqrt{a+a \cos (x)} \tan \left (\frac{x}{2}\right )+2 x^3 \sqrt{a+a \cos (x)} \tan \left (\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0550919, size = 33, normalized size = 0.49 \[ 2 \left (6 \left (x^2-8\right )+x \left (x^2-24\right ) \tan \left (\frac{x}{2}\right )\right ) \sqrt{a (\cos (x)+1)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.214, size = 87, normalized size = 1.3 \begin{align*}{\frac{-i\sqrt{2} \left ( 6\,i{x}^{2}{{\rm e}^{ix}}+{x}^{3}{{\rm e}^{ix}}+6\,i{x}^{2}-{x}^{3}-48\,i{{\rm e}^{ix}}-24\,x{{\rm e}^{ix}}-48\,i+24\,x \right ) }{{{\rm e}^{ix}}+1}\sqrt{a \left ({{\rm e}^{ix}}+1 \right ) ^{2}{{\rm e}^{-ix}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.33819, size = 65, normalized size = 0.96 \begin{align*} 2 \,{\left (\sqrt{2} x^{3} \sin \left (\frac{1}{2} \, x\right ) + 6 \, \sqrt{2} x^{2} \cos \left (\frac{1}{2} \, x\right ) - 24 \, \sqrt{2} x \sin \left (\frac{1}{2} \, x\right ) - 48 \, \sqrt{2} \cos \left (\frac{1}{2} \, x\right )\right )} \sqrt{a} \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^{3} \sqrt{a \left (\cos{\left (x \right )} + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cos \left (x\right ) + a} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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