Optimal. Leaf size=88 \[ -\frac {16 \tan \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a}}{d^3}+\frac {8 x \sqrt {a \cos (c+d x)+a}}{d^2}+\frac {2 x^2 \tan \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a}}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 88, normalized size of antiderivative = 1.00, 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} \[ \frac {8 x \sqrt {a \cos (c+d x)+a}}{d^2}-\frac {16 \tan \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a}}{d^3}+\frac {2 x^2 \tan \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3319
Rubi steps
\begin {align*} \int x^2 \sqrt {a+a \cos (c+d x)} \, dx &=\left (\sqrt {a+a \cos (c+d x)} \csc \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x^2 \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx\\ &=\frac {2 x^2 \sqrt {a+a \cos (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {\left (4 \sqrt {a+a \cos (c+d x)} \csc \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x \sin \left (\frac {c}{2}+\frac {d x}{2}\right ) \, dx}{d}\\ &=\frac {8 x \sqrt {a+a \cos (c+d x)}}{d^2}+\frac {2 x^2 \sqrt {a+a \cos (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {\left (8 \sqrt {a+a \cos (c+d x)} \csc \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \, dx}{d^2}\\ &=\frac {8 x \sqrt {a+a \cos (c+d x)}}{d^2}-\frac {16 \sqrt {a+a \cos (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3}+\frac {2 x^2 \sqrt {a+a \cos (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 44, normalized size = 0.50 \[ \frac {2 \left (\left (d^2 x^2-8\right ) \tan \left (\frac {1}{2} (c+d x)\right )+4 d x\right ) \sqrt {a (\cos (c+d x)+1)}}{d^3} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 77, normalized size = 0.88 \[ 2 \, \sqrt {2} \sqrt {a} {\left (\frac {4 \, x \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d^{2}} + \frac {{\left (d^{2} x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 8 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 105, normalized size = 1.19 \[ -\frac {i \sqrt {2}\, \sqrt {a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2} {\mathrm e}^{-i \left (d x +c \right )}}\, \left (d^{2} x^{2} {\mathrm e}^{i \left (d x +c \right )}+4 i d x \,{\mathrm e}^{i \left (d x +c \right )}-d^{2} x^{2}+4 i d x -8 \,{\mathrm e}^{i \left (d x +c \right )}+8\right )}{\left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.56, size = 122, normalized size = 1.39 \[ \frac {2 \, {\left (\sqrt {2} \sqrt {a} c^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, {\left (\sqrt {2} {\left (d x + c\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a} c + {\left (\sqrt {2} {\left (d x + c\right )}^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, \sqrt {2} {\left (d x + c\right )} \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}\right )}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 63, normalized size = 0.72 \[ \frac {2\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}\,\left (4\,d\,x-8\,\sin \left (c+d\,x\right )+d^2\,x^2\,\sin \left (c+d\,x\right )+4\,d\,x\,\cos \left (c+d\,x\right )\right )}{d^3\,\left (\cos \left (c+d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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