Optimal. Leaf size=67 \[ -\frac {\sqrt {a+a \cos (x)}}{2 x^2}-\frac {1}{8} \sqrt {a+a \cos (x)} \text {CosIntegral}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {\sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )}{4 x} \]
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Rubi [A]
time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3400, 3378,
3383} \begin {gather*} -\frac {1}{8} \text {CosIntegral}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {\sqrt {a \cos (x)+a}}{2 x^2}+\frac {\tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3383
Rule 3400
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \cos (x)}}{x^3} \, dx &=\left (\sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\cos \left (\frac {x}{2}\right )}{x^3} \, dx\\ &=-\frac {\sqrt {a+a \cos (x)}}{2 x^2}-\frac {1}{4} \left (\sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x^2} \, dx\\ &=-\frac {\sqrt {a+a \cos (x)}}{2 x^2}+\frac {\sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )}{4 x}-\frac {1}{8} \left (\sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\cos \left (\frac {x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a+a \cos (x)}}{2 x^2}-\frac {1}{8} \sqrt {a+a \cos (x)} \text {Ci}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {\sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )}{4 x}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 44, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {a (1+\cos (x))} \left (4+x^2 \text {CosIntegral}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )-2 x \tan \left (\frac {x}{2}\right )\right )}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +a \cos \left (x \right )}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.54, size = 19, normalized size = 0.28 \begin {gather*} \frac {1}{8} \, \sqrt {2} \sqrt {a} {\left (\Gamma \left (-2, \frac {1}{2} i \, x\right ) + \Gamma \left (-2, -\frac {1}{2} i \, x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a \left (\cos {\left (x \right )} + 1\right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 48, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {2} {\left (x^{2} \operatorname {Ci}\left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 2 \, x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) + 4 \, \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a}}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+a\,\cos \left (x\right )}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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