Optimal. Leaf size=109 \[ -\frac {a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x^2}-\frac {3}{16} a \sqrt {a+a \cos (x)} \text {CosIntegral}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )-\frac {9}{16} a \sqrt {a+a \cos (x)} \text {CosIntegral}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {3 a \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )}{2 x} \]
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Rubi [A]
time = 0.12, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3400, 3395,
3383, 3393} \begin {gather*} -\frac {3}{16} a \text {CosIntegral}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {9}{16} a \text {CosIntegral}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}}{x^2}+\frac {3 a \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 3393
Rule 3395
Rule 3400
Rubi steps
\begin {align*} \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx &=\left (2 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\cos ^3\left (\frac {x}{2}\right )}{x^3} \, dx\\ &=-\frac {a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x^2}+\frac {3 a \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )}{2 x}+\frac {1}{2} \left (3 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\cos \left (\frac {x}{2}\right )}{x} \, dx-\frac {1}{4} \left (9 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\cos ^3\left (\frac {x}{2}\right )}{x} \, dx\\ &=-\frac {a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x^2}+\frac {3}{2} a \sqrt {a+a \cos (x)} \text {Ci}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {3 a \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )}{2 x}-\frac {1}{4} \left (9 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \left (\frac {3 \cos \left (\frac {x}{2}\right )}{4 x}+\frac {\cos \left (\frac {3 x}{2}\right )}{4 x}\right ) \, dx\\ &=-\frac {a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x^2}+\frac {3}{2} a \sqrt {a+a \cos (x)} \text {Ci}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {3 a \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )}{2 x}-\frac {1}{16} \left (9 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\cos \left (\frac {3 x}{2}\right )}{x} \, dx-\frac {1}{16} \left (27 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \frac {\cos \left (\frac {x}{2}\right )}{x} \, dx\\ &=-\frac {a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x^2}-\frac {3}{16} a \sqrt {a+a \cos (x)} \text {Ci}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )-\frac {9}{16} a \sqrt {a+a \cos (x)} \text {Ci}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {3 a \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )}{2 x}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 66, normalized size = 0.61 \begin {gather*} -\frac {(a (1+\cos (x)))^{3/2} \left (16+3 x^2 \text {CosIntegral}\left (\frac {x}{2}\right ) \sec ^3\left (\frac {x}{2}\right )+9 x^2 \text {CosIntegral}\left (\frac {3 x}{2}\right ) \sec ^3\left (\frac {x}{2}\right )-24 x \tan \left (\frac {x}{2}\right )\right )}{32 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.53, size = 33, normalized size = 0.30 \begin {gather*} \frac {3}{16} \, \sqrt {2} a^{\frac {3}{2}} {\left (3 \, \Gamma \left (-2, \frac {3}{2} i \, x\right ) + \Gamma \left (-2, \frac {1}{2} i \, x\right ) + \Gamma \left (-2, -\frac {1}{2} i \, x\right ) + 3 \, \Gamma \left (-2, -\frac {3}{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 {\left (a \left (\cos {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 92, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {2} {\left (9 \, a x^{2} \operatorname {Ci}\left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 3 \, a x^{2} \operatorname {Ci}\left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 6 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {3}{2} \, x\right ) - 6 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) + 4 \, a \cos \left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 12 \, a \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a}}{16 \, 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 {{\left (a+a\,\cos \left (x\right )\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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