Optimal. Leaf size=55 \[ \frac {3}{2} a \sqrt {a+a \cos (x)} \text {CosIntegral}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {1}{2} a \sqrt {a+a \cos (x)} \text {CosIntegral}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 = {3400, 3393,
3383} \begin {gather*} \frac {3}{2} a \text {CosIntegral}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {1}{2} a \text {CosIntegral}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 3393
Rule 3400
Rubi steps
\begin {align*} \int \frac {(a+a \cos (x))^{3/2}}{x} \, 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} \, dx\\ &=\left (2 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 {1}{2} \left (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}{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 {3}{2} a \sqrt {a+a \cos (x)} \text {Ci}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {1}{2} a \sqrt {a+a \cos (x)} \text {Ci}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.65 \begin {gather*} \frac {1}{2} a \sqrt {a (1+\cos (x))} \left (3 \text {CosIntegral}\left (\frac {x}{2}\right )+\text {CosIntegral}\left (\frac {3 x}{2}\right )\right ) \sec \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}}{x}\, dx\]
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 = 29, normalized size = 0.53 \begin {gather*} \frac {1}{4} \, \sqrt {2} a^{\frac {3}{2}} {\left ({\rm Ei}\left (\frac {3}{2} i \, x\right ) + 3 \, {\rm Ei}\left (\frac {1}{2} i \, x\right ) + 3 \, {\rm Ei}\left (-\frac {1}{2} i \, x\right ) + {\rm Ei}\left (-\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}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 32, normalized size = 0.58 \begin {gather*} \frac {1}{2} \, \sqrt {2} {\left (a \operatorname {Ci}\left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 3 \, a \operatorname {Ci}\left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \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 = -0.02 \begin {gather*} \int \frac {{\left (a+a\,\cos \left (x\right )\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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