Optimal. Leaf size=61 \[ -\frac{9}{8} \text{Unintegrable}\left (\frac{\cos ^{\frac{3}{2}}(x)}{x},x\right )+\frac{3}{8} \text{Unintegrable}\left (\frac{1}{x \sqrt{\cos (x)}},x\right )-\frac{\cos ^{\frac{3}{2}}(x)}{2 x^2}+\frac{3 \sin (x) \sqrt{\cos (x)}}{4 x} \]
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Rubi [A] time = 0.0768357, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx &=-\frac{\cos ^{\frac{3}{2}}(x)}{2 x^2}+\frac{3 \sqrt{\cos (x)} \sin (x)}{4 x}+\frac{3}{8} \int \frac{1}{x \sqrt{\cos (x)}} \, dx-\frac{9}{8} \int \frac{\cos ^{\frac{3}{2}}(x)}{x} \, dx\\ \end{align*}
Mathematica [A] time = 5.12425, size = 0, normalized size = 0. \[ \int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( \cos \left ( x \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (x\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (x\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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