Optimal. Leaf size=24 \[ \frac{2 x \sin (x)}{3 \cos ^{\frac{3}{2}}(x)}-\frac{4}{3 \sqrt{\cos (x)}} \]
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Rubi [A] time = 0.0459401, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3315} \[ \frac{2 x \sin (x)}{3 \cos ^{\frac{3}{2}}(x)}-\frac{4}{3 \sqrt{\cos (x)}} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\cos ^{\frac{5}{2}}(x)}-\frac{x}{3 \sqrt{\cos (x)}}\right ) \, dx &=-\left (\frac{1}{3} \int \frac{x}{\sqrt{\cos (x)}} \, dx\right )+\int \frac{x}{\cos ^{\frac{5}{2}}(x)} \, dx\\ &=-\frac{4}{3 \sqrt{\cos (x)}}+\frac{2 x \sin (x)}{3 \cos ^{\frac{3}{2}}(x)}\\ \end{align*}
Mathematica [A] time = 0.0751972, size = 17, normalized size = 0.71 \[ -\frac{8-4 x \tan (x)}{6 \sqrt{\cos (x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.213, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cos \left ( x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{x}{3}{\frac{1}{\sqrt{\cos \left ( x \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x}{3 \, \sqrt{\cos \left (x\right )}} + \frac{x}{\cos \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59847, size = 54, normalized size = 2.25 \begin{align*} \frac{2 \,{\left (x \sin \left (x\right ) - 2 \, \cos \left (x\right )\right )}}{3 \, \cos \left (x\right )^{\frac{3}{2}}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x}{3 \, \sqrt{\cos \left (x\right )}} + \frac{x}{\cos \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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