Optimal. Leaf size=47 \[ -\frac{4}{15 \cos ^{\frac{3}{2}}(x)}+\frac{12 \sqrt{\cos (x)}}{5}+\frac{2 x \sin (x)}{5 \cos ^{\frac{5}{2}}(x)}+\frac{6 x \sin (x)}{5 \sqrt{\cos (x)}} \]
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Rubi [A] time = 0.0610706, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3315} \[ -\frac{4}{15 \cos ^{\frac{3}{2}}(x)}+\frac{12 \sqrt{\cos (x)}}{5}+\frac{2 x \sin (x)}{5 \cos ^{\frac{5}{2}}(x)}+\frac{6 x \sin (x)}{5 \sqrt{\cos (x)}} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\cos ^{\frac{7}{2}}(x)}+\frac{3}{5} x \sqrt{\cos (x)}\right ) \, dx &=\frac{3}{5} \int x \sqrt{\cos (x)} \, dx+\int \frac{x}{\cos ^{\frac{7}{2}}(x)} \, dx\\ &=-\frac{4}{15 \cos ^{\frac{3}{2}}(x)}+\frac{2 x \sin (x)}{5 \cos ^{\frac{5}{2}}(x)}+\frac{3}{5} \int \frac{x}{\cos ^{\frac{3}{2}}(x)} \, dx+\frac{3}{5} \int x \sqrt{\cos (x)} \, dx\\ &=-\frac{4}{15 \cos ^{\frac{3}{2}}(x)}+\frac{12 \sqrt{\cos (x)}}{5}+\frac{2 x \sin (x)}{5 \cos ^{\frac{5}{2}}(x)}+\frac{6 x \sin (x)}{5 \sqrt{\cos (x)}}\\ \end{align*}
Mathematica [A] time = 0.129401, size = 33, normalized size = 0.7 \[ \frac{21 x \sin (x)+9 x \sin (3 x)+46 \cos (x)+18 \cos (3 x)}{30 \cos ^{\frac{5}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.298, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cos \left ( x \right ) \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,x}{5}\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{3}{5} \, x \sqrt{\cos \left (x\right )} + \frac{x}{\cos \left (x\right )^{\frac{7}{2}}}\,{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3}{5} \, x \sqrt{\cos \left (x\right )} + \frac{x}{\cos \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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