Optimal. Leaf size=38 \[ \frac{4 \sqrt{\cos (a+b x)}}{b^2}+\frac{2 x \sin (a+b x)}{b \sqrt{\cos (a+b x)}} \]
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Rubi [A] time = 0.055347, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {3315} \[ \frac{4 \sqrt{\cos (a+b x)}}{b^2}+\frac{2 x \sin (a+b x)}{b \sqrt{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\cos ^{\frac{3}{2}}(a+b x)}+x \sqrt{\cos (a+b x)}\right ) \, dx &=\int \frac{x}{\cos ^{\frac{3}{2}}(a+b x)} \, dx+\int x \sqrt{\cos (a+b x)} \, dx\\ &=\frac{4 \sqrt{\cos (a+b x)}}{b^2}+\frac{2 x \sin (a+b x)}{b \sqrt{\cos (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.387592, size = 33, normalized size = 0.87 \[ \frac{2 (b x \sin (a+b x)+2 \cos (a+b x))}{b^2 \sqrt{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.22, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cos \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}+x\sqrt{\cos \left ( bx+a \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 x \sqrt{\cos \left (b x + a\right )} + \frac{x}{\cos \left (b x + a\right )^{\frac{3}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (\cos ^{2}{\left (a + b x \right )} + 1\right )}{\cos ^{\frac{3}{2}}{\left (a + b x \right )}}\, dx \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 x \sqrt{\cos \left (b x + a\right )} + \frac{x}{\cos \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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