Optimal. Leaf size=38 \[ \frac{4 \sqrt{\sin (e+f x)}}{f^2}-\frac{2 x \cos (e+f x)}{f \sqrt{\sin (e+f x)}} \]
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Rubi [A] time = 0.0613543, 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} \[ \frac{4 \sqrt{\sin (e+f x)}}{f^2}-\frac{2 x \cos (e+f x)}{f \sqrt{\sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\sin ^{\frac{3}{2}}(e+f x)}+x \sqrt{\sin (e+f x)}\right ) \, dx &=\int \frac{x}{\sin ^{\frac{3}{2}}(e+f x)} \, dx+\int x \sqrt{\sin (e+f x)} \, dx\\ &=-\frac{2 x \cos (e+f x)}{f \sqrt{\sin (e+f x)}}+\frac{4 \sqrt{\sin (e+f x)}}{f^2}\\ \end{align*}
Mathematica [A] time = 0.408233, size = 33, normalized size = 0.87 \[ \frac{4 \sin (e+f x)-2 f x \cos (e+f x)}{f^2 \sqrt{\sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}+x\sqrt{\sin \left ( fx+e \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{\sin \left (f x + e\right )} + \frac{x}{\sin \left (f x + e\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 (\sin ^{2}{\left (e + f x \right )} + 1\right )}{\sin ^{\frac{3}{2}}{\left (e + f 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{\sin \left (f x + e\right )} + \frac{x}{\sin \left (f x + e\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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