Optimal. Leaf size=42 \[ -\frac{4}{3 f^2 \sqrt{\sin (e+f x)}}-\frac{2 x \cos (e+f x)}{3 f \sin ^{\frac{3}{2}}(e+f x)} \]
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Rubi [A] time = 0.0603062, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {3315} \[ -\frac{4}{3 f^2 \sqrt{\sin (e+f x)}}-\frac{2 x \cos (e+f x)}{3 f \sin ^{\frac{3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\sin ^{\frac{5}{2}}(e+f x)}-\frac{x}{3 \sqrt{\sin (e+f x)}}\right ) \, dx &=-\left (\frac{1}{3} \int \frac{x}{\sqrt{\sin (e+f x)}} \, dx\right )+\int \frac{x}{\sin ^{\frac{5}{2}}(e+f x)} \, dx\\ &=-\frac{2 x \cos (e+f x)}{3 f \sin ^{\frac{3}{2}}(e+f x)}-\frac{4}{3 f^2 \sqrt{\sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.404757, size = 35, normalized size = 0.83 \[ -\frac{2 (2 \sin (e+f x)+f x \cos (e+f x))}{3 f^2 \sin ^{\frac{3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{x}{3}{\frac{1}{\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 -\frac{x}{3 \, \sqrt{\sin \left (f x + e\right )}} + \frac{x}{\sin \left (f x + e\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 = 1.72272, size = 117, normalized size = 2.79 \begin{align*} \frac{2 \,{\left (f x \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sqrt{\sin \left (f x + e\right )}}{3 \,{\left (f^{2} \cos \left (f x + e\right )^{2} - f^{2}\right )}} \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*} - \frac{\int - \frac{3 x}{\sin ^{\frac{5}{2}}{\left (e + f x \right )}}\, dx + \int \frac{x}{\sqrt{\sin{\left (e + f x \right )}}}\, dx}{3} \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{\sin \left (f x + e\right )}} + \frac{x}{\sin \left (f x + e\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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