Optimal. Leaf size=42 \[ \frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}} \]
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Rubi [A] time = 0.123271, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4187, 4189} \[ \frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin{align*} \int \left (\frac{x}{\csc ^{\frac{3}{2}}(e+f x)}-\frac{1}{3} x \sqrt{\csc (e+f x)}\right ) \, dx &=-\left (\frac{1}{3} \int x \sqrt{\csc (e+f x)} \, dx\right )+\int \frac{x}{\csc ^{\frac{3}{2}}(e+f x)} \, dx\\ &=\frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}}+\frac{1}{3} \int x \sqrt{\csc (e+f x)} \, dx-\frac{1}{3} \left (\sqrt{\csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{x}{\sqrt{\sin (e+f x)}} \, dx\\ &=\frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.487129, size = 29, normalized size = 0.69 \[ -\frac{2 (3 f x \cot (e+f x)-2)}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \csc \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{3}\sqrt{\csc \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{1}{3} \, x \sqrt{\csc \left (f x + e\right )} + \frac{x}{\csc \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*} - \frac{\int - \frac{3 x}{\csc ^{\frac{3}{2}}{\left (e + f x \right )}}\, dx + \int x \sqrt{\csc{\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{1}{3} \, x \sqrt{\csc \left (f x + e\right )} + \frac{x}{\csc \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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