Optimal. Leaf size=83 \[ \frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}+\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}} \]
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Rubi [A] time = 0.133373, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4187, 4189} \[ \frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}+\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin{align*} \int \left (\frac{x}{\csc ^{\frac{7}{2}}(e+f x)}-\frac{5}{21} x \sqrt{\csc (e+f x)}\right ) \, dx &=-\left (\frac{5}{21} \int x \sqrt{\csc (e+f x)} \, dx\right )+\int \frac{x}{\csc ^{\frac{7}{2}}(e+f x)} \, dx\\ &=\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}+\frac{5}{7} \int \frac{x}{\csc ^{\frac{3}{2}}(e+f x)} \, dx-\frac{1}{21} \left (5 \sqrt{\csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{x}{\sqrt{\sin (e+f x)}} \, dx\\ &=\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}+\frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}}+\frac{5}{21} \int x \sqrt{\csc (e+f x)} \, dx-\frac{1}{21} \left (5 \sqrt{\csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{x}{\sqrt{\sin (e+f x)}} \, dx\\ &=\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}+\frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.22757, size = 57, normalized size = 0.69 \[ \frac{-36 \cos (2 (e+f x))-483 f x \cot (e+f x)+63 f x \cos (3 (e+f x)) \csc (e+f x)+316}{882 f^2 \csc ^{\frac{3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \csc \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,x}{21}\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{5}{21} \, x \sqrt{\csc \left (f x + e\right )} + \frac{x}{\csc \left (f x + e\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{5}{21} \, x \sqrt{\csc \left (f x + e\right )} + \frac{x}{\csc \left (f x + e\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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