Optimal. Leaf size=83 \[ -\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}+\frac{12 \sqrt{\sin (e+f x)}}{5 f^2}-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{6 x \cos (e+f x)}{5 f \sqrt{\sin (e+f x)}} \]
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Rubi [A] time = 0.0858376, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {3315} \[ -\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}+\frac{12 \sqrt{\sin (e+f x)}}{5 f^2}-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{6 x \cos (e+f x)}{5 f \sqrt{\sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\sin ^{\frac{7}{2}}(e+f x)}+\frac{3}{5} x \sqrt{\sin (e+f x)}\right ) \, dx &=\frac{3}{5} \int x \sqrt{\sin (e+f x)} \, dx+\int \frac{x}{\sin ^{\frac{7}{2}}(e+f x)} \, dx\\ &=-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}+\frac{3}{5} \int \frac{x}{\sin ^{\frac{3}{2}}(e+f x)} \, dx+\frac{3}{5} \int x \sqrt{\sin (e+f x)} \, dx\\ &=-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}-\frac{6 x \cos (e+f x)}{5 f \sqrt{\sin (e+f x)}}+\frac{12 \sqrt{\sin (e+f x)}}{5 f^2}\\ \end{align*}
Mathematica [A] time = 0.585283, size = 58, normalized size = 0.7 \[ \frac{46 \sin (e+f x)-18 \sin (3 (e+f x))-21 f x \cos (e+f x)+9 f x \cos (3 (e+f x))}{30 f^2 \sin ^{\frac{5}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,x}{5}\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{3}{5} \, x \sqrt{\sin \left (f x + e\right )} + \frac{x}{\sin \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{3}{5} \, x \sqrt{\sin \left (f x + e\right )} + \frac{x}{\sin \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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