Optimal. Leaf size=221 \[ \frac{3}{2} a \sin \left (\frac{1}{4} (2 e+\pi )\right ) \text{CosIntegral}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{1}{2} a \cos \left (\frac{3}{4} (2 e-\pi )\right ) \text{CosIntegral}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{1}{2} a \sin \left (\frac{3}{4} (2 e-\pi )\right ) \text{Si}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{3}{2} a \cos \left (\frac{1}{4} (2 e+\pi )\right ) \text{Si}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a} \]
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Rubi [A] time = 0.275993, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3319, 3312, 3303, 3299, 3302} \[ \frac{3}{2} a \sin \left (\frac{1}{4} (2 e+\pi )\right ) \text{CosIntegral}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{1}{2} a \cos \left (\frac{3}{4} (2 e-\pi )\right ) \text{CosIntegral}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{1}{2} a \sin \left (\frac{3}{4} (2 e-\pi )\right ) \text{Si}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{3}{2} a \cos \left (\frac{1}{4} (2 e+\pi )\right ) \text{Si}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3312
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^{3/2}}{x} \, dx &=\left (2 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \frac{\sin ^3\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{x} \, dx\\ &=\left (2 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \left (\frac{3 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{4 x}+\frac{\sin \left (\frac{3 e}{2}-\frac{\pi }{4}+\frac{3 f x}{2}\right )}{4 x}\right ) \, dx\\ &=\frac{1}{2} \left (a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \frac{\sin \left (\frac{3 e}{2}-\frac{\pi }{4}+\frac{3 f x}{2}\right )}{x} \, dx+\frac{1}{2} \left (3 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \frac{\sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{x} \, dx\\ &=\frac{1}{2} \left (a \cos \left (\frac{3}{4} (2 e-\pi )\right ) \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \frac{\cos \left (\frac{3 f x}{2}\right )}{x} \, dx+\frac{1}{2} \left (a \cos \left (\frac{3 e}{2}-\frac{\pi }{4}\right ) \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \frac{\sin \left (\frac{3 f x}{2}\right )}{x} \, dx+\frac{1}{2} \left (3 a \cos \left (\frac{1}{4} (2 e+\pi )\right ) \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \frac{\sin \left (\frac{f x}{2}\right )}{x} \, dx+\frac{1}{2} \left (3 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{1}{4} (2 e+\pi )\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \frac{\cos \left (\frac{f x}{2}\right )}{x} \, dx\\ &=\frac{1}{2} a \cos \left (\frac{3}{4} (2 e-\pi )\right ) \text{Ci}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}+\frac{3}{2} a \text{Ci}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{1}{4} (2 e+\pi )\right ) \sqrt{a+a \sin (e+f x)}+\frac{3}{2} a \cos \left (\frac{1}{4} (2 e+\pi )\right ) \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)} \text{Si}\left (\frac{f x}{2}\right )-\frac{1}{2} a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{3}{4} (2 e-\pi )\right ) \sqrt{a+a \sin (e+f x)} \text{Si}\left (\frac{3 f x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.662187, size = 127, normalized size = 0.57 \[ \frac{(a (\sin (e+f x)+1))^{3/2} \left (3 \text{CosIntegral}\left (\frac{f x}{2}\right ) \left (\sin \left (\frac{e}{2}\right )+\cos \left (\frac{e}{2}\right )\right )+\text{CosIntegral}\left (\frac{3 f x}{2}\right ) \left (\sin \left (\frac{3 e}{2}\right )-\cos \left (\frac{3 e}{2}\right )\right )+\left (\cos \left (\frac{e}{2}\right )-\sin \left (\frac{e}{2}\right )\right ) \left ((2 \sin (e)+1) \text{Si}\left (\frac{3 f x}{2}\right )+3 \text{Si}\left (\frac{f x}{2}\right )\right )\right )}{2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}\, 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{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{x}\,{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{\left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}{x}\, 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 \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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