Optimal. Leaf size=332 \[ -\frac{3}{16} a f^2 \sin \left (\frac{1}{4} (2 e+\pi )\right ) \text{CosIntegral}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{9}{16} a f^2 \cos \left (\frac{3}{4} (2 e-\pi )\right ) \text{CosIntegral}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{9}{16} a f^2 \sin \left (\frac{3}{4} (2 e-\pi )\right ) \text{Si}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{3}{16} a f^2 \cos \left (\frac{1}{4} (2 e+\pi )\right ) \text{Si}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{a \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{x^2}-\frac{3 a f \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{2 x} \]
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Rubi [A] time = 0.375586, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3319, 3314, 3303, 3299, 3302, 3312} \[ -\frac{3}{16} a f^2 \sin \left (\frac{1}{4} (2 e+\pi )\right ) \text{CosIntegral}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{9}{16} a f^2 \cos \left (\frac{3}{4} (2 e-\pi )\right ) \text{CosIntegral}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}+\frac{9}{16} a f^2 \sin \left (\frac{3}{4} (2 e-\pi )\right ) \text{Si}\left (\frac{3 f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{3}{16} a f^2 \cos \left (\frac{1}{4} (2 e+\pi )\right ) \text{Si}\left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}-\frac{a \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{x^2}-\frac{3 a f \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{2 x} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3314
Rule 3303
Rule 3299
Rule 3302
Rule 3312
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^{3/2}}{x^3} \, 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^3} \, dx\\ &=-\frac{3 a f \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{2 x}-\frac{a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x^2}+\frac{1}{2} \left (3 a f^2 \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}{4} \left (9 a f^2 \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\\ &=-\frac{3 a f \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{2 x}-\frac{a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x^2}-\frac{1}{4} \left (9 a f^2 \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 (3 a f^2 \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 f^2 \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{3}{2} a f^2 \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 a f \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{2 x}-\frac{a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x^2}+\frac{3}{2} a f^2 \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}{16} \left (9 a f^2 \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}{16} \left (27 a f^2 \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{3}{2} a f^2 \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 a f \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{2 x}-\frac{a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x^2}+\frac{3}{2} a f^2 \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}{16} \left (9 a f^2 \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}{16} \left (9 a f^2 \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}{16} \left (27 a f^2 \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}{16} \left (27 a f^2 \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{9}{16} a f^2 \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}{16} a f^2 \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 a f \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{2 x}-\frac{a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{x^2}-\frac{3}{16} a f^2 \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{9}{16} a f^2 \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 [C] time = 0.875693, size = 295, normalized size = 0.89 \[ -\frac{i \left (-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2\right )^{3/2} \left (3 i f^2 x^2 e^{i e+\frac{3 i f x}{2}} \text{Ei}\left (-\frac{1}{2} i f x\right )+3 f^2 x^2 e^{2 i e+\frac{3 i f x}{2}} \text{Ei}\left (\frac{i f x}{2}\right )-9 i f^2 x^2 e^{\frac{3}{2} i (2 e+f x)} \text{Ei}\left (\frac{3 i f x}{2}\right )+6 f x e^{i (e+f x)}+6 i f x e^{2 i (e+f x)}+6 f x e^{3 i (e+f x)}+12 i e^{i (e+f x)}+12 e^{2 i (e+f x)}-4 i e^{3 i (e+f x)}-9 f^2 x^2 e^{\frac{3 i f x}{2}} \text{Ei}\left (-\frac{3}{2} i f x\right )+6 i f x-4\right )}{16 \sqrt{2} x^2 \left (e^{i (e+f x)}+i\right )^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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^{3}}\,{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^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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