Optimal. Leaf size=273 \[ -c f \sin (e) \text{CosIntegral}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+c f \cos (2 e) \text{CosIntegral}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f \sin (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f \cos (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{x}-\frac{c \sin (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x} \]
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Rubi [A] time = 0.664795, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4604, 6741, 12, 6742, 3297, 3303, 3299, 3302} \[ -c f \sin (e) \text{CosIntegral}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+c f \cos (2 e) \text{CosIntegral}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f \sin (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f \cos (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{x}-\frac{c \sin (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x} \]
Antiderivative was successfully verified.
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Rule 4604
Rule 6741
Rule 12
Rule 6742
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx &=\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (e+f x) (c+c \sin (e+f x))}{x^2} \, dx\\ &=\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{c \cos (e+f x) (1+\sin (e+f x))}{x^2} \, dx\\ &=\left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (e+f x) (1+\sin (e+f x))}{x^2} \, dx\\ &=\left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \left (\frac{\cos (e+f x)}{x^2}+\frac{\sin (2 e+2 f x)}{2 x^2}\right ) \, dx\\ &=\frac{1}{2} \left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (2 e+2 f x)}{x^2} \, dx+\left (c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (e+f x)}{x^2} \, dx\\ &=-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x}-\frac{c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \sin (2 e+2 f x)}{2 x}+\left (c f \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (2 e+2 f x)}{x} \, dx-\left (c f \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (e+f x)}{x} \, dx\\ &=-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x}-\frac{c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \sin (2 e+2 f x)}{2 x}-\left (c f \cos (e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (f x)}{x} \, dx+\left (c f \cos (2 e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (2 f x)}{x} \, dx-\left (c f \sec (e+f x) \sin (e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (f x)}{x} \, dx-\left (c f \sec (e+f x) \sin (2 e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (2 f x)}{x} \, dx\\ &=-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x}+c f \cos (2 e) \text{Ci}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}-c f \text{Ci}(f x) \sec (e+f x) \sin (e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}-\frac{c \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \sin (2 e+2 f x)}{2 x}-c f \cos (e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \text{Si}(f x)-c f \sec (e+f x) \sin (2 e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \text{Si}(2 f x)\\ \end{align*}
Mathematica [C] time = 1.48438, size = 231, normalized size = 0.85 \[ \frac{c e^{-i (e+f x)} \sqrt{-i c e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2} \left (-2 i f x e^{i (e+2 f x)} \text{ExpIntegralEi}(-i f x)+2 i f x e^{3 i e+2 i f x} \text{ExpIntegralEi}(i f x)+2 f x e^{2 i (2 e+f x)} \text{ExpIntegralEi}(2 i f x)-2 e^{i (e+f x)}-2 e^{3 i (e+f x)}+i e^{4 i (e+f x)}+2 f x e^{2 i f x} \text{ExpIntegralEi}(-2 i f x)-i\right ) \sqrt{a-a \sin (e+f x)}}{2 \sqrt{2} x \left (1+e^{2 i (e+f x)}\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( c+c\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a-a\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{\sqrt{-a \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{x^{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{\sqrt{-a \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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