Optimal. Leaf size=176 \[ -\frac{1}{2} f^2 \cos (e) \text{CosIntegral}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} f^2 \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x^2}+\frac{f \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x} \]
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Rubi [A] time = 0.225648, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {4604, 3297, 3303, 3299, 3302} \[ -\frac{1}{2} f^2 \cos (e) \text{CosIntegral}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} f^2 \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x^2}+\frac{f \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x} \]
Antiderivative was successfully verified.
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Rule 4604
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x^3} \, 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)}{x^3} \, dx\\ &=-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x^2}-\frac{1}{2} \left (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^2} \, dx\\ &=-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x^2}+\frac{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{2 x}-\frac{1}{2} \left (f^2 \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} \, dx\\ &=-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x^2}+\frac{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{2 x}-\frac{1}{2} \left (f^2 \cos (e) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\cos (f x)}{x} \, dx+\frac{1}{2} \left (f^2 \sec (e+f x) \sin (e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}\right ) \int \frac{\sin (f x)}{x} \, dx\\ &=-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{2 x^2}-\frac{1}{2} f^2 \cos (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}+\frac{1}{2} f^2 \sec (e+f x) \sin (e) \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \text{Si}(f x)+\frac{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \tan (e+f x)}{2 x}\\ \end{align*}
Mathematica [A] time = 0.290416, size = 87, normalized size = 0.49 \[ \frac{\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} \left (-f^2 x^2 \cos (e) \text{CosIntegral}(f x)+f^2 x^2 \sin (e) \text{Si}(f x)+f x \sin (e+f x)-\cos (e+f x)\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{c+c\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} \sqrt{c \sin \left (f x + e\right ) + c}}{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{\sqrt{c \left (\sin{\left (e + f x \right )} + 1\right )} \sqrt{- a \left (\sin{\left (e + f x \right )} - 1\right )}}{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{\sqrt{-a \sin \left (f x + e\right ) + a} \sqrt{c \sin \left (f x + e\right ) + c}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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