Integrand size = 33, antiderivative size = 86 \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x} \, dx=\cos (e) \operatorname {CosIntegral}(f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}-\sec (e+f x) \sin (e) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \text {Si}(f x) \]
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Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4700, 3384, 3380, 3383} \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x} \, dx=\cos (e) \operatorname {CosIntegral}(f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}-\sin (e) \text {Si}(f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 4700
Rubi steps \begin{align*} \text {integral}& = \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} \, dx \\ & = \left (\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-\left (\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 \\ & = \cos (e) \operatorname {CosIntegral}(f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}-\sec (e+f x) \sin (e) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \text {Si}(f x) \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x} \, dx=\sec (e+f x) \sqrt {c (1+\sin (e+f x))} \sqrt {a-a \sin (e+f x)} (\cos (e) \operatorname {CosIntegral}(f x)-\sin (e) \text {Si}(f x)) \]
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\[\int \frac {\sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {c +c \sin \left (f x +e \right )}}{x}d x\]
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Exception generated. \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x} \, dx=\int \frac {\sqrt {c \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}{x}\, dx \]
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\[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x} \, dx=\int { \frac {\sqrt {-a \sin \left (f x + e\right ) + a} \sqrt {c \sin \left (f x + e\right ) + c}}{x} \,d x } \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.19 \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x} \, dx=\frac {{\left (\Re \left ( \operatorname {Ci}\left (f x\right ) \right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (\frac {1}{2} \, e\right )^{2} + \Re \left ( \operatorname {Ci}\left (-f x\right ) \right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (\frac {1}{2} \, e\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (f x\right ) \right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (\frac {1}{2} \, e\right ) - 2 \, \Im \left ( \operatorname {Ci}\left (-f x\right ) \right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (\frac {1}{2} \, e\right ) + 4 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \operatorname {Si}\left (f x\right ) \tan \left (\frac {1}{2} \, e\right ) - \Re \left ( \operatorname {Ci}\left (f x\right ) \right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \Re \left ( \operatorname {Ci}\left (-f x\right ) \right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{2 \, {\left (\tan \left (\frac {1}{2} \, e\right )^{2} + 1\right )}} \]
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Timed out. \[ \int \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x} \, dx=\int \frac {\sqrt {a-a\,\sin \left (e+f\,x\right )}\,\sqrt {c+c\,\sin \left (e+f\,x\right )}}{x} \,d x \]
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