Integrand size = 33, antiderivative size = 118 \[ \int x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx=\frac {2 x \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{f^2}-\frac {2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \tan (e+f x)}{f^3}+\frac {x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \tan (e+f x)}{f} \]
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Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4700, 3377, 2717} \[ \int x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx=-\frac {2 \tan (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{f^3}+\frac {2 x \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{f^2}+\frac {x^2 \tan (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{f} \]
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Rule 2717
Rule 3377
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 x^2 \cos (e+f x) \, dx \\ & = \frac {x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \tan (e+f x)}{f}-\frac {\left (2 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}\right ) \int x \sin (e+f x) \, dx}{f} \\ & = \frac {2 x \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{f^2}+\frac {x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \tan (e+f x)}{f}-\frac {\left (2 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}\right ) \int \cos (e+f x) \, dx}{f^2} \\ & = \frac {2 x \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{f^2}-\frac {2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \tan (e+f x)}{f^3}+\frac {x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \tan (e+f x)}{f} \\ \end{align*}
Time = 1.51 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.46 \[ \int x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx=\frac {\sqrt {c (1+\sin (e+f x))} \sqrt {a-a \sin (e+f x)} \left (2 f x+\left (-2+f^2 x^2\right ) \tan (e+f x)\right )}{f^3} \]
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\[\int x^{2} \sqrt {a -\sin \left (f x +e \right ) a}\, \sqrt {c +c \sin \left (f x +e \right )}d x\]
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Exception generated. \[ \int x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx=\int x^{2} \sqrt {c \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}\, dx \]
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\[ \int x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx=\int { \sqrt {-a \sin \left (f x + e\right ) + a} \sqrt {c \sin \left (f x + e\right ) + c} x^{2} \,d x } \]
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none
Time = 0.32 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01 \[ \int x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx=-{\left (\frac {2 \, x \cos \left (f x + e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f^{2}} + \frac {{\left (f^{2} x^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (f x + e\right )}{f^{3}}\right )} \sqrt {a} \sqrt {c} \]
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Time = 27.42 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.73 \[ \int x^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx=\frac {\sqrt {-a\,\left (\sin \left (e+f\,x\right )-1\right )}\,\sqrt {c\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (2\,f\,x-2\,\sin \left (2\,e+2\,f\,x\right )+2\,f\,x\,\left (2\,{\cos \left (e+f\,x\right )}^2-1\right )+f^2\,x^2\,\sin \left (2\,e+2\,f\,x\right )\right )}{2\,f^3\,{\cos \left (e+f\,x\right )}^2} \]
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