Integrand size = 33, antiderivative size = 273 \[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx=-\frac {c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{x}+c f \cos (2 e) \operatorname {CosIntegral}(2 f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}-c f \operatorname {CosIntegral}(f x) \sec (e+f x) \sin (e) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}-\frac {c \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \sin (2 e+2 f x)}{2 x}-c f \cos (e) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \text {Si}(f x)-c f \sec (e+f x) \sin (2 e) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \text {Si}(2 f x) \]
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Time = 0.83 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4700, 6873, 12, 6874, 3378, 3384, 3380, 3383} \[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx=-c f \sin (e) \operatorname {CosIntegral}(f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}+c f \cos (2 e) \operatorname {CosIntegral}(2 f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}-c f \sin (2 e) \text {Si}(2 f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}-c f \cos (e) \text {Si}(f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}-\frac {c \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{x}-\frac {c \sin (2 e+2 f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{2 x} \]
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Rule 12
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 4700
Rule 6873
Rule 6874
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) (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) \operatorname {CosIntegral}(2 f x) \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}-c f \operatorname {CosIntegral}(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*}
Result contains complex when optimal does not.
Time = 2.71 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx=\frac {c e^{-i (e+f x)} \sqrt {-i c e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2} \left (-i-2 e^{i (e+f x)}-2 e^{3 i (e+f x)}+i e^{4 i (e+f x)}-2 i e^{i (e+2 f x)} f x \operatorname {ExpIntegralEi}(-i f x)+2 i e^{3 i e+2 i f x} f x \operatorname {ExpIntegralEi}(i f x)+2 e^{2 i f x} f x \operatorname {ExpIntegralEi}(-2 i f x)+2 e^{2 i (2 e+f x)} f x \operatorname {ExpIntegralEi}(2 i f x)\right ) \sqrt {a-a \sin (e+f x)}}{2 \sqrt {2} \left (1+e^{2 i (e+f x)}\right ) x} \]
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\[\int \frac {\left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a -\sin \left (f x +e \right ) a}}{x^{2}}d x\]
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Exception generated. \[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx=\int \frac {\left (c \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx=\int { \frac {\sqrt {-a \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (247) = 494\).
Time = 0.38 (sec) , antiderivative size = 608, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx=\int \frac {\sqrt {a-a\,\sin \left (e+f\,x\right )}\,{\left (c+c\,\sin \left (e+f\,x\right )\right )}^{3/2}}{x^2} \,d x \]
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