Integrand size = 38, antiderivative size = 81 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 a^2 (7 A+3 B) c^3 \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a^2 B c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}} \]
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Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {3046, 2935, 2752} \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 a^2 c^3 (7 A+3 B) \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a^2 B c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}} \]
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Rule 2752
Rule 2935
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = -\frac {2 a^2 B c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}}+\frac {1}{7} \left (a^2 (7 A+3 B) c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {2 a^2 (7 A+3 B) c^3 \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a^2 B c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}} \\ \end{align*}
Time = 0.82 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.10 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (7 A-2 B+5 B \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{35 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 1.37 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (5 B \sin \left (f x +e \right )+7 A -2 B \right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(65\) |
parts | \(-\frac {2 A \,a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right ) c}{\cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 B \,a^{2} \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (5 \left (\sin ^{3}\left (f x +e \right )\right )-6 \left (\sin ^{2}\left (f x +e \right )\right )+8 \sin \left (f x +e \right )-16\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{2} \left (A +2 B \right ) \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )+8\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{2} \left (2 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-2\right )}{3 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(256\) |
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (73) = 146\).
Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.38 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 \, {\left (5 \, B a^{2} \cos \left (f x + e\right )^{4} - {\left (7 \, A + 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - {\left (21 \, A + 19 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 2 \, {\left (7 \, A + 3 \, B\right )} a^{2} \cos \left (f x + e\right ) + 4 \, {\left (7 \, A + 3 \, B\right )} a^{2} - {\left (5 \, B a^{2} \cos \left (f x + e\right )^{3} + {\left (7 \, A + 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (7 \, A + 3 \, B\right )} a^{2} \cos \left (f x + e\right ) - 4 \, {\left (7 \, A + 3 \, B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{35 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=a^{2} \left (\int A \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int 2 A \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int A \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\, dx + \int B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int 2 B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\, dx + \int B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \]
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\[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2} \sqrt {-c \sin \left (f x + e\right ) + c} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (73) = 146\).
Time = 0.54 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.47 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=-\frac {\sqrt {2} {\left (5 \, B a^{2} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 35 \, {\left (4 \, A a^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 35 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 7 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right )\right )} \sqrt {c}}{140 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,\sqrt {c-c\,\sin \left (e+f\,x\right )} \,d x \]
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