Integrand size = 30, antiderivative size = 43 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {c \cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \]
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Rule 2817
Rubi steps \begin{align*} \text {integral}& = \frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 5.63 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.91 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {27 \sqrt {3} \sec (e+f x) \sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)} (-28 \cos (2 (e+f x))+\cos (4 (e+f x))+56 \sin (e+f x)-8 \sin (3 (e+f x)))}{32 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(37)=74\).
Time = 3.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, a^{3} \left (\cos ^{3}\left (f x +e \right )-4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-8 \cos \left (f x +e \right )+8 \tan \left (f x +e \right )+7 \sec \left (f x +e \right )\right )}{4 f}\) | \(81\) |
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (37) = 74\).
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.21 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {{\left (a^{3} \cos \left (f x + e\right )^{4} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 7 \, a^{3} - 4 \, {\left (a^{3} \cos \left (f x + e\right )^{2} - 2 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{4 \, f \cos \left (f x + e\right )} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\text {Timed out} \]
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\[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} \,d x } \]
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none
Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {4 \, a^{\frac {7}{2}} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \]
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Time = 8.62 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.30 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {a^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (28\,\cos \left (e+f\,x\right )+27\,\cos \left (3\,e+3\,f\,x\right )-\cos \left (5\,e+5\,f\,x\right )-48\,\sin \left (2\,e+2\,f\,x\right )+8\,\sin \left (4\,e+4\,f\,x\right )\right )}{32\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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