Integrand size = 28, antiderivative size = 33 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{135 c^2 f} \]
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Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2815, 2752} \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^2 f} \]
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Rule 2752
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^6(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{a^3 c^3} \\ & = -\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^2 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).
Time = 0.66 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \sqrt {c-c \sin (e+f x)}}{135 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
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Time = 0.63 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48
method | result | size |
default | \(\frac {2 c \left (\sin \left (f x +e \right )-1\right )}{5 a^{3} \left (\sin \left (f x +e \right )+1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(49\) |
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none
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^3} \, dx=\frac {2 \, \sqrt {-c \sin \left (f x + e\right ) + c}}{5 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} - 2 \, a^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} f \cos \left (f x + e\right )\right )}} \]
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\[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^3} \, dx=\frac {\int \frac {\sqrt {- c \sin {\left (e + f x \right )} + c}}{\sin ^{3}{\left (e + f x \right )} + 3 \sin ^{2}{\left (e + f x \right )} + 3 \sin {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (32) = 64\).
Time = 0.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 6.61 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^3} \, dx=\frac {2 \, {\left (\sqrt {c} + \frac {3 \, \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sqrt {c} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {\sqrt {c} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )}}{5 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} f \sqrt {\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (32) = 64\).
Time = 0.32 (sec) , antiderivative size = 157, normalized size of antiderivative = 4.76 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^3} \, dx=\frac {\sqrt {2} \sqrt {c} {\left (\frac {10 \, {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {5 \, {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}} + \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )}}{10 \, a^{3} f {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}^{5}} \]
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Time = 9.66 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.73 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^3} \, dx=\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {c-c\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,16{}\mathrm {i}}{5\,a^3\,f\,{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}^5\,\left (1+{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}\right )} \]
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