Integrand size = 29, antiderivative size = 89 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 \cos (e+f x)}{(c+d) f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 \cos (e+f x)}{(c+d)^2 f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2851, 2850} \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}} \]
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Rule 2850
Rule 2851
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos (e+f x)}{3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {2 \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 (c+d)} \\ & = -\frac {2 a \cos (e+f x)}{3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 a \cos (e+f x)}{3 (c+d)^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {1+\sin (e+f x)} (3 c+d+2 d \sin (e+f x))}{\sqrt {3} (c+d)^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(83)=166\).
Time = 3.87 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.21
method | result | size |
default | \(-\frac {2 \sec \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (2 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) d^{3}+\left (\sin ^{3}\left (f x +e \right )\right ) c \,d^{2}+\left (\sin ^{3}\left (f x +e \right )\right ) d^{3}+\left (\sin ^{2}\left (f x +e \right )\right ) c \,d^{2}-d^{3} \left (\sin ^{2}\left (f x +e \right )\right )-4 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d -3 c^{3} \sin \left (f x +e \right )-5 c^{2} d \sin \left (f x +e \right )-2 \sin \left (f x +e \right ) c \,d^{2}+3 c^{3}+5 c^{2} d \right )}{3 f {\left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right )}^{2} \left (c +d \right )^{2}}\) | \(197\) |
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Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (83) = 166\).
Time = 0.29 (sec) , antiderivative size = 300, normalized size of antiderivative = 3.37 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 \, {\left (2 \, d \cos \left (f x + e\right )^{2} + {\left (3 \, c + d\right )} \cos \left (f x + e\right ) + {\left (2 \, d \cos \left (f x + e\right ) - 3 \, c + d\right )} \sin \left (f x + e\right ) + 3 \, c - d\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{3 \, {\left ({\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f \cos \left (f x + e\right )^{3} + {\left (2 \, c^{3} d + 5 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{4} + 2 \, c^{3} d + 2 \, c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f \cos \left (f x + e\right ) - {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} f + {\left ({\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{3} d + 2 \, c^{2} d^{2} + c d^{3}\right )} f \cos \left (f x + e\right ) - {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} f\right )} \sin \left (f x + e\right )\right )}} \]
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\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (83) = 166\).
Time = 0.34 (sec) , antiderivative size = 340, normalized size of antiderivative = 3.82 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 \, {\left ({\left (3 \, c^{2} + c d\right )} \sqrt {a} - \frac {{\left (3 \, c^{2} - 9 \, c d - 2 \, d^{2}\right )} \sqrt {a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, {\left (3 \, c^{2} - 4 \, c d + 3 \, d^{2}\right )} \sqrt {a} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, {\left (3 \, c^{2} - 4 \, c d + 3 \, d^{2}\right )} \sqrt {a} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {{\left (3 \, c^{2} - 9 \, c d - 2 \, d^{2}\right )} \sqrt {a} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {{\left (3 \, c^{2} + c d\right )} \sqrt {a} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{2}}{3 \, {\left (c^{2} + 2 \, c d + d^{2} + \frac {2 \, {\left (c^{2} + 2 \, c d + d^{2}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (c^{2} + 2 \, c d + d^{2}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {5}{2}} f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (83) = 166\).
Time = 0.83 (sec) , antiderivative size = 471, normalized size of antiderivative = 5.29 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {4 \, \sqrt {2} {\left ({\left (\frac {3 \, {\left (c^{6} d^{4} - 2 \, c^{5} d^{5} - c^{4} d^{6} + 4 \, c^{3} d^{7} - c^{2} d^{8} - 2 \, c d^{9} + d^{10}\right )} \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2}}{c^{7} d^{4} - c^{6} d^{5} - 3 \, c^{5} d^{6} + 3 \, c^{4} d^{7} + 3 \, c^{3} d^{8} - 3 \, c^{2} d^{9} - c d^{10} + d^{11}} + \frac {2 \, {\left (3 \, c^{6} d^{4} - 14 \, c^{5} d^{5} + 21 \, c^{4} d^{6} - 4 \, c^{3} d^{7} - 19 \, c^{2} d^{8} + 18 \, c d^{9} - 5 \, d^{10}\right )}}{c^{7} d^{4} - c^{6} d^{5} - 3 \, c^{5} d^{6} + 3 \, c^{4} d^{7} + 3 \, c^{3} d^{8} - 3 \, c^{2} d^{9} - c d^{10} + d^{11}}\right )} \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + \frac {3 \, {\left (c^{6} d^{4} - 2 \, c^{5} d^{5} - c^{4} d^{6} + 4 \, c^{3} d^{7} - c^{2} d^{8} - 2 \, c d^{9} + d^{10}\right )}}{c^{7} d^{4} - c^{6} d^{5} - 3 \, c^{5} d^{6} + 3 \, c^{4} d^{7} + 3 \, c^{3} d^{8} - 3 \, c^{2} d^{9} - c d^{10} + d^{11}}\right )} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )}{3 \, {\left (c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + 2 \, c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} - 6 \, d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + c + d\right )}^{\frac {3}{2}} f} \]
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Time = 15.15 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.97 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,8{}\mathrm {i}}{3\,d\,f\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^2}+\frac {8\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,d\,f\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^2}+\frac {8\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{d^2\,f\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^2}+\frac {c\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,8{}\mathrm {i}}{d^2\,f\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^2}\right )}{{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}-\frac {{\left (c+d\right )}^2\,1{}\mathrm {i}}{{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^2}-\frac {2\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (2\,c^2+2\,c\,d+d^2\right )}{d^2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (4\,c+d\right )}{d}+\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c+d\right )}^2\,\left (2\,c^2+2\,c\,d+d^2\right )\,2{}\mathrm {i}}{d^2\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^2}-\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c+d\right )}^2\,\left (4\,c+d\right )\,1{}\mathrm {i}}{d\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^2}} \]
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