Integrand size = 27, antiderivative size = 242 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {d (c+3 d) \cos (e+f x)}{3 (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{(c-d) f (3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {(c+3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 (c-d)^2 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 (c-d) f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.22 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2847, 2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {d (c+3 d) \cos (e+f x)}{a f (c-d)^2 (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}+\frac {\sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f (c-d) \sqrt {c+d \sin (e+f x)}}-\frac {(c+3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2833
Rule 2847
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {d \int \frac {-\frac {3 a}{2}+\frac {1}{2} a \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{a^2 (c-d)} \\ & = -\frac {d (c+3 d) \cos (e+f x)}{a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {(2 d) \int \frac {\frac {1}{4} a (3 c+d)+\frac {1}{4} a (c+3 d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{a^2 (c-d)^2 (c+d)} \\ & = -\frac {d (c+3 d) \cos (e+f x)}{a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {\int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 a (c-d)}-\frac {(c+3 d) \int \sqrt {c+d \sin (e+f x)} \, dx}{2 a (c-d)^2 (c+d)} \\ & = -\frac {d (c+3 d) \cos (e+f x)}{a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\left ((c+3 d) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{2 a (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{2 a (c-d) \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {d (c+3 d) \cos (e+f x)}{a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {(c+3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{a (c-d)^2 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a (c-d) f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 1.48 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (-\frac {2 \left ((c+d)^2 \cos \left (\frac {1}{2} (e+f x)\right )+d (2 (c+d)+(c+3 d) \cos (e+f x)) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}+(c+3 d) (c+d \sin (e+f x))+\left (c^2+4 c d+3 d^2\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-\left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{3 (c-d)^2 (c+d) f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(924\) vs. \(2(298)=596\).
Time = 1.58 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.82
method | result | size |
default | \(\frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) d \sin \left (f x +e \right )+c \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, E\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}+3 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, E\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, E\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}-3 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, E\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-4 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, F\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d +4 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, F\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-c \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}-3 \left (\cos ^{2}\left (f x +e \right )\right ) d^{3}+c^{2} d \sin \left (f x +e \right )-d^{3} \sin \left (f x +e \right )-c^{2} d +d^{3}\right )}{d \left (c^{2}-d^{2}\right ) \sqrt {-\left (c +d \sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+1\right )}\, \left (c -d \right ) a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(925\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 1172, normalized size of antiderivative = 4.84 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\int \frac {1}{c \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c \sqrt {c + d \sin {\left (e + f x \right )}} + d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}\, dx}{a} \]
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\[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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