Integrand size = 27, antiderivative size = 180 \[ \int \frac {1}{(3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{(c-d) f (3+3 \sin (e+f x))}-\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 (c-d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2847, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {1}{(3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}+\frac {\sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f (c-d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2847
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{(c-d) f (a+a \sin (e+f x))}+\frac {d \int \frac {-\frac {a}{2}-\frac {1}{2} a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{a^2 (c-d)} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{(c-d) f (a+a \sin (e+f x))}+\frac {\int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 a}-\frac {\int \sqrt {c+d \sin (e+f x)} \, dx}{2 a (c-d)} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{(c-d) f (a+a \sin (e+f x))}-\frac {\sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{2 a (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 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{(c-d) f (a+a \sin (e+f x))}-\frac {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) 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 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 \sin \left (\frac {1}{2} (e+f x)\right ) (c+d \sin (e+f x))-\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (c+d \sin (e+f x)-(c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+(c-d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )\right )}{3 (c-d) f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \]
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Time = 1.27 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.46
method | result | size |
default | \(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {-d \left (\sin ^{2}\left (f x +e \right )\right )-c \sin \left (f x +e \right )+d \sin \left (f x +e \right )+c}{\left (c -d \right ) \sqrt {\left (\sin \left (f x +e \right )+1\right ) \left (\sin \left (f x +e \right )-1\right ) \left (-d \sin \left (f x +e \right )-c \right )}}-\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (2 c -2 d \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (c -d \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(443\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 582, normalized size of antiderivative = 3.23 \[ \int \frac {1}{(3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\frac {{\left (\sqrt {2} {\left (2 \, c - 3 \, d\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, c - 3 \, d\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, c - 3 \, d\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (\sqrt {2} {\left (2 \, c - 3 \, d\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, c - 3 \, d\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, c - 3 \, d\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, {\left (i \, \sqrt {2} d \cos \left (f x + e\right ) + i \, \sqrt {2} d \sin \left (f x + e\right ) + i \, \sqrt {2} d\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (-i \, \sqrt {2} d \cos \left (f x + e\right ) - i \, \sqrt {2} d \sin \left (f x + e\right ) - i \, \sqrt {2} d\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) - 6 \, {\left (d \cos \left (f x + e\right ) - d \sin \left (f x + e\right ) + d\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{6 \, {\left ({\left (a c d - a d^{2}\right )} f \cos \left (f x + e\right ) + {\left (a c d - a d^{2}\right )} f \sin \left (f x + e\right ) + {\left (a c d - a d^{2}\right )} f\right )}} \]
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\[ \int \frac {1}{(3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\int \frac {1}{\sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx}{a} \]
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\[ \int \frac {1}{(3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {1}{(3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
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