Integrand size = 27, antiderivative size = 169 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (3+3 \sin (e+f x))}-\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.16 (sec) , antiderivative size = 170, 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 = {2848, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}+\frac {(c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2848
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}+\frac {d \int \frac {a-a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 a^2} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {\int \sqrt {c+d \sin (e+f x)} \, dx}{2 a}+\frac {(c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 a} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \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)}}{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 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \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 = 0.87 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 \sin \left (\frac {1}{2} (e+f x)\right ) (c+d \sin (e+f x))-\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (c+d \sin (e+f x)-(c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+(c+d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )\right )}{3 f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \]
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Time = 1.33 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.26
method | result | size |
default | \(\frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) d \sin \left (f x +e \right )+c \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, E\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, E\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c d \sin \left (f x +e \right )-\sin \left (f x +e \right ) d^{2}-c d +d^{2}\right )}{d \sqrt {-\left (c +d \sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+1\right )}\, a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(382\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.28 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \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 (a d f \cos \left (f x + e\right ) + a d f \sin \left (f x + e\right ) + a d f\right )}} \]
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\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
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\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{a \sin \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{a \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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