Integrand size = 27, antiderivative size = 185 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{3+3 \sin (e+f x)} \, dx=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (3+3 \sin (e+f x))}-\frac {(c-3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.19 (sec) , antiderivative size = 186, 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 = {2846, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{3+3 \sin (e+f x)} \, dx=\frac {\left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f \sqrt {c+d \sin (e+f x)}}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}-\frac {(c-3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2846
Rubi steps \begin{align*} \text {integral}& = -\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {d \int \frac {-\frac {1}{2} a (3 c-d)+\frac {1}{2} a (c-3 d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{a^2} \\ & = -\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {(c-3 d) \int \sqrt {c+d \sin (e+f x)} \, dx}{2 a}+\frac {\left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 a} \\ & = -\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \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 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (\left (c^2-d^2\right ) \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 {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \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 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (c^2-d^2\right ) \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 = 1.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.21 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{3+3 \sin (e+f x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 (c-d) \sin \left (\frac {1}{2} (e+f x)\right ) (c+d \sin (e+f x))-\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\left (\left (c^2-2 c d-3 d^2\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )+(c-d) \left (c+d \sin (e+f x)+(c+d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )\right )\right )}{3 f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(924\) vs. \(2(242)=484\).
Time = 1.78 (sec) , antiderivative size = 925, normalized size of antiderivative = 5.00
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 (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}}\, 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 -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}}\, 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}+\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}-c \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+\left (\cos ^{2}\left (f x +e \right )\right ) d^{3}+c^{2} d \sin \left (f x +e \right )-2 \sin \left (f x +e \right ) c \,d^{2}+d^{3} \sin \left (f x +e \right )-c^{2} d +2 c \,d^{2}-d^{3}\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}\) | \(925\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 674, normalized size of antiderivative = 3.64 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{3+3 \sin (e+f x)} \, dx=\frac {{\left (\sqrt {2} {\left (2 \, c^{2} + 3 \, c d - 3 \, d^{2}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, c^{2} + 3 \, c d - 3 \, d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, c^{2} + 3 \, c d - 3 \, d^{2}\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^{2} + 3 \, c d - 3 \, d^{2}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, c^{2} + 3 \, c d - 3 \, d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, c^{2} + 3 \, c d - 3 \, d^{2}\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 (\sqrt {2} {\left (-i \, c d + 3 i \, d^{2}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (-i \, c d + 3 i \, d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-i \, c d + 3 i \, d^{2}\right )}\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 (\sqrt {2} {\left (i \, c d - 3 i \, d^{2}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (i \, c d - 3 i \, d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (i \, c d - 3 i \, d^{2}\right )}\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 (c d - d^{2} + {\left (c d - d^{2}\right )} \cos \left (f x + e\right ) - {\left (c d - d^{2}\right )} \sin \left (f x + e\right )\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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\[ \int \frac {(c+d \sin (e+f x))^{3/2}}{3+3 \sin (e+f x)} \, dx=\frac {\int \frac {c \sqrt {c + d \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
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\[ \int \frac {(c+d \sin (e+f x))^{3/2}}{3+3 \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {(c+d \sin (e+f x))^{3/2}}{3+3 \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{3+3 \sin (e+f x)} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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