Integrand size = 27, antiderivative size = 302 \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=-\frac {36 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d^2 f}+\frac {216 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) (27+27 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {36 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{35 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {36 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{35 d^3 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.40 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2842, 3047, 3102, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=-\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{105 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 2842
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \int (a+a \sin (e+f x)) \left (a^2 (c+5 d)-2 a^2 (c-4 d) \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)} \, dx}{7 d} \\ & = -\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \int \sqrt {c+d \sin (e+f x)} \left (a^3 (c+5 d)+\left (-2 a^3 (c-4 d)+a^3 (c+5 d)\right ) \sin (e+f x)-2 a^3 (c-4 d) \sin ^2(e+f x)\right ) \, dx}{7 d} \\ & = \frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {4 \int \sqrt {c+d \sin (e+f x)} \left (-\frac {1}{2} a^3 (c-49 d) d+\frac {1}{2} a^3 \left (4 c^2-21 c d+65 d^2\right ) \sin (e+f x)\right ) \, dx}{35 d^2} \\ & = -\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {8 \int \frac {\frac {1}{4} a^3 d \left (c^2+126 c d+65 d^2\right )+\frac {1}{4} a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^2} \\ & = -\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}-\frac {\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^3}+\frac {\left (2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{105 d^3} \\ & = -\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {\left (2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{105 d^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 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}{105 d^3 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {4 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {4 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) 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)}}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 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}}}{105 d^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75 \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\frac {9 \left (-8 \left (d^2 \left (c^2+126 c d+65 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (\left (16 c^2-84 c d-535 d^2\right ) \cos (e+f x)+3 d (5 d \cos (3 (e+f x))-2 (c+21 d) \sin (2 (e+f x)))\right )\right )}{70 d^3 f \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1315\) vs. \(2(360)=720\).
Time = 6.14 (sec) , antiderivative size = 1316, normalized size of antiderivative = 4.36
method | result | size |
default | \(\text {Expression too large to display}\) | \(1316\) |
parts | \(\text {Expression too large to display}\) | \(2443\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.07 \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (8 \, a^{3} c^{4} - 42 \, a^{3} c^{3} d + 121 \, a^{3} c^{2} d^{2} - 84 \, a^{3} c d^{3} - 195 \, a^{3} d^{4}\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 ) + \sqrt {2} {\left (8 \, a^{3} c^{4} - 42 \, a^{3} c^{3} d + 121 \, a^{3} c^{2} d^{2} - 84 \, a^{3} c d^{3} - 195 \, a^{3} d^{4}\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 \, \sqrt {2} {\left (4 i \, a^{3} c^{3} d - 21 i \, a^{3} c^{2} d^{2} + 62 i \, a^{3} c d^{3} + 147 i \, a^{3} d^{4}\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 \, \sqrt {2} {\left (-4 i \, a^{3} c^{3} d + 21 i \, a^{3} c^{2} d^{2} - 62 i \, a^{3} c d^{3} - 147 i \, a^{3} d^{4}\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 (15 \, a^{3} d^{4} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} c d^{3} + 21 \, a^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (4 \, a^{3} c^{2} d^{2} - 21 \, a^{3} c d^{3} - 145 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{315 \, d^{4} f} \]
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\[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=a^{3} \left (\int 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}}\, dx\right ) \]
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\[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]
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