Integrand size = 27, antiderivative size = 371 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {12 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d^2 f}-\frac {12 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}+\frac {24 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d^2 f}-\frac {2 \cos (e+f x) (27+27 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {12 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{35 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {12 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{35 d^3 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.54 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.05, number of steps used = 10, 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} \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac {4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{315 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{315 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f} \]
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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))^{5/2}}{9 d f}+\frac {2 \int (a+a \sin (e+f x)) \left (a^2 (c+7 d)-2 a^2 (c-5 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2} \, dx}{9 d} \\ & = -\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {2 \int (c+d \sin (e+f x))^{3/2} \left (a^3 (c+7 d)+\left (-2 a^3 (c-5 d)+a^3 (c+7 d)\right ) \sin (e+f x)-2 a^3 (c-5 d) \sin ^2(e+f x)\right ) \, dx}{9 d} \\ & = \frac {8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 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))^{5/2}}{9 d f}+\frac {4 \int (c+d \sin (e+f x))^{3/2} \left (-\frac {3}{2} a^3 (c-33 d) d+\frac {1}{2} a^3 \left (4 c^2-27 c d+119 d^2\right ) \sin (e+f x)\right ) \, dx}{63 d^2} \\ & = -\frac {4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 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))^{5/2}}{9 d f}+\frac {8 \int \sqrt {c+d \sin (e+f x)} \left (-\frac {3}{4} a^3 d \left (c^2-138 c d-119 d^2\right )+\frac {3}{4} a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sin (e+f x)\right ) \, dx}{315 d^2} \\ & = -\frac {4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}-\frac {4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 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))^{5/2}}{9 d f}+\frac {16 \int \frac {\frac {3}{8} a^3 d \left (c^3+387 c^2 d+471 c d^2+165 d^3\right )+\frac {3}{8} a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{945 d^2} \\ & = -\frac {4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}-\frac {4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 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))^{5/2}}{9 d f}-\frac {\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{315 d^3}+\frac {\left (2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{315 d^3} \\ & = -\frac {4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}-\frac {4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 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))^{5/2}}{9 d f}+\frac {\left (2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{315 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^3-27 c^2 d+114 c d^2+165 d^3\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}{315 d^3 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {4 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}-\frac {4 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 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))^{5/2}}{9 d f}+\frac {4 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\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)}}{315 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^3-27 c^2 d+114 c d^2+165 d^3\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}}}{315 d^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 0.96 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.76 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\frac {3 \left (-16 \left (d^2 \left (c^3+387 c^2 d+471 c d^2+165 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (\left (32 c^3-216 c^2 d-3828 c d^2-2910 d^3\right ) \cos (e+f x)+2 d \left (5 d (10 c+27 d) \cos (3 (e+f x))-\left (6 c^2+432 c d+511 d^2-35 d^2 \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )\right )\right )}{140 d^3 f \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1612\) vs. \(2(428)=856\).
Time = 7.22 (sec) , antiderivative size = 1613, normalized size of antiderivative = 4.35
method | result | size |
default | \(\text {Expression too large to display}\) | \(1613\) |
parts | \(\text {Expression too large to display}\) | \(3589\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.94 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (8 \, a^{3} c^{5} - 54 \, a^{3} c^{4} d + 219 \, a^{3} c^{3} d^{2} - 3 \, a^{3} c^{2} d^{3} - 699 \, a^{3} c d^{4} - 495 \, a^{3} d^{5}\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^{5} - 54 \, a^{3} c^{4} d + 219 \, a^{3} c^{3} d^{2} - 3 \, a^{3} c^{2} d^{3} - 699 \, a^{3} c d^{4} - 495 \, a^{3} d^{5}\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^{4} d - 27 i \, a^{3} c^{3} d^{2} + 111 i \, a^{3} c^{2} d^{3} + 579 i \, a^{3} c d^{4} + 357 i \, a^{3} d^{5}\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^{4} d + 27 i \, a^{3} c^{3} d^{2} - 111 i \, a^{3} c^{2} d^{3} - 579 i \, a^{3} c d^{4} - 357 i \, a^{3} d^{5}\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 (5 \, {\left (10 \, a^{3} c d^{4} + 27 \, a^{3} d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (4 \, a^{3} c^{3} d^{2} - 27 \, a^{3} c^{2} d^{3} - 516 \, a^{3} c d^{4} - 465 \, a^{3} d^{5}\right )} \cos \left (f x + e\right ) + {\left (35 \, a^{3} d^{5} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} c^{2} d^{3} + 72 \, a^{3} c d^{4} + 91 \, a^{3} d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{945 \, d^{4} f} \]
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\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=a^{3} \left (\int c \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 3 c \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 c \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int c \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 3 d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \]
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\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
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