Integrand size = 27, antiderivative size = 312 \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=-\frac {12 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x)}{385 d^2 f \sqrt {3+3 \sin (e+f x)}}-\frac {8 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{385 d f}-\frac {4 (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (3+3 \sin (e+f x))^{3/2}}{385 f}-\frac {6 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{77 d^2 f \sqrt {3+3 \sin (e+f x)}}+\frac {6 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{11 d^2 f \sqrt {3+3 \sin (e+f x)}}-\frac {18 \cos (e+f x) \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f} \]
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Time = 0.45 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2842, 3060, 2849, 2840, 2830, 2725} \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=-\frac {2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x)}{3465 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {8 a^2 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3465 d f}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}-\frac {4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f} \]
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Rule 2725
Rule 2830
Rule 2840
Rule 2842
Rule 2849
Rule 3060
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac {2 \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a^2 (c+19 d)-\frac {1}{2} a^2 (3 c-23 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3 \, dx}{11 d} \\ & = \frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac {\left (a^2 \left (3 c^2-38 c d+355 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx}{99 d^2} \\ & = -\frac {2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac {\left (2 a^2 (c+d) \left (3 c^2-38 c d+355 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{231 d^2} \\ & = -\frac {4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}-\frac {2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac {\left (4 a (c+d) \left (3 c^2-38 c d+355 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{1155 d^2} \\ & = -\frac {8 a^2 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d f}-\frac {4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}-\frac {2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac {\left (2 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx}{3465 d^2} \\ & = -\frac {4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x)}{3465 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {8 a^2 (5 c-d) (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d f}-\frac {4 a (c+d) \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}-\frac {2 a^3 \left (3 c^2-38 c d+355 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f} \\ \end{align*}
Time = 8.52 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.79 \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=-\frac {\sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{5/2} \left (164472 c^3+411840 c^2 d+373098 c d^2+114640 d^3-8 \left (693 c^3+5940 c^2 d+8382 c d^2+3250 d^3\right ) \cos (2 (e+f x))+70 d^2 (33 c+32 d) \cos (4 (e+f x))+51744 c^3 \sin (e+f x)+199980 c^2 d \sin (e+f x)+205656 c d^2 \sin (e+f x)+69890 d^3 \sin (e+f x)-5940 c^2 d \sin (3 (e+f x))-17160 c d^2 \sin (3 (e+f x))-8675 d^3 \sin (3 (e+f x))+315 d^3 \sin (5 (e+f x))\right )}{3080 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
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Time = 37.72 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )+1\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (315 \left (\sin ^{5}\left (f x +e \right )\right ) d^{3}+1155 \left (\sin ^{4}\left (f x +e \right )\right ) c \,d^{2}+1120 \left (\sin ^{4}\left (f x +e \right )\right ) d^{3}+1485 \left (\sin ^{3}\left (f x +e \right )\right ) c^{2} d +4290 \left (\sin ^{3}\left (f x +e \right )\right ) c \,d^{2}+1775 \left (\sin ^{3}\left (f x +e \right )\right ) d^{3}+693 \left (\sin ^{2}\left (f x +e \right )\right ) c^{3}+5940 \left (\sin ^{2}\left (f x +e \right )\right ) c^{2} d +7227 \left (\sin ^{2}\left (f x +e \right )\right ) c \,d^{2}+2130 d^{3} \left (\sin ^{2}\left (f x +e \right )\right )+3234 c^{3} \sin \left (f x +e \right )+11385 c^{2} d \sin \left (f x +e \right )+9636 \sin \left (f x +e \right ) c \,d^{2}+2840 d^{3} \sin \left (f x +e \right )+9933 c^{3}+22770 c^{2} d +19272 c \,d^{2}+5680 d^{3}\right )}{3465 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(249\) |
parts | \(\frac {2 c^{3} \left (\sin \left (f x +e \right )+1\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )+14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d^{3} \left (\sin \left (f x +e \right )+1\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (63 \left (\sin ^{5}\left (f x +e \right )\right )+224 \left (\sin ^{4}\left (f x +e \right )\right )+355 \left (\sin ^{3}\left (f x +e \right )\right )+426 \left (\sin ^{2}\left (f x +e \right )\right )+568 \sin \left (f x +e \right )+1136\right )}{693 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 c^{2} d \left (\sin \left (f x +e \right )+1\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (3 \left (\sin ^{3}\left (f x +e \right )\right )+12 \left (\sin ^{2}\left (f x +e \right )\right )+23 \sin \left (f x +e \right )+46\right )}{7 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 c \,d^{2} \left (\sin \left (f x +e \right )+1\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )+130 \left (\sin ^{3}\left (f x +e \right )\right )+219 \left (\sin ^{2}\left (f x +e \right )\right )+292 \sin \left (f x +e \right )+584\right )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(332\) |
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Time = 0.28 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.58 \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=-\frac {2 \, {\left (315 \, a^{2} d^{3} \cos \left (f x + e\right )^{6} + 35 \, {\left (33 \, a^{2} c d^{2} + 32 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} + 7392 \, a^{2} c^{3} + 15840 \, a^{2} c^{2} d + 13728 \, a^{2} c d^{2} + 4000 \, a^{2} d^{3} - 5 \, {\left (297 \, a^{2} c^{2} d + 627 \, a^{2} c d^{2} + 320 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} - {\left (693 \, a^{2} c^{3} + 5940 \, a^{2} c^{2} d + 9537 \, a^{2} c d^{2} + 4370 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (2541 \, a^{2} c^{3} + 8415 \, a^{2} c^{2} d + 8679 \, a^{2} c d^{2} + 2965 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5313 \, a^{2} c^{3} + 14355 \, a^{2} c^{2} d + 13827 \, a^{2} c d^{2} + 4465 \, a^{2} d^{3}\right )} \cos \left (f x + e\right ) + {\left (315 \, a^{2} d^{3} \cos \left (f x + e\right )^{5} - 7392 \, a^{2} c^{3} - 15840 \, a^{2} c^{2} d - 13728 \, a^{2} c d^{2} - 4000 \, a^{2} d^{3} - 35 \, {\left (33 \, a^{2} c d^{2} + 23 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} - 5 \, {\left (297 \, a^{2} c^{2} d + 858 \, a^{2} c d^{2} + 481 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (231 \, a^{2} c^{3} + 1485 \, a^{2} c^{2} d + 1749 \, a^{2} c d^{2} + 655 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (1617 \, a^{2} c^{3} + 6435 \, a^{2} c^{2} d + 6963 \, a^{2} c d^{2} + 2465 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3465 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{3}\, dx \]
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\[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{3} \,d x } \]
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Time = 0.45 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.56 \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=\frac {\sqrt {2} {\left (315 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) + 6930 \, {\left (40 \, a^{2} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 90 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 78 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 23 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2310 \, {\left (20 \, a^{2} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 66 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 60 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 19 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 693 \, {\left (8 \, a^{2} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 60 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 72 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 25 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 495 \, {\left (12 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 30 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 13 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 385 \, {\left (6 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right )\right )} \sqrt {a}}{55440 \, f} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
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