Integrand size = 27, antiderivative size = 150 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx=-\frac {24 \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x)}{35 f \sqrt {3+3 \sin (e+f x)}}-\frac {2 \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{35 f}-\frac {4 (7 c-d) d \cos (e+f x) (3+3 \sin (e+f x))^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (3+3 \sin (e+f x))^{5/2}}{21 f} \]
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Time = 0.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2840, 2830, 2726, 2725} \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx=-\frac {8 a^2 \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x)}{105 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{105 f}-\frac {4 d (7 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 a f} \]
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Rule 2725
Rule 2726
Rule 2830
Rule 2840
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac {2 \int (a+a \sin (e+f x))^{3/2} \left (\frac {1}{2} a \left (7 c^2+5 d^2\right )+a (7 c-d) d \sin (e+f x)\right ) \, dx}{7 a} \\ & = -\frac {4 (7 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac {1}{35} \left (35 c^2+42 c d+19 d^2\right ) \int (a+a \sin (e+f x))^{3/2} \, dx \\ & = -\frac {2 a \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {4 (7 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac {1}{105} \left (4 a \left (35 c^2+42 c d+19 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx \\ & = -\frac {8 a^2 \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {4 (7 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f} \\ \end{align*}
Time = 3.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.92 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx=-\frac {\sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{3/2} \left (700 c^2+1092 c d+494 d^2-6 d (14 c+13 d) \cos (2 (e+f x))+\left (140 c^2+504 c d+253 d^2\right ) \sin (e+f x)-15 d^2 \sin (3 (e+f x))\right )}{70 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
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Time = 2.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (15 d^{2} \left (\sin ^{3}\left (f x +e \right )\right )+42 c d \left (\sin ^{2}\left (f x +e \right )\right )+39 \left (\sin ^{2}\left (f x +e \right )\right ) d^{2}+35 c^{2} \sin \left (f x +e \right )+126 c d \sin \left (f x +e \right )+52 \sin \left (f x +e \right ) d^{2}+175 c^{2}+252 c d +104 d^{2}\right )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(130\) |
parts | \(\frac {2 c^{2} \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+5\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d^{2} \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (15 \left (\sin ^{3}\left (f x +e \right )\right )+39 \left (\sin ^{2}\left (f x +e \right )\right )+52 \sin \left (f x +e \right )+104\right )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {4 c d \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin ^{2}\left (f x +e \right )+3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(198\) |
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Time = 0.26 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.53 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx=\frac {2 \, {\left (15 \, a d^{2} \cos \left (f x + e\right )^{4} + 3 \, {\left (14 \, a c d + 13 \, a d^{2}\right )} \cos \left (f x + e\right )^{3} - 140 \, a c^{2} - 168 \, a c d - 76 \, a d^{2} - {\left (35 \, a c^{2} + 84 \, a c d + 43 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (175 \, a c^{2} + 294 \, a c d + 143 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (15 \, a d^{2} \cos \left (f x + e\right )^{3} + 140 \, a c^{2} + 168 \, a c d + 76 \, a d^{2} - 6 \, {\left (7 \, a c d + 4 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (35 \, a c^{2} + 126 \, a c d + 67 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{105 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{2}\, dx \]
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\[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{2} \,d x } \]
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Time = 0.37 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.58 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx=\frac {\sqrt {2} {\left (15 \, a d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 105 \, {\left (12 \, a c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 16 \, a c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 7 \, a d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 35 \, {\left (4 \, a c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 12 \, a c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 21 \, {\left (4 \, a c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, a d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right )\right )} \sqrt {a}}{420 \, f} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]
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