Integrand size = 28, antiderivative size = 100 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx=\frac {192 c^6 \cos ^7(e+f x)}{77 f (c-c \sin (e+f x))^{7/2}}+\frac {48 c^5 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{5/2}}+\frac {54 c^4 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2815, 2753, 2752} \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx=\frac {64 a^3 c^6 \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}+\frac {16 a^3 c^5 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}} \]
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Rule 2752
Rule 2753
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {2 a^3 c^4 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}+\frac {1}{11} \left (8 a^3 c^4\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {16 a^3 c^5 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}+\frac {1}{99} \left (32 a^3 c^5\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx \\ & = \frac {64 a^3 c^6 \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}+\frac {16 a^3 c^5 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}} \\ \end{align*}
Time = 4.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx=-\frac {3 c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (-365+63 \cos (2 (e+f x))+364 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{77 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 37.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (\sin \left (f x +e \right )+1\right )^{4} a^{3} \left (63 \left (\sin ^{2}\left (f x +e \right )\right )-182 \sin \left (f x +e \right )+151\right )}{693 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(71\) |
parts | \(-\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (\sin \left (f x +e \right )+1\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )-14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (\sin \left (f x +e \right )+1\right ) \left (63 \left (\sin ^{5}\left (f x +e \right )\right )-224 \left (\sin ^{4}\left (f x +e \right )\right )+355 \left (\sin ^{3}\left (f x +e \right )\right )-426 \left (\sin ^{2}\left (f x +e \right )\right )+568 \sin \left (f x +e \right )-1136\right )}{693 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (\sin \left (f x +e \right )+1\right ) \left (3 \left (\sin ^{3}\left (f x +e \right )\right )-12 \left (\sin ^{2}\left (f x +e \right )\right )+23 \sin \left (f x +e \right )-46\right )}{7 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (\sin \left (f x +e \right )+1\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )-130 \left (\sin ^{3}\left (f x +e \right )\right )+219 \left (\sin ^{2}\left (f x +e \right )\right )-292 \sin \left (f x +e \right )+584\right )}{105 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(334\) |
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Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (97) = 194\).
Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.34 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (63 \, a^{3} c^{2} \cos \left (f x + e\right )^{6} - 7 \, a^{3} c^{2} \cos \left (f x + e\right )^{5} + 10 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} - 16 \, a^{3} c^{2} \cos \left (f x + e\right )^{3} + 32 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} - 128 \, a^{3} c^{2} \cos \left (f x + e\right ) - 256 \, a^{3} c^{2} - {\left (63 \, a^{3} c^{2} \cos \left (f x + e\right )^{5} + 70 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} + 80 \, a^{3} c^{2} \cos \left (f x + e\right )^{3} + 96 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} + 128 \, a^{3} c^{2} \cos \left (f x + e\right ) + 256 \, a^{3} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{693 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx=a^{3} \left (\int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\right )\, dx + \int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\, dx + \int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{5}{\left (e + f x \right )}\, dx\right ) \]
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\[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (97) = 194\).
Time = 0.42 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.10 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx=-\frac {\sqrt {2} {\left (6930 \, a^{3} c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2310 \, a^{3} c^{2} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 693 \, a^{3} c^{2} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 495 \, a^{3} c^{2} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 77 \, a^{3} c^{2} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 63 \, a^{3} c^{2} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{11088 \, f} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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