Integrand size = 38, antiderivative size = 217 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {5 a^3 (A-15 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{128 \sqrt {2} c^{9/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}} \]
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Time = 0.39 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3046, 2938, 2759, 2728, 212} \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {5 a^3 (A-15 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{128 \sqrt {2} c^{9/2} f}+\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 c (A-15 B) \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}} \]
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Rule 212
Rule 2728
Rule 2759
Rule 2938
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {1}{16} \left (a^3 (A-15 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{13/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {1}{96} \left (5 a^3 (A-15 B)\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{9/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {\left (5 a^3 (A-15 B)\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx}{128 c^2} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (5 a^3 (A-15 B)\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{256 c^4} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (5 a^3 (A-15 B)\right ) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{128 c^4 f} \\ & = -\frac {5 a^3 (A-15 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{128 \sqrt {2} c^{9/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 13.41 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.64 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (1765 A \cos \left (\frac {1}{2} (e+f x)\right )+405 B \cos \left (\frac {1}{2} (e+f x)\right )-895 A \cos \left (\frac {3}{2} (e+f x)\right )-2703 B \cos \left (\frac {3}{2} (e+f x)\right )-397 A \cos \left (\frac {5}{2} (e+f x)\right )+579 B \cos \left (\frac {5}{2} (e+f x)\right )+15 A \cos \left (\frac {7}{2} (e+f x)\right )+543 B \cos \left (\frac {7}{2} (e+f x)\right )+(120+120 i) \sqrt [4]{-1} (A-15 B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8+1765 A \sin \left (\frac {1}{2} (e+f x)\right )+405 B \sin \left (\frac {1}{2} (e+f x)\right )+895 A \sin \left (\frac {3}{2} (e+f x)\right )+2703 B \sin \left (\frac {3}{2} (e+f x)\right )-397 A \sin \left (\frac {5}{2} (e+f x)\right )+579 B \sin \left (\frac {5}{2} (e+f x)\right )-15 A \sin \left (\frac {7}{2} (e+f x)\right )-543 B \sin \left (\frac {7}{2} (e+f x)\right )\right )}{3072 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{9/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(431\) vs. \(2(190)=380\).
Time = 4.52 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.99
method | result | size |
default | \(-\frac {a^{3} \left (-15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} \left (A -15 B \right ) \left (\cos ^{4}\left (f x +e \right )\right )-60 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} \left (A -15 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+120 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} \left (A -15 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+120 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} \left (A -15 B \right ) \sin \left (f x +e \right )+30 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {c}+292 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}-440 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}+240 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {7}{2}}+1086 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {c}-4380 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}+6600 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}-3600 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {7}{2}}-120 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}+1800 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{768 c^{\frac {17}{2}} \left (\sin \left (f x +e \right )-1\right )^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(432\) |
parts | \(\text {Expression too large to display}\) | \(1529\) |
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Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (190) = 380\).
Time = 0.30 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.92 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {15 \, \sqrt {2} {\left ({\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 5 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 8 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 20 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 8 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right ) + 16 \, {\left (A - 15 \, B\right )} a^{3} - {\left ({\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 4 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 12 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 8 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right ) + 16 \, {\left (A - 15 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (3 \, {\left (5 \, A + 181 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - {\left (191 \, A - 561 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 2 \, {\left (169 \, A + 537 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 12 \, {\left (21 \, A - 59 \, B\right )} a^{3} \cos \left (f x + e\right ) + 384 \, {\left (A + B\right )} a^{3} - {\left (3 \, {\left (5 \, A + 181 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 2 \, {\left (103 \, A - 9 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 12 \, {\left (11 \, A + 91 \, B\right )} a^{3} \cos \left (f x + e\right ) - 384 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{1536 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (190) = 380\).
Time = 0.49 (sec) , antiderivative size = 928, normalized size of antiderivative = 4.28 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]
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