Integrand size = 40, antiderivative size = 264 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^4 (A+7 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.46 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3051, 2818, 2819, 2816, 2746, 31} \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {a^4 (A+7 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}+\frac {a^2 (A+7 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {a (A+7 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}} \]
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Rule 31
Rule 2746
Rule 2816
Rule 2818
Rule 2819
Rule 3051
Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx}{6 c} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {(a (A+7 B)) \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{4 c^2} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (a^2 (A+7 B)\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{2 c^3} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (a^3 (A+7 B)\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^3} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (a^4 (A+7 B) \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c^2 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}+\frac {\left (a^4 (A+7 B) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^4 (A+7 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 12.49 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.92 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2} \left (8 (A+B)-6 (3 A+5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+18 (A+3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+6 (A+7 B) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6+3 B \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sin (e+f x)\right )}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(236)=472\).
Time = 4.48 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.06
method | result | size |
default | \(\frac {a^{3} \sec \left (f x +e \right ) \left (3 A \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-6 A \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+21 B \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-42 B \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-3 B \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right )-9 A \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+18 A \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-8 \left (\sin ^{3}\left (f x +e \right )\right ) A -63 B \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+126 B \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-41 B \left (\sin ^{3}\left (f x +e \right )\right )-12 A \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+24 A \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+6 \left (\sin ^{2}\left (f x +e \right )\right ) A -84 B \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+168 B \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+54 B \left (\sin ^{2}\left (f x +e \right )\right )+12 A \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-24 A \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-6 A \sin \left (f x +e \right )+84 B \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-168 B \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-21 B \sin \left (f x +e \right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{3 c^{3} f \left (\cos ^{2}\left (f x +e \right )+2 \sin \left (f x +e \right )-2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(545\) |
parts | \(-\frac {A \sec \left (f x +e \right ) \left (6 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-3 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-18 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+9 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-8 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-24 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )+12 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+6 \left (\cos ^{2}\left (f x +e \right )\right )+24 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-12 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+14 \sin \left (f x +e \right )-6\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3}}{3 f \left (\cos ^{2}\left (f x +e \right )+2 \sin \left (f x +e \right )-2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{3}}-\frac {B \sec \left (f x +e \right ) \left (42 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-21 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-3 \left (\cos ^{4}\left (f x +e \right )\right )-126 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+63 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-41 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-168 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )+84 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+57 \left (\cos ^{2}\left (f x +e \right )\right )+168 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-84 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+62 \sin \left (f x +e \right )-54\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3}}{3 f \left (\cos ^{2}\left (f x +e \right )+2 \sin \left (f x +e \right )-2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{3}}\) | \(594\) |
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\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (236) = 472\).
Time = 0.36 (sec) , antiderivative size = 749, normalized size of antiderivative = 2.84 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \]
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Time = 0.46 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.36 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {2} {\left (\frac {12 \, \sqrt {2} B a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c^{\frac {7}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {6 \, \sqrt {2} {\left (A a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 7 \, B a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \log \left (-192 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 192\right )}{c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (11 \, A a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 41 \, B a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 18 \, {\left (A a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, {\left (9 \, A a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 31 \, B a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \sqrt {a}}{12 \, f} \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]
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