Integrand size = 38, antiderivative size = 200 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {128 (7 A-9 B) c^4 \cos (e+f x)}{35 a f \sqrt {c-c \sin (e+f x)}}-\frac {32 (7 A-9 B) c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{35 a f}-\frac {12 (7 A-9 B) c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac {(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f} \]
[Out]
Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2934, 2726, 2725} \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {128 c^4 (7 A-9 B) \cos (e+f x)}{35 a f \sqrt {c-c \sin (e+f x)}}-\frac {32 c^3 (7 A-9 B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{35 a f}-\frac {12 c^2 (7 A-9 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac {c (7 A-9 B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f} \]
[In]
[Out]
Rule 2725
Rule 2726
Rule 2934
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx}{a c} \\ & = -\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}-\frac {(7 A-9 B) \int (c-c \sin (e+f x))^{7/2} \, dx}{2 a} \\ & = -\frac {(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}-\frac {(6 (7 A-9 B) c) \int (c-c \sin (e+f x))^{5/2} \, dx}{7 a} \\ & = -\frac {12 (7 A-9 B) c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac {(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}-\frac {\left (48 (7 A-9 B) c^2\right ) \int (c-c \sin (e+f x))^{3/2} \, dx}{35 a} \\ & = -\frac {32 (7 A-9 B) c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{35 a f}-\frac {12 (7 A-9 B) c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac {(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}-\frac {\left (64 (7 A-9 B) c^3\right ) \int \sqrt {c-c \sin (e+f x)} \, dx}{35 a} \\ & = -\frac {128 (7 A-9 B) c^4 \cos (e+f x)}{35 a f \sqrt {c-c \sin (e+f x)}}-\frac {32 (7 A-9 B) c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{35 a f}-\frac {12 (7 A-9 B) c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac {(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f} \\ \end{align*}
Time = 14.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.78 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (4900 A-6125 B+196 (A-2 B) \cos (2 (e+f x))+5 B \cos (4 (e+f x))+2450 A \sin (e+f x)-3430 B \sin (e+f x)-14 A \sin (3 (e+f x))+58 B \sin (3 (e+f x)))}{140 a f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))} \]
[In]
[Out]
Time = 3.19 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {2 c^{4} \left (\sin \left (f x +e \right )-1\right ) \left (5 B \left (\cos ^{4}\left (f x +e \right )\right )+\left (-7 A +29 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (49 A -103 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (308 A -436 B \right ) \sin \left (f x +e \right )+588 A -716 B \right )}{35 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(111\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.58 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {2 \, {\left (5 \, B c^{3} \cos \left (f x + e\right )^{4} + {\left (49 \, A - 103 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (147 \, A - 179 \, B\right )} c^{3} - {\left ({\left (7 \, A - 29 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 4 \, {\left (77 \, A - 109 \, B\right )} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{35 \, a f \cos \left (f x + e\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (182) = 364\).
Time = 0.34 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.39 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (91 \, c^{\frac {7}{2}} + \frac {86 \, c^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {336 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {266 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {490 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {266 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {336 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {86 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {91 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} A}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} - \frac {2 \, {\left (407 \, c^{\frac {7}{2}} + \frac {407 \, c^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {1442 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {1337 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {2030 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {1337 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {1442 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {407 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {407 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} B}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}}\right )}}{35 \, f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (182) = 364\).
Time = 0.51 (sec) , antiderivative size = 777, normalized size of antiderivative = 3.88 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
[In]
[Out]