Integrand size = 36, antiderivative size = 244 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\frac {768 c^3 \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (7+2 m) (9+2 m) \left (15+16 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}+\frac {192 c^2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt {c-c \sin (e+f x)}}{a f (9+2 m) \left (35+24 m+4 m^2\right )}+\frac {24 c \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2}}{a f \left (63+32 m+4 m^2\right )}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)} \]
[Out]
Time = 0.46 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2920, 2819, 2817} \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\frac {768 c^3 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right ) \sqrt {c-c \sin (e+f x)}}+\frac {192 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9) \left (4 m^2+24 m+35\right )}+\frac {24 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f \left (4 m^2+32 m+63\right )}+\frac {2 \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9)} \]
[In]
[Out]
Rule 2817
Rule 2819
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{7/2} \, dx}{a c} \\ & = \frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)}+\frac {12 \int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2} \, dx}{a (9+2 m)} \\ & = \frac {24 c \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2}}{a f \left (63+32 m+4 m^2\right )}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)}+\frac {(96 c) \int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2} \, dx}{a \left (63+32 m+4 m^2\right )} \\ & = \frac {192 c^2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt {c-c \sin (e+f x)}}{a f (5+2 m) \left (63+32 m+4 m^2\right )}+\frac {24 c \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2}}{a f \left (63+32 m+4 m^2\right )}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)}+\frac {\left (384 c^2\right ) \int (a+a \sin (e+f x))^{1+m} \sqrt {c-c \sin (e+f x)} \, dx}{a (5+2 m) \left (63+32 m+4 m^2\right )} \\ & = \frac {768 c^3 \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (3+2 m) (5+2 m) \left (63+32 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}+\frac {192 c^2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt {c-c \sin (e+f x)}}{a f (5+2 m) \left (63+32 m+4 m^2\right )}+\frac {24 c \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2}}{a f \left (63+32 m+4 m^2\right )}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 13.24 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.85 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\frac {(a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^{5/2} \left (\frac {\left (2205+590 m+108 m^2+8 m^3\right ) \left (\left (\frac {3}{8}+\frac {3 i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {3}{8}-\frac {3 i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (2205+590 m+108 m^2+8 m^3\right ) \left (\left (\frac {3}{8}-\frac {3 i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {3}{8}+\frac {3 i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (191 m+48 m^2+4 m^3\right ) \left ((1-i) \cos \left (\frac {3}{2} (e+f x)\right )-(1+i) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (191 m+48 m^2+4 m^3\right ) \left ((1+i) \cos \left (\frac {3}{2} (e+f x)\right )-(1-i) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {(21+2 m) \left (\left (\frac {3}{2}+\frac {3 i}{2}\right ) \cos \left (\frac {5}{2} (e+f x)\right )+\left (\frac {3}{2}-\frac {3 i}{2}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(5+2 m) (7+2 m) (9+2 m)}+\frac {(21+2 m) \left (\left (\frac {3}{2}-\frac {3 i}{2}\right ) \cos \left (\frac {5}{2} (e+f x)\right )+\left (\frac {3}{2}+\frac {3 i}{2}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(5+2 m) (7+2 m) (9+2 m)}+\frac {(15+2 m) \left (\left (\frac {3}{16}-\frac {3 i}{16}\right ) \cos \left (\frac {7}{2} (e+f x)\right )-\left (\frac {3}{16}+\frac {3 i}{16}\right ) \sin \left (\frac {7}{2} (e+f x)\right )\right )}{(7+2 m) (9+2 m)}+\frac {(15+2 m) \left (\left (\frac {3}{16}+\frac {3 i}{16}\right ) \cos \left (\frac {7}{2} (e+f x)\right )-\left (\frac {3}{16}-\frac {3 i}{16}\right ) \sin \left (\frac {7}{2} (e+f x)\right )\right )}{(7+2 m) (9+2 m)}+\frac {\left (-\frac {1}{16}+\frac {i}{16}\right ) \cos \left (\frac {9}{2} (e+f x)\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \sin \left (\frac {9}{2} (e+f x)\right )}{9+2 m}+\frac {\left (-\frac {1}{16}-\frac {i}{16}\right ) \cos \left (\frac {9}{2} (e+f x)\right )-\left (\frac {1}{16}-\frac {i}{16}\right ) \sin \left (\frac {9}{2} (e+f x)\right )}{9+2 m}\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
[In]
[Out]
\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}d x\]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.62 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{2} m^{3} + 60 \, c^{2} m^{2} + 142 \, c^{2} m + 105 \, c^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (8 \, c^{2} m^{3} + 108 \, c^{2} m^{2} + 334 \, c^{2} m + 285 \, c^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, c^{2} m^{3} + 84 \, c^{2} m^{2} + 334 \, c^{2} m + 339 \, c^{2}\right )} \cos \left (f x + e\right )^{3} - 384 \, c^{2} \cos \left (f x + e\right ) - 96 \, {\left (2 \, c^{2} m - c^{2}\right )} \cos \left (f x + e\right )^{2} - 768 \, c^{2} + {\left ({\left (8 \, c^{2} m^{3} + 60 \, c^{2} m^{2} + 142 \, c^{2} m + 105 \, c^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (8 \, c^{2} m^{3} + 84 \, c^{2} m^{2} + 238 \, c^{2} m + 195 \, c^{2}\right )} \cos \left (f x + e\right )^{3} - 384 \, c^{2} \cos \left (f x + e\right ) - 96 \, {\left (2 \, c^{2} m + 3 \, c^{2}\right )} \cos \left (f x + e\right )^{2} - 768 \, c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + {\left (16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + 945 \, f\right )} \cos \left (f x + e\right ) - {\left (16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + 945 \, f\right )} \sin \left (f x + e\right ) + 945 \, f} \]
[In]
[Out]
Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (232) = 464\).
Time = 0.41 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.29 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left ({\left (8 \, m^{3} + 108 \, m^{2} + 526 \, m + 957\right )} a^{m} c^{\frac {5}{2}} - \frac {3 \, {\left (8 \, m^{3} + 76 \, m^{2} + 142 \, m - 315\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {24 \, {\left (4 \, m^{2} + 16 \, m - 81\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {16 \, {\left (4 \, m^{3} + 36 \, m^{2} + 95 \, m + 315\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {6 \, {\left (8 \, m^{3} + 60 \, m^{2} + 206 \, m - 567\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {6 \, {\left (8 \, m^{3} + 60 \, m^{2} + 206 \, m - 567\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {16 \, {\left (4 \, m^{3} + 36 \, m^{2} + 95 \, m + 315\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {24 \, {\left (4 \, m^{2} + 16 \, m - 81\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {3 \, {\left (8 \, m^{3} + 76 \, m^{2} + 142 \, m - 315\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {{\left (8 \, m^{3} + 108 \, m^{2} + 526 \, m + 957\right )} a^{m} c^{\frac {5}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (16 \, m^{4} + 192 \, m^{3} + 824 \, m^{2} + 1488 \, m + \frac {2 \, {\left (16 \, m^{4} + 192 \, m^{3} + 824 \, m^{2} + 1488 \, m + 945\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (16 \, m^{4} + 192 \, m^{3} + 824 \, m^{2} + 1488 \, m + 945\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 945\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \]
[In]
[Out]
\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]
[In]
[Out]
Time = 16.07 (sec) , antiderivative size = 1060, normalized size of antiderivative = 4.34 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx=\text {Too large to display} \]
[In]
[Out]