Optimal. Leaf size=244 \[ \frac {768 c^3 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right ) \sqrt {c-c \sin (e+f x)}}+\frac {192 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9) \left (4 m^2+24 m+35\right )}+\frac {24 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f \left (4 m^2+32 m+63\right )}+\frac {2 \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9)} \]
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Rubi [A] time = 0.62, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2841, 2740, 2738} \[ \frac {192 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9) \left (4 m^2+24 m+35\right )}+\frac {768 c^3 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right ) \sqrt {c-c \sin (e+f x)}}+\frac {24 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f \left (4 m^2+32 m+63\right )}+\frac {2 \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9)} \]
Antiderivative was successfully verified.
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Rule 2738
Rule 2740
Rule 2841
Rubi steps
\begin {align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx &=\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*}
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Mathematica [C] time = 6.56, size = 695, normalized size = 2.85 \[ \frac {(c-c \sin (e+f x))^{5/2} (a (\sin (e+f x)+1))^m \left (\frac {\left (8 m^3+108 m^2+590 m+2205\right ) \left (\left (\frac {3}{8}-\frac {3 i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )+\left (\frac {3}{8}+\frac {3 i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac {\left (8 m^3+108 m^2+590 m+2205\right ) \left (\left (\frac {3}{8}+\frac {3 i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )+\left (\frac {3}{8}-\frac {3 i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac {\left (4 m^3+48 m^2+191 m\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 )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac {\left (4 m^3+48 m^2+191 m\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 )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac {(2 m+21) \left (\left (\frac {3}{2}-\frac {3 i}{2}\right ) \sin \left (\frac {5}{2} (e+f x)\right )+\left (\frac {3}{2}+\frac {3 i}{2}\right ) \cos \left (\frac {5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7) (2 m+9)}+\frac {(2 m+21) \left (\left (\frac {3}{2}+\frac {3 i}{2}\right ) \sin \left (\frac {5}{2} (e+f x)\right )+\left (\frac {3}{2}-\frac {3 i}{2}\right ) \cos \left (\frac {5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7) (2 m+9)}+\frac {(2 m+15) \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 )}{(2 m+7) (2 m+9)}+\frac {(2 m+15) \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 )}{(2 m+7) (2 m+9)}+\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 )}{2 m+9}+\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 )}{2 m+9}\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 395, normalized size = 1.62 \[ -\frac {2 \, {\left ({\left (8 \, c^{2} m^{3} + 60 \, c^{2} m^{2} + 142 \, c^{2} m + 105 \, c^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (8 \, c^{2} m^{3} + 108 \, c^{2} m^{2} + 334 \, c^{2} m + 285 \, c^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, c^{2} m^{3} + 84 \, c^{2} m^{2} + 334 \, c^{2} m + 339 \, c^{2}\right )} \cos \left (f x + e\right )^{3} - 384 \, c^{2} \cos \left (f x + e\right ) - 96 \, {\left (2 \, c^{2} m - c^{2}\right )} \cos \left (f x + e\right )^{2} - 768 \, c^{2} + {\left ({\left (8 \, c^{2} m^{3} + 60 \, c^{2} m^{2} + 142 \, c^{2} m + 105 \, c^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (8 \, c^{2} m^{3} + 84 \, c^{2} m^{2} + 238 \, c^{2} m + 195 \, c^{2}\right )} \cos \left (f x + e\right )^{3} - 384 \, c^{2} \cos \left (f x + e\right ) - 96 \, {\left (2 \, c^{2} m + 3 \, c^{2}\right )} \cos \left (f x + e\right )^{2} - 768 \, c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + {\left (16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + 945 \, f\right )} \cos \left (f x + e\right ) - {\left (16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + 945 \, f\right )} \sin \left (f x + e\right ) + 945 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.80, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 558, normalized size = 2.29 \[ -\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}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.97, size = 1060, normalized size = 4.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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