Optimal. Leaf size=384 \[ \frac{16 c^2 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right )}+\frac{64 c^3 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left (4 m^2+8 m+3\right ) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac{4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)} \]
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Rubi [A] time = 0.932094, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3040, 2973, 2740, 2738} \[ \frac{16 c^2 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right )}+\frac{64 c^3 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left (4 m^2+8 m+3\right ) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac{4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)} \]
Antiderivative was successfully verified.
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Rule 3040
Rule 2973
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx &=\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}-\frac{2 \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (-\frac{1}{2} a c (C (7-2 m)+A (9+2 m))-a c C (1+2 m) \sin (e+f x)\right ) \, dx}{a c (9+2 m)}\\ &=-\frac{4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}+\frac{\left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx}{(7+2 m) (9+2 m)}\\ &=\frac{2 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac{4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}+\frac{\left (8 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx}{(5+2 m) (7+2 m) (9+2 m)}\\ &=\frac{16 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac{2 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac{4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}+\frac{\left (32 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right )\right ) \int (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \, dx}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}\\ &=\frac{64 c^3 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m) \sqrt{c-c \sin (e+f x)}}+\frac{16 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac{2 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac{4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}\\ \end{align*}
Mathematica [C] time = 6.967, size = 899, normalized size = 2.34 \[ \frac{(a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{5/2} \left (\frac{\left (64 A m^4+16 C m^4+896 A m^3+224 C m^3+5280 A m^2+1416 C m^2+15648 A m+648 C m+18900 A+12285 C\right ) \left (\left (\frac{1}{8}+\frac{i}{8}\right ) \cos \left (\frac{1}{2} (e+f x)\right )+\left (\frac{1}{8}-\frac{i}{8}\right ) \sin \left (\frac{1}{2} (e+f x)\right )\right )}{(2 m+1) (2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left (64 A m^4+16 C m^4+896 A m^3+224 C m^3+5280 A m^2+1416 C m^2+15648 A m+648 C m+18900 A+12285 C\right ) \left (\left (\frac{1}{8}-\frac{i}{8}\right ) \cos \left (\frac{1}{2} (e+f x)\right )+\left (\frac{1}{8}+\frac{i}{8}\right ) \sin \left (\frac{1}{2} (e+f x)\right )\right )}{(2 m+1) (2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left (24 A m^3+8 C m^3+292 A m^2+100 C m^2+1178 A m+414 C m+1575 A+1575 C\right ) \left (\left (\frac{1}{4}-\frac{i}{4}\right ) \cos \left (\frac{3}{2} (e+f x)\right )-\left (\frac{1}{4}+\frac{i}{4}\right ) \sin \left (\frac{3}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left (24 A m^3+8 C m^3+292 A m^2+100 C m^2+1178 A m+414 C m+1575 A+1575 C\right ) \left (\left (\frac{1}{4}+\frac{i}{4}\right ) \cos \left (\frac{3}{2} (e+f x)\right )-\left (\frac{1}{4}-\frac{i}{4}\right ) \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 A m^2+4 C m^2+32 A m+44 C m+63 A+189 C\right ) \left (\left (-\frac{1}{4}+\frac{i}{4}\right ) \cos \left (\frac{5}{2} (e+f x)\right )-\left (\frac{1}{4}+\frac{i}{4}\right ) \sin \left (\frac{5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7) (2 m+9)}+\frac{\left (4 A m^2+4 C m^2+32 A m+44 C m+63 A+189 C\right ) \left (\left (-\frac{1}{4}-\frac{i}{4}\right ) \cos \left (\frac{5}{2} (e+f x)\right )-\left (\frac{1}{4}-\frac{i}{4}\right ) \sin \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 ) C \sin \left (\frac{7}{2} (e+f x)\right )-\left (\frac{3}{16}+\frac{3 i}{16}\right ) C \cos \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 ) C \sin \left (\frac{7}{2} (e+f x)\right )-\left (\frac{3}{16}-\frac{3 i}{16}\right ) C \cos \left (\frac{7}{2} (e+f x)\right )\right )}{(2 m+7) (2 m+9)}+\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) C \cos \left (\frac{9}{2} (e+f x)\right )+\left (\frac{1}{16}-\frac{i}{16}\right ) C \sin \left (\frac{9}{2} (e+f x)\right )}{2 m+9}+\frac{\left (\frac{1}{16}-\frac{i}{16}\right ) C \cos \left (\frac{9}{2} (e+f x)\right )+\left (\frac{1}{16}+\frac{i}{16}\right ) C \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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Maple [F] time = 0.698, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( A+C \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.8332, size = 1204, normalized size = 3.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40549, size = 1982, normalized size = 5.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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