Optimal. Leaf size=166 \[ -\frac {2 B c^2 \cos (e+f x) (a \sin (e+f x)+a)^{m+2}}{a^2 f (2 m+5) \sqrt {c-c \sin (e+f x)}}+\frac {4 c^2 (A-B) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}}-\frac {2 c^2 (A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.35, antiderivative size = 192, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2973, 2740, 2738} \[ -\frac {8 c^2 (B (3-2 m)-A (2 m+5)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}-\frac {2 c (B (3-2 m)-A (2 m+5)) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5)}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)} \]
Antiderivative was successfully verified.
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Rule 2738
Rule 2740
Rule 2973
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx &=-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m)}+\frac {\left (B c \left (-\frac {3}{2}+m\right )+A c \left (\frac {5}{2}+m\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx}{c \left (\frac {5}{2}+m\right )}\\ &=-\frac {2 c (B (3-2 m)-A (5+2 m)) \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)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m)}-\frac {(4 c (B (3-2 m)-A (5+2 m))) \int (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx}{(3+2 m) (5+2 m)}\\ &=-\frac {8 c^2 (B (3-2 m)-A (5+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+2 m) (3+2 m) (5+2 m) \sqrt {c-c \sin (e+f x)}}-\frac {2 c (B (3-2 m)-A (5+2 m)) \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)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m)}\\ \end {align*}
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Mathematica [A] time = 1.77, size = 174, normalized size = 1.05 \[ \frac {c \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (a (\sin (e+f x)+1))^m \left (-2 (2 m+1) (2 A m+5 A-2 B m-9 B) \sin (e+f x)+8 A m^2+40 A m+50 A+B \left (4 m^2+8 m+3\right ) \cos (2 (e+f x))-4 B m^2-16 B m-39 B\right )}{f (2 m+1) (2 m+3) (2 m+5) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 313, normalized size = 1.89 \[ \frac {2 \, {\left ({\left (4 \, B c m^{2} + 8 \, B c m + 3 \, B c\right )} \cos \left (f x + e\right )^{3} + 8 \, {\left (A + B\right )} c m + {\left (4 \, A c m^{2} + 12 \, {\left (A - B\right )} c m + {\left (5 \, A - 6 \, B\right )} c\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (5 \, A - 3 \, B\right )} c + {\left (4 \, {\left (A - B\right )} c m^{2} + 4 \, {\left (5 \, A - 3 \, B\right )} c m + {\left (25 \, A - 21 \, B\right )} c\right )} \cos \left (f x + e\right ) + {\left (8 \, {\left (A + B\right )} c m + {\left (4 \, B c m^{2} + 8 \, B c m + 3 \, B c\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (5 \, A - 3 \, B\right )} c - {\left (4 \, {\left (A - B\right )} c m^{2} + 4 \, {\left (3 \, A - 5 \, B\right )} c m + {\left (5 \, A - 9 \, B\right )} c\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + {\left (8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + 15 \, f\right )} \cos \left (f x + e\right ) - {\left (8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + 15 \, f\right )} \sin \left (f x + e\right ) + 15 \, f} \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.28, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 498, normalized size = 3.00 \[ -\frac {2 \, {\left (\frac {{\left (a^{m} c^{\frac {3}{2}} {\left (2 \, m + 5\right )} - \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m - 3\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m - 3\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m + 5\right )} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} A 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 (4 \, m^{2} + 8 \, m + 3\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a^{m} c^{\frac {3}{2}} {\left (2 \, m + 9\right )} - \frac {2 \, {\left (2 \, m^{2} + 9 \, m\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (4 \, m^{2} + 15\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (4 \, m^{2} + 15\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {2 \, {\left (2 \, m^{2} + 9 \, m\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m + 9\right )} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} B 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 (8 \, m^{3} + 36 \, m^{2} + 46 \, m + \frac {{\left (8 \, m^{3} + 36 \, m^{2} + 46 \, m + 15\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 15\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 19.20, size = 480, normalized size = 2.89 \[ \frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {c\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (45\,A-30\,B+28\,A\,m+4\,B\,m+4\,A\,m^2\right )}{f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}+\frac {c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,45{}\mathrm {i}-B\,30{}\mathrm {i}+A\,m\,28{}\mathrm {i}+B\,m\,4{}\mathrm {i}+A\,m^2\,4{}\mathrm {i}\right )}{f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}+\frac {B\,c\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (m^2\,4{}\mathrm {i}+m\,8{}\mathrm {i}+3{}\mathrm {i}\right )}{2\,f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}+\frac {B\,c\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )}{2\,f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}+\frac {c\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (10\,A-15\,B+4\,A\,m-2\,B\,m\right )}{2\,f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}+\frac {c\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,10{}\mathrm {i}-B\,15{}\mathrm {i}+A\,m\,4{}\mathrm {i}-B\,m\,2{}\mathrm {i}\right )}{2\,f\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}\right )}{{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}+\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (8\,m^3+36\,m^2+46\,m+15\right )}{m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}}} \]
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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