Optimal. Leaf size=139 \[ \frac {2^{\frac {1}{2}+m} a^2 c (B (1-m)-A (2+m)) \cos ^3(e+f x) \, _2F_1\left (\frac {3}{2},\frac {1}{2}-m;\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-2+m}}{3 f (2+m)}-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^{-1+m}}{f (2+m)} \]
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Rubi [A]
time = 0.20, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3046, 2939,
2768, 72, 71} \begin {gather*} \frac {a^2 c 2^{m+\frac {1}{2}} (B (1-m)-A (m+2)) \cos ^3(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left (\frac {3}{2},\frac {1}{2}-m;\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{3 f (m+2)}-\frac {a B c \cos ^3(e+f x) (a \sin (e+f x)+a)^{m-1}}{f (m+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 2768
Rule 2939
Rule 3046
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx &=(a c) \int \cos ^2(e+f x) (a+a \sin (e+f x))^{-1+m} (A+B \sin (e+f x)) \, dx\\ &=-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^{-1+m}}{f (2+m)}+\left (a c \left (A-\frac {B (1-m)}{2+m}\right )\right ) \int \cos ^2(e+f x) (a+a \sin (e+f x))^{-1+m} \, dx\\ &=-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^{-1+m}}{f (2+m)}+\frac {\left (a^3 c \left (A-\frac {B (1-m)}{2+m}\right ) \cos ^3(e+f x)\right ) \text {Subst}\left (\int \sqrt {a-a x} (a+a x)^{-\frac {1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}\\ &=-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^{-1+m}}{f (2+m)}+\frac {\left (2^{-\frac {1}{2}+m} a^3 c \left (A-\frac {B (1-m)}{2+m}\right ) \cos ^3(e+f x) (a+a \sin (e+f x))^{-2+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m} \sqrt {a-a x} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{3/2}}\\ &=-\frac {2^{\frac {1}{2}+m} a^2 c \left (A-\frac {B (1-m)}{2+m}\right ) \cos ^3(e+f x) \, _2F_1\left (\frac {3}{2},\frac {1}{2}-m;\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-2+m}}{3 f}-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^{-1+m}}{f (2+m)}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.66, size = 462, normalized size = 3.32 \begin {gather*} \frac {i 4^{-1-m} c e^{i f m x} \left (1+i e^{-i (e+f x)}\right )^{-2 m} \left (-(-1)^{3/4} e^{-\frac {1}{2} i (e+f x)} \left (i+e^{i (e+f x)}\right )\right )^{2 m} \left (-\frac {i B e^{-i (2 e+f (2+m) x)} \, _2F_1\left (-2-m,-2 m;-1-m;-i e^{-i (e+f x)}\right )}{2+m}+\frac {2 (-i A+B) e^{-i (e+f (1+m) x)} \, _2F_1\left (-1-m,-2 m;-m;-i e^{-i (e+f x)}\right )}{1+m}+\frac {2 i A e^{i (e-f (-1+m) x)} \, _2F_1\left (1-m,-2 m;2-m;-i e^{-i (e+f x)}\right )}{-1+m}+\frac {2 B e^{i (e-f (-1+m) x)} \, _2F_1\left (1-m,-2 m;2-m;-i e^{-i (e+f x)}\right )}{-1+m}+\frac {i B e^{2 i e-i f (-2+m) x} \, _2F_1\left (2-m,-2 m;3-m;-i e^{-i (e+f x)}\right )}{-2+m}+\frac {4 A e^{-i f m x} \, _2F_1\left (-2 m,-m;1-m;-i e^{-i (e+f x)}\right )}{m}\right ) (-1+\sin (e+f x)) (a (1+\sin (e+f x)))^m \sin ^{-2 m}\left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.31, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c \left (\int \left (- A \left (a \sin {\left (e + f x \right )} + a\right )^{m}\right )\, dx + \int A \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx + \int \left (- B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\right )\, dx + \int B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (c-c\,\sin \left (e+f\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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