Optimal. Leaf size=145 \[ \frac {a^3 c^2 2^{m+\frac {1}{2}} (B (2-m)-A (m+3)) \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left (\frac {5}{2},\frac {1}{2}-m;\frac {7}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{5 f (m+3)}-\frac {a^2 B c^2 \cos ^5(e+f x) (a \sin (e+f x)+a)^{m-2}}{f (m+3)} \]
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Rubi [A] time = 0.33, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {2967, 2860, 2689, 70, 69} \[ \frac {a^3 c^2 2^{m+\frac {1}{2}} (B (2-m)-A (m+3)) \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left (\frac {5}{2},\frac {1}{2}-m;\frac {7}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{5 f (m+3)}-\frac {a^2 B c^2 \cos ^5(e+f x) (a \sin (e+f x)+a)^{m-2}}{f (m+3)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
Rule 2860
Rule 2967
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (a+a \sin (e+f x))^{-2+m} (A+B \sin (e+f x)) \, dx\\ &=-\frac {a^2 B c^2 \cos ^5(e+f x) (a+a \sin (e+f x))^{-2+m}}{f (3+m)}+\left (a^2 c^2 \left (A-\frac {B (2-m)}{3+m}\right )\right ) \int \cos ^4(e+f x) (a+a \sin (e+f x))^{-2+m} \, dx\\ &=-\frac {a^2 B c^2 \cos ^5(e+f x) (a+a \sin (e+f x))^{-2+m}}{f (3+m)}+\frac {\left (a^4 c^2 \left (A-\frac {B (2-m)}{3+m}\right ) \cos ^5(e+f x)\right ) \operatorname {Subst}\left (\int (a-a x)^{3/2} (a+a x)^{-\frac {1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}\\ &=-\frac {a^2 B c^2 \cos ^5(e+f x) (a+a \sin (e+f x))^{-2+m}}{f (3+m)}+\frac {\left (2^{-\frac {1}{2}+m} a^4 c^2 \left (A-\frac {B (2-m)}{3+m}\right ) \cos ^5(e+f x) (a+a \sin (e+f x))^{-3+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m} (a-a x)^{3/2} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{5/2}}\\ &=-\frac {2^{\frac {1}{2}+m} a^3 c^2 \left (A-\frac {B (2-m)}{3+m}\right ) \cos ^5(e+f x) \, _2F_1\left (\frac {5}{2},\frac {1}{2}-m;\frac {7}{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))^{-3+m}}{5 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (a+a \sin (e+f x))^{-2+m}}{f (3+m)}\\ \end {align*}
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Mathematica [F] time = 180.08, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left ({\left (A - 2 \, B\right )} c^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (A - B\right )} c^{2} + {\left (B c^{2} \cos \left (f x + e\right )^{2} + 2 \, {\left (A - B\right )} c^{2}\right )} \sin \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.31, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{2}\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]
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 \[ c^{2} \left (\int A \left (a \sin {\left (e + f x \right )} + a\right )^{m}\, dx + \int \left (- 2 A \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\right )\, dx + \int A \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx + \int B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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