Optimal. Leaf size=145 \[ \frac{a^4 c^3 2^{m+\frac{1}{2}} (B (3-m)-A (m+4)) \cos ^7(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left (\frac{7}{2},\frac{1}{2}-m;\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{7 f (m+4)}-\frac{a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f (m+4)} \]
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Rubi [A] time = 0.341807, antiderivative size = 145, normalized size of antiderivative = 1., 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} \[ \frac{a^4 c^3 2^{m+\frac{1}{2}} (B (3-m)-A (m+4)) \cos ^7(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left (\frac{7}{2},\frac{1}{2}-m;\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{7 f (m+4)}-\frac{a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f (m+4)} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2860
Rule 2689
Rule 70
Rule 69
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 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (a+a \sin (e+f x))^{-3+m} (A+B \sin (e+f x)) \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)}+\left (a^3 c^3 \left (A-\frac{B (3-m)}{4+m}\right )\right ) \int \cos ^6(e+f x) (a+a \sin (e+f x))^{-3+m} \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)}+\frac{\left (a^5 c^3 \left (A-\frac{B (3-m)}{4+m}\right ) \cos ^7(e+f x)\right ) \operatorname{Subst}\left (\int (a-a x)^{5/2} (a+a x)^{-\frac{1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{7/2} (a+a \sin (e+f x))^{7/2}}\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)}+\frac{\left (2^{-\frac{1}{2}+m} a^5 c^3 \left (A-\frac{B (3-m)}{4+m}\right ) \cos ^7(e+f x) (a+a \sin (e+f x))^{-4+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)^{5/2} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{7/2}}\\ &=-\frac{2^{\frac{1}{2}+m} a^4 c^3 \left (A-\frac{B (3-m)}{4+m}\right ) \cos ^7(e+f x) \, _2F_1\left (\frac{7}{2},\frac{1}{2}-m;\frac{9}{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))^{-4+m}}{7 f}-\frac{a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)}\\ \end{align*}
Mathematica [F] time = 180.084, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 3.02, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c-c\sin \left ( fx+e \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (B \sin \left (f x + e\right ) + A\right )}{\left (c \sin \left (f x + e\right ) - c\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (B c^{3} \cos \left (f x + e\right )^{4} +{\left (3 \, A - 5 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 4 \,{\left (A - B\right )} c^{3} -{\left ({\left (A - 3 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 4 \,{\left (A - B\right )} c^{3}\right )} \sin \left (f x + e\right )\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \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*} \int -{\left (B \sin \left (f x + e\right ) + A\right )}{\left (c \sin \left (f x + e\right ) - c\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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