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 = 0.34, 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} \[ \frac {a^4 c^3 2^{m+\frac {1}{2}} (B (3-m)-A (m+4)) \cos ^7(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left (\frac {7}{2},\frac {1}{2}-m;\frac {9}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{7 f (m+4)}-\frac {a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f (m+4)} \]
Antiderivative was successfully verified.
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Rule 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))^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*}
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Mathematica [F] time = 180.06, 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 \[ {\rm integral}\left (-{\left (B c^{3} \cos \left (f x + e\right )^{4} + {\left (3 \, A - 5 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 4 \, {\left (A - B\right )} c^{3} - {\left ({\left (A - 3 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 4 \, {\left (A - B\right )} c^{3}\right )} \sin \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int -{\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.10, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{3}\, dx \]
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 \[ -\int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
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 \[ \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
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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