Optimal. Leaf size=182 \[ \frac {2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c^3 f (2 m+7) \left (4 m^2+16 m+15\right )}+\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-3}}{a c^2 f \left (4 m^2+24 m+35\right )}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-4}}{a c f (2 m+7)} \]
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Rubi [A] time = 0.45, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2841, 2743, 2742} \[ \frac {2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-3}}{a c^2 f \left (4 m^2+24 m+35\right )}+\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c^3 f (2 m+7) \left (4 m^2+16 m+15\right )}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-4}}{a c f (2 m+7)} \]
Antiderivative was successfully verified.
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Rule 2742
Rule 2743
Rule 2841
Rubi steps
\begin {align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx &=\frac {\int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m} \, dx}{a c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m}}{a c f (7+2 m)}+\frac {2 \int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-3-m} \, dx}{a c^2 (7+2 m)}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m}}{a c f (7+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-3-m}}{a c^2 f (5+2 m) (7+2 m)}+\frac {2 \int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-2-m} \, dx}{a c^3 (5+2 m) (7+2 m)}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m}}{a c f (7+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-3-m}}{a c^2 f (5+2 m) (7+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-2-m}}{a c^3 f (3+2 m) (5+2 m) (7+2 m)}\\ \end {align*}
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Mathematica [A] time = 17.03, size = 176, normalized size = 0.97 \[ \frac {2^{-m-2} \cos ^3\left (\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )\right ) \sin ^{-2 m-7}\left (\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )\right ) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-5} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{-2 (-m-5)} \left (-2 (2 m+5) \sin (e+f x)+\cos \left (2 \left (-e-f x+\frac {\pi }{2}\right )\right )+4 \left (m^2+5 m+6\right )\right )}{f (2 m+3) (2 m+5) (2 m+7)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 105, normalized size = 0.58 \[ -\frac {{\left (2 \, \cos \left (f x + e\right )^{5} + 2 \, {\left (2 \, m + 5\right )} \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - {\left (4 \, m^{2} + 20 \, m + 25\right )} \cos \left (f x + e\right )^{3}\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 5}}{8 \, f m^{3} + 60 \, f m^{2} + 142 \, f m + 105 \, f} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 5} \cos \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.37, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-5-m}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 5} \cos \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.75, size = 334, normalized size = 1.84 \[ \frac {\cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (24\,m^2+120\,m+140\right )}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )}-\frac {\cos \left (5\,e+5\,f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )}+\frac {\cos \left (3\,e+3\,f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (8\,m^2+40\,m+45\right )}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )}+\frac {\sin \left (4\,e+4\,f\,x\right )\,\left (m\,4{}\mathrm {i}+10{}\mathrm {i}\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,1{}\mathrm {i}}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )}+\frac {\sin \left (2\,e+2\,f\,x\right )\,\left (m\,8{}\mathrm {i}+20{}\mathrm {i}\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,1{}\mathrm {i}}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )} \]
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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