Optimal. Leaf size=76 \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+\frac {3}{2};m+\frac {5}{2};\frac {1}{2} (\sin (e+f x)+1)\right )}{a c f (2 m+3) \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.36, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2841, 2745, 2667, 68} \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+\frac {3}{2};m+\frac {5}{2};\frac {1}{2} (\sin (e+f x)+1)\right )}{a c f (2 m+3) \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 68
Rule 2667
Rule 2745
Rule 2841
Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx &=\frac {\int \frac {(a+a \sin (e+f x))^{1+m}}{\sqrt {c-c \sin (e+f x)}} \, dx}{a c}\\ &=\frac {\cos (e+f x) \int \sec (e+f x) (a+a \sin (e+f x))^{\frac {3}{2}+m} \, dx}{a c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {\cos (e+f x) \operatorname {Subst}\left (\int \frac {(a+x)^{\frac {1}{2}+m}}{a-x} \, dx,x,a \sin (e+f x)\right )}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {\cos (e+f x) \, _2F_1\left (1,\frac {3}{2}+m;\frac {5}{2}+m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{a c f (3+2 m) \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 6.66, size = 218, normalized size = 2.87 \[ -\frac {2^{-2 m-\frac {5}{2}} \cos ^2\left (\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a \sin (e+f x)+a)^m \left (\sec ^4\left (\frac {1}{4} \left (-e-f x+\frac {\pi }{2}\right )\right ) \sec ^2\left (\frac {1}{4} \left (-e-f x+\frac {\pi }{2}\right )\right )^{2 m} \, _2F_1\left (2 m+2,2 m+2;2 m+3;\frac {1}{2} \left (1-\tan ^2\left (\frac {1}{4} \left (-e-f x+\frac {\pi }{2}\right )\right )\right )\right )-4^{m+1} \, _2F_1\left (1,2 m+2;2 m+3;\cos \left (\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )\right )\right )\right )}{f (m+1) (c-c \sin (e+f x))^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c^{2} \sin \left (f x + e\right ) - 2 \, c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cos ^{2}{\left (e + f x \right )}}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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