Optimal. Leaf size=76 \[ \frac {\cos (e+f x) \, _2F_1\left (1,\frac {3}{2}+m;\frac {5}{2}+m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{a c f (3+2 m) \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A]
time = 0.24, 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 = {2920, 2824,
2746, 70} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 2746
Rule 2824
Rule 2920
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) \text {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] Leaf count is larger than twice the leaf count of optimal. \(218\) vs. \(2(76)=152\).
time = 6.98, size = 218, normalized size = 2.87 \begin {gather*} -\frac {2^{-\frac {5}{2}-2 m} \cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \left (-4^{1+m} \, _2F_1\left (1,2+2 m;3+2 m;\cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )\right )+\, _2F_1\left (2+2 m,2+2 m;3+2 m;\frac {1}{2} \left (1-\tan ^2\left (\frac {1}{4} \left (-e+\frac {\pi }{2}-f x\right )\right )\right )\right ) \sec ^4\left (\frac {1}{4} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sec ^2\left (\frac {1}{4} \left (-e+\frac {\pi }{2}-f x\right )\right )^{2 m}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a+a \sin (e+f x))^m}{f (1+m) (c-c \sin (e+f x))^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.14, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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