Optimal. Leaf size=134 \[ \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m}{4 f (c-c \sin (e+f x))^{5/2}}+\frac {(A (3-2 m)-B (5+2 m)) \cos (e+f x) \, _2F_1\left (2,\frac {1}{2}+m;\frac {3}{2}+m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^m}{16 c^2 f (1+2 m) \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A]
time = 0.22, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3051, 2824,
2746, 70} \begin {gather*} \frac {(A (3-2 m)-B (2 m+5)) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left (2,m+\frac {1}{2};m+\frac {3}{2};\frac {1}{2} (\sin (e+f x)+1)\right )}{16 c^2 f (2 m+1) \sqrt {c-c \sin (e+f x)}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m}{4 f (c-c \sin (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 2746
Rule 2824
Rule 3051
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m}{4 f (c-c \sin (e+f x))^{5/2}}+\frac {\left (B c \left (-\frac {5}{2}-m\right )-A c \left (-\frac {3}{2}+m\right )\right ) \int \frac {(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx}{4 c^2}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m}{4 f (c-c \sin (e+f x))^{5/2}}+\frac {\left (\left (B c \left (-\frac {5}{2}-m\right )-A c \left (-\frac {3}{2}+m\right )\right ) \cos (e+f x)\right ) \int \sec ^3(e+f x) (a+a \sin (e+f x))^{\frac {3}{2}+m} \, dx}{4 a c^3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m}{4 f (c-c \sin (e+f x))^{5/2}}+\frac {\left (a^2 \left (B c \left (-\frac {5}{2}-m\right )-A c \left (-\frac {3}{2}+m\right )\right ) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+x)^{-\frac {1}{2}+m}}{(a-x)^2} \, dx,x,a \sin (e+f x)\right )}{4 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^m}{4 f (c-c \sin (e+f x))^{5/2}}+\frac {(A (3-2 m)-B (5+2 m)) \cos (e+f x) \, _2F_1\left (2,\frac {1}{2}+m;\frac {3}{2}+m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^m}{16 c^2 f (1+2 m) \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 6.70, size = 8147, normalized size = 60.80 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}\, dx\]
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} \left (A + B \sin {\left (e + f x \right )}\right )}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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