Optimal. Leaf size=148 \[ \frac {B \sec ^3(e+f x) (a \sin (e+f x)+a)^{m+2}}{a^2 c^2 f (1-m)}+\frac {2^{m+\frac {1}{2}} (A (1-m)-B (m+2)) \sec ^3(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {1}{2}-m;-\frac {1}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{3 a c^2 f (1-m)} \]
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Rubi [A] time = 0.33, antiderivative size = 148, 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} \[ \frac {B \sec ^3(e+f x) (a \sin (e+f x)+a)^{m+2}}{a^2 c^2 f (1-m)}+\frac {2^{m+\frac {1}{2}} (A (1-m)-B (m+2)) \sec ^3(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {1}{2}-m;-\frac {1}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{3 a c^2 f (1-m)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
Rule 2860
Rule 2967
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))^2} \, dx &=\frac {\int \sec ^4(e+f x) (a+a \sin (e+f x))^{2+m} (A+B \sin (e+f x)) \, dx}{a^2 c^2}\\ &=\frac {B \sec ^3(e+f x) (a+a \sin (e+f x))^{2+m}}{a^2 c^2 f (1-m)}+\frac {\left (A-\frac {B (2+m)}{1-m}\right ) \int \sec ^4(e+f x) (a+a \sin (e+f x))^{2+m} \, dx}{a^2 c^2}\\ &=\frac {B \sec ^3(e+f x) (a+a \sin (e+f x))^{2+m}}{a^2 c^2 f (1-m)}+\frac {\left (\left (A-\frac {B (2+m)}{1-m}\right ) \sec ^3(e+f x) (a-a \sin (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{(a-a x)^{5/2}} \, dx,x,\sin (e+f x)\right )}{c^2 f}\\ &=\frac {B \sec ^3(e+f x) (a+a \sin (e+f x))^{2+m}}{a^2 c^2 f (1-m)}+\frac {\left (2^{-\frac {1}{2}+m} \left (A-\frac {B (2+m)}{1-m}\right ) \sec ^3(e+f x) (a-a \sin (e+f x))^{3/2} (a+a \sin (e+f x))^{1+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m}}{(a-a x)^{5/2}} \, dx,x,\sin (e+f x)\right )}{c^2 f}\\ &=\frac {2^{\frac {1}{2}+m} \left (A-\frac {B (2+m)}{1-m}\right ) \, _2F_1\left (-\frac {3}{2},\frac {1}{2}-m;-\frac {1}{2};\frac {1}{2} (1-\sin (e+f x))\right ) \sec ^3(e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{1+m}}{3 a c^2 f}+\frac {B \sec ^3(e+f x) (a+a \sin (e+f x))^{2+m}}{a^2 c^2 f (1-m)}\\ \end {align*}
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Mathematica [C] time = 6.93, size = 9240, normalized size = 62.43 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c^{2} \sin \left (f x + e\right ) - 2 \, c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )}{\left (c -c \sin \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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 \[ \frac {\int \frac {A \left (a \sin {\left (e + f x \right )} + a\right )^{m}}{\sin ^{2}{\left (e + f x \right )} - 2 \sin {\left (e + f x \right )} + 1}\, dx + \int \frac {B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}}{\sin ^{2}{\left (e + f x \right )} - 2 \sin {\left (e + f x \right )} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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