Optimal. Leaf size=148 \[ \frac{2^{m+\frac{1}{2}} (A (2-m)-B (m+3)) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{1}{2}-m;-\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{5 a^2 c^3 f (2-m)}+\frac{B \sec ^5(e+f x) (a \sin (e+f x)+a)^{m+3}}{a^3 c^3 f (2-m)} \]
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Rubi [A] time = 0.331781, antiderivative size = 148, normalized size of antiderivative = 1., 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} \[ \frac{2^{m+\frac{1}{2}} (A (2-m)-B (m+3)) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{1}{2}-m;-\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{5 a^2 c^3 f (2-m)}+\frac{B \sec ^5(e+f x) (a \sin (e+f x)+a)^{m+3}}{a^3 c^3 f (2-m)} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2860
Rule 2689
Rule 70
Rule 69
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))^3} \, dx &=\frac{\int \sec ^6(e+f x) (a+a \sin (e+f x))^{3+m} (A+B \sin (e+f x)) \, dx}{a^3 c^3}\\ &=\frac{B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac{\left (A-\frac{B (3+m)}{2-m}\right ) \int \sec ^6(e+f x) (a+a \sin (e+f x))^{3+m} \, dx}{a^3 c^3}\\ &=\frac{B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac{\left (\left (A-\frac{B (3+m)}{2-m}\right ) \sec ^5(e+f x) (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{1}{2}+m}}{(a-a x)^{7/2}} \, dx,x,\sin (e+f x)\right )}{a c^3 f}\\ &=\frac{B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac{\left (2^{-\frac{1}{2}+m} \left (A-\frac{B (3+m)}{2-m}\right ) \sec ^5(e+f x) (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{2+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)^{7/2}} \, dx,x,\sin (e+f x)\right )}{a c^3 f}\\ &=\frac{2^{\frac{1}{2}+m} \left (A-\frac{B (3+m)}{2-m}\right ) \, _2F_1\left (-\frac{5}{2},\frac{1}{2}-m;-\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right ) \sec ^5(e+f x) (1+\sin (e+f x))^{\frac{1}{2}-m} (a+a \sin (e+f x))^{2+m}}{5 a^2 c^3 f}+\frac{B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}\\ \end{align*}
Mathematica [C] time = 25.9016, size = 9702, normalized size = 65.55 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.931, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) }{ \left ( c-c\sin \left ( fx+e \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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}}{3 \, c^{3} \cos \left (f x + e\right )^{2} - 4 \, c^{3} -{\left (c^{3} \cos \left (f x + e\right )^{2} - 4 \, c^{3}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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