Optimal. Leaf size=80 \[ \frac {(A+B) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {1}{2} (\sin (e+f x)+1)\right )}{4 a f (m+1)}+\frac {(A-B) (a \sin (e+f x)+a)^m}{2 f m} \]
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Rubi [A] time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 79, 68} \[ \frac {(A+B) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {1}{2} (\sin (e+f x)+1)\right )}{4 a f (m+1)}+\frac {(A-B) (a \sin (e+f x)+a)^m}{2 f m} \]
Antiderivative was successfully verified.
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Rule 68
Rule 79
Rule 2836
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=\frac {a \operatorname {Subst}\left (\int \frac {(a+x)^{-1+m} \left (A+\frac {B x}{a}\right )}{a-x} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=\frac {(A-B) (a+a \sin (e+f x))^m}{2 f m}+\frac {(A+B) \operatorname {Subst}\left (\int \frac {(a+x)^m}{a-x} \, dx,x,a \sin (e+f x)\right )}{2 f}\\ &=\frac {(A-B) (a+a \sin (e+f x))^m}{2 f m}+\frac {(A+B) \, _2F_1\left (1,1+m;2+m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{4 a f (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 71, normalized size = 0.89 \[ \frac {(a (\sin (e+f x)+1))^m \left (m (A+B) (\sin (e+f x)+1) \, _2F_1\left (1,m+1;m+2;\frac {1}{2} (\sin (e+f x)+1)\right )+2 (m+1) (A-B)\right )}{4 f m (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sec \left (f x + e\right ) \sin \left (f x + e\right ) + A \sec \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.12, size = 0, normalized size = 0.00 \[ \int \sec \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{\cos \left (e+f\,x\right )} \,d x \]
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 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \sin {\left (e + f x \right )}\right ) \sec {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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