Optimal. Leaf size=100 \[ \frac {a^2 (A+B) (a \sin (e+f x)+a)^{m-1}}{2 f (a-a \sin (e+f x))}-\frac {a (A (2-m)-B m) (a \sin (e+f x)+a)^{m-1} \, _2F_1\left (1,m-1;m;\frac {1}{2} (\sin (e+f x)+1)\right )}{4 f (1-m)} \]
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Rubi [A] time = 0.13, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2836, 78, 68} \[ \frac {a^2 (A+B) (a \sin (e+f x)+a)^{m-1}}{2 f (a-a \sin (e+f x))}-\frac {a (A (2-m)-B m) (a \sin (e+f x)+a)^{m-1} \, _2F_1\left (1,m-1;m;\frac {1}{2} (\sin (e+f x)+1)\right )}{4 f (1-m)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 78
Rule 2836
Rubi steps
\begin {align*} \int \sec ^3(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {(a+x)^{-2+m} \left (A+\frac {B x}{a}\right )}{(a-x)^2} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=\frac {a^2 (A+B) (a+a \sin (e+f x))^{-1+m}}{2 f (a-a \sin (e+f x))}+\frac {\left (a^2 (A (2-m)-B m)\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{-2+m}}{a-x} \, dx,x,a \sin (e+f x)\right )}{2 f}\\ &=-\frac {a (A (2-m)-B m) \, _2F_1\left (1,-1+m;m;\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{-1+m}}{4 f (1-m)}+\frac {a^2 (A+B) (a+a \sin (e+f x))^{-1+m}}{2 f (a-a \sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 82, normalized size = 0.82 \[ -\frac {a (a (\sin (e+f x)+1))^{m-1} \left ((A (m-2)+B m) (\sin (e+f x)-1) \, _2F_1\left (1,m-1;m;\frac {1}{2} (\sin (e+f x)+1)\right )+2 (m-1) (A+B)\right )}{4 f (m-1) (\sin (e+f x)-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sec \left (f x + e\right )^{3} \sin \left (f x + e\right ) + A \sec \left (f x + e\right )^{3}\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.75, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{3}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{3}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\cos \left (e+f\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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