Optimal. Leaf size=135 \[ \frac {2^{m-\frac {5}{2}} (A (5-m)-B m) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left (-\frac {5}{2},\frac {7}{2}-m;-\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{5 a^2 f (5-m)}+\frac {B \sec ^5(e+f x) (a \sin (e+f x)+a)^m}{f (5-m)} \]
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Rubi [A] time = 0.19, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2860, 2689, 70, 69} \[ \frac {2^{m-\frac {5}{2}} (A (5-m)-B m) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left (-\frac {5}{2},\frac {7}{2}-m;-\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{5 a^2 f (5-m)}+\frac {B \sec ^5(e+f x) (a \sin (e+f x)+a)^m}{f (5-m)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
Rule 2860
Rubi steps
\begin {align*} \int \sec ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^m}{f (5-m)}+\left (A-\frac {B m}{5-m}\right ) \int \sec ^6(e+f x) (a+a \sin (e+f x))^m \, dx\\ &=\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^m}{f (5-m)}+\frac {\left (a^2 \left (A-\frac {B m}{5-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 {7}{2}+m}}{(a-a x)^{7/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^m}{f (5-m)}+\frac {\left (2^{-\frac {7}{2}+m} \left (A-\frac {B m}{5-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 {7}{2}+m}}{(a-a x)^{7/2}} \, dx,x,\sin (e+f x)\right )}{a f}\\ &=\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^m}{f (5-m)}+\frac {2^{-\frac {5}{2}+m} \left (A-\frac {B m}{5-m}\right ) \, _2F_1\left (-\frac {5}{2},\frac {7}{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 f}\\ \end {align*}
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Mathematica [F] time = 3.91, size = 0, normalized size = 0.00 \[ \int \sec ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sec \left (f x + e\right )^{6} \sin \left (f x + e\right ) + A \sec \left (f x + e\right )^{6}\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.94, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{6}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )^{6}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\cos \left (e+f\,x\right )}^6} \,d x \]
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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