Integrand size = 12, antiderivative size = 104 \[ \int (3+b \sin (e+f x))^m \, dx=-\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{3+b}\right ) \cos (e+f x) (3+b \sin (e+f x))^m \left (\frac {3+b \sin (e+f x)}{3+b}\right )^{-m}}{f \sqrt {1+\sin (e+f x)}} \]
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Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2744, 144, 143} \[ \int (3+b \sin (e+f x))^m \, dx=-\frac {\sqrt {2} \cos (e+f x) (a+b \sin (e+f x))^m \left (\frac {a+b \sin (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{a+b}\right )}{f \sqrt {\sin (e+f x)+1}} \]
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Rule 143
Rule 144
Rule 2744
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \text {Subst}\left (\int \frac {(a+b x)^m}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = \frac {\left (\cos (e+f x) (a+b \sin (e+f x))^m \left (-\frac {a+b \sin (e+f x)}{-a-b}\right )^{-m}\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = -\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{a+b}\right ) \cos (e+f x) (a+b \sin (e+f x))^m \left (\frac {a+b \sin (e+f x)}{a+b}\right )^{-m}}{f \sqrt {1+\sin (e+f x)}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.12 \[ \int (3+b \sin (e+f x))^m \, dx=\frac {\operatorname {AppellF1}\left (1+m,\frac {1}{2},\frac {1}{2},2+m,-\frac {3+b \sin (e+f x)}{-3+b},\frac {3+b \sin (e+f x)}{3+b}\right ) \sec (e+f x) \sqrt {-\frac {b (-1+\sin (e+f x))}{3+b}} \sqrt {\frac {b (1+\sin (e+f x))}{-3+b}} (3+b \sin (e+f x))^{1+m}}{b f (1+m)} \]
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\[\int \left (a +b \sin \left (f x +e \right )\right )^{m}d x\]
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\[ \int (3+b \sin (e+f x))^m \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (3+b \sin (e+f x))^m \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{m}\, dx \]
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\[ \int (3+b \sin (e+f x))^m \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (3+b \sin (e+f x))^m \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (3+b \sin (e+f x))^m \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
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