Integrand size = 12, antiderivative size = 72 \[ \int (3+4 \sin (c+d x))^n \, dx=-\frac {\sqrt {2} 7^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {4}{7} (1-\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt {1+\sin (c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2744, 143} \[ \int (3+4 \sin (c+d x))^n \, dx=-\frac {\sqrt {2} 7^n \cos (c+d x) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {4}{7} (1-\sin (c+d x))\right )}{d \sqrt {\sin (c+d x)+1}} \]
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Rule 143
Rule 2744
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \text {Subst}\left (\int \frac {(3+4 x)^n}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}} \\ & = -\frac {\sqrt {2} 7^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {4}{7} (1-\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt {1+\sin (c+d x)}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.15 \[ \int (3+4 \sin (c+d x))^n \, dx=\frac {\operatorname {AppellF1}\left (1+n,\frac {1}{2},\frac {1}{2},2+n,-3-4 \sin (c+d x),\frac {1}{7} (3+4 \sin (c+d x))\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x) (3+4 \sin (c+d x))^{1+n}}{\sqrt {7} d (1+n)} \]
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\[\int \left (3+4 \sin \left (d x +c \right )\right )^{n}d x\]
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\[ \int (3+4 \sin (c+d x))^n \, dx=\int { {\left (4 \, \sin \left (d x + c\right ) + 3\right )}^{n} \,d x } \]
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\[ \int (3+4 \sin (c+d x))^n \, dx=\int \left (4 \sin {\left (c + d x \right )} + 3\right )^{n}\, dx \]
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\[ \int (3+4 \sin (c+d x))^n \, dx=\int { {\left (4 \, \sin \left (d x + c\right ) + 3\right )}^{n} \,d x } \]
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\[ \int (3+4 \sin (c+d x))^n \, dx=\int { {\left (4 \, \sin \left (d x + c\right ) + 3\right )}^{n} \,d x } \]
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Timed out. \[ \int (3+4 \sin (c+d x))^n \, dx=\int {\left (4\,\sin \left (c+d\,x\right )+3\right )}^n \,d x \]
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