Integrand size = 12, antiderivative size = 69 \[ \int (4-3 \sin (c+d x))^n \, dx=\frac {\sqrt {2} 7^n \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2},\frac {3}{2},\frac {3}{7} (1+\sin (c+d x)),\frac {1}{2} (1+\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt {1-\sin (c+d x)}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 69, 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} \[ \int (4-3 \sin (c+d x))^n \, dx=\frac {\sqrt {2} 7^n \cos (c+d x) \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2},\frac {3}{2},\frac {3}{7} (\sin (c+d x)+1),\frac {1}{2} (\sin (c+d x)+1)\right )}{d \sqrt {1-\sin (c+d x)}} \]
[In]
[Out]
Rule 143
Rule 2744
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \text {Subst}\left (\int \frac {(4-3 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},-n,\frac {1}{2},\frac {3}{2},\frac {3}{7} (1+\sin (c+d x)),\frac {1}{2} (1+\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt {1-\sin (c+d x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.39 \[ \int (4-3 \sin (c+d x))^n \, dx=-\frac {\operatorname {AppellF1}\left (1+n,\frac {1}{2},\frac {1}{2},2+n,\frac {1}{7} (4-3 \sin (c+d x)),4-3 \sin (c+d x)\right ) \sec (c+d x) (4-3 \sin (c+d x))^{1+n} \sqrt {-1+\sin (c+d x)} \sqrt {1+\sin (c+d x)}}{\sqrt {7} d (1+n)} \]
[In]
[Out]
\[\int \left (4-3 \sin \left (d x +c \right )\right )^{n}d x\]
[In]
[Out]
\[ \int (4-3 \sin (c+d x))^n \, dx=\int { {\left (-3 \, \sin \left (d x + c\right ) + 4\right )}^{n} \,d x } \]
[In]
[Out]
\[ \int (4-3 \sin (c+d x))^n \, dx=\int \left (4 - 3 \sin {\left (c + d x \right )}\right )^{n}\, dx \]
[In]
[Out]
\[ \int (4-3 \sin (c+d x))^n \, dx=\int { {\left (-3 \, \sin \left (d x + c\right ) + 4\right )}^{n} \,d x } \]
[In]
[Out]
\[ \int (4-3 \sin (c+d x))^n \, dx=\int { {\left (-3 \, \sin \left (d x + c\right ) + 4\right )}^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int (4-3 \sin (c+d x))^n \, dx=\int {\left (4-3\,\sin \left (c+d\,x\right )\right )}^n \,d x \]
[In]
[Out]