Integrand size = 25, antiderivative size = 103 \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx=-\frac {2^{4+\frac {p}{2}} a^4 (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-6-p),\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-p/2}}{d e (1+p) \sqrt {a+a \sin (c+d x)}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2768, 72, 71} \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx=-\frac {a^4 2^{\frac {p}{2}+4} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-p-6),\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d e (p+1) \sqrt {a \sin (c+d x)+a}} \]
[In]
[Out]
Rule 71
Rule 72
Rule 2768
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 (e \cos (c+d x))^{1+p} (a-a \sin (c+d x))^{\frac {1}{2} (-1-p)} (a+a \sin (c+d x))^{\frac {1}{2} (-1-p)}\right ) \text {Subst}\left (\int (a-a x)^{\frac {1}{2} (-1+p)} (a+a x)^{\frac {7}{2}+\frac {1}{2} (-1+p)} \, dx,x,\sin (c+d x)\right )}{d e} \\ & = \frac {\left (2^{3+\frac {p}{2}} a^5 (e \cos (c+d x))^{1+p} (a-a \sin (c+d x))^{\frac {1}{2} (-1-p)} (a+a \sin (c+d x))^{\frac {1}{2} (-1-p)+\frac {p}{2}} \left (\frac {a+a \sin (c+d x)}{a}\right )^{-p/2}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {7}{2}+\frac {1}{2} (-1+p)} (a-a x)^{\frac {1}{2} (-1+p)} \, dx,x,\sin (c+d x)\right )}{d e} \\ & = -\frac {2^{4+\frac {p}{2}} a^4 (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-6-p),\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-p/2}}{d e (1+p) \sqrt {a+a \sin (c+d x)}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99 \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx=-\frac {2^{4+\frac {p}{2}} a^4 \cos (c+d x) (e \cos (c+d x))^p \operatorname {Hypergeometric2F1}\left (-3-\frac {p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-p/2}}{d (1+p) \sqrt {a (1+\sin (c+d x))}} \]
[In]
[Out]
\[\int \left (e \cos \left (d x +c \right )\right )^{p} \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}d x\]
[In]
[Out]
\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
[In]
[Out]
Timed out. \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
[In]
[Out]
\[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
[In]
[Out]
Timed out. \[ \int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^p\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
[In]
[Out]