Integrand size = 29, antiderivative size = 134 \[ \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{a d (1+n) \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{a d (2+n) \sqrt {\cos ^2(c+d x)}} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2918, 2657} \[ \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cos (c+d x) \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(c+d x)\right )}{a d (n+1) \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(c+d x)\right )}{a d (n+2) \sqrt {\cos ^2(c+d x)}} \]
[In]
[Out]
Rule 2657
Rule 2918
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^2(c+d x) \sin ^n(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^{1+n}(c+d x) \, dx}{a} \\ & = \frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{a d (1+n) \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{a d (2+n) \sqrt {\cos ^2(c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(441\) vs. \(2(134)=268\).
Time = 11.84 (sec) , antiderivative size = 441, normalized size of antiderivative = 3.29 \[ \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2^{1+n} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}\right )^n \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^n \left (\frac {\operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},4+n,\frac {3+n}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{1+n}+\tan \left (\frac {1}{2} (c+d x)\right ) \left (-\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {2+n}{2},4+n,\frac {4+n}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{2+n}+\tan \left (\frac {1}{2} (c+d x)\right ) \left (-\frac {\operatorname {Hypergeometric2F1}\left (\frac {3+n}{2},4+n,\frac {5+n}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{3+n}+\frac {4 \operatorname {Hypergeometric2F1}\left (\frac {4+n}{2},4+n,\frac {6+n}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{4+n}-\frac {\operatorname {Hypergeometric2F1}\left (4+n,\frac {5+n}{2},\frac {7+n}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )}{5+n}-\frac {2 \operatorname {Hypergeometric2F1}\left (4+n,\frac {6+n}{2},\frac {8+n}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan ^3\left (\frac {1}{2} (c+d x)\right )}{6+n}+\frac {\operatorname {Hypergeometric2F1}\left (4+n,\frac {7+n}{2},\frac {9+n}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan ^4\left (\frac {1}{2} (c+d x)\right )}{7+n}\right )\right )\right )}{d (a+a \sin (c+d x))} \]
[In]
[Out]
\[\int \frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{a +a \sin \left (d x +c \right )}d x\]
[In]
[Out]
\[ \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}}{a \sin \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}}{a \sin \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}}{a \sin \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^n}{a+a\,\sin \left (c+d\,x\right )} \,d x \]
[In]
[Out]