Integrand size = 23, antiderivative size = 106 \[ \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=-\frac {2 a^2 (5+4 n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{f (3+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {a+a \sin (e+f x)}} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2842, 21, 2855, 67} \[ \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=-\frac {2 a^2 (4 n+5) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{f (2 n+3) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt {a \sin (e+f x)+a}} \]
[In]
[Out]
Rule 21
Rule 67
Rule 2842
Rule 2855
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {a+a \sin (e+f x)}}+\frac {2 \int \frac {\sin ^n(e+f x) \left (\frac {1}{2} a^2 (5+4 n)+\frac {1}{2} a^2 (5+4 n) \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{3+2 n} \\ & = -\frac {2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {a+a \sin (e+f x)}}+\frac {(a (5+4 n)) \int \sin ^n(e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{3+2 n} \\ & = -\frac {2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^3 (5+4 n) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {x^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f (3+2 n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {2 a^2 (5+4 n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{f (3+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 7.11 (sec) , antiderivative size = 5111, normalized size of antiderivative = 48.22 \[ \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=\text {Result too large to show} \]
[In]
[Out]
\[\int \left (\sin ^{n}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
[In]
[Out]
\[ \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{n} \,d x } \]
[In]
[Out]
\[ \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sin ^{n}{\left (e + f x \right )}\, dx \]
[In]
[Out]
\[ \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{n} \,d x } \]
[In]
[Out]
\[ \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=\int {\sin \left (e+f\,x\right )}^n\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
[In]
[Out]