Integrand size = 27, antiderivative size = 99 \[ \int \frac {(c+d \sin (e+f x))^n}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d},\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{f \sqrt {3+3 \sin (e+f x)}} \]
-AppellF1(1/2,-n,1,3/2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin(f*x+e))*cos(f*x+ e)*(c+d*sin(f*x+e))^n/f/(((c+d*sin(f*x+e))/(c+d))^n)/(a+a*sin(f*x+e))^(1/2 )
Leaf count is larger than twice the leaf count of optimal. \(237\) vs. \(2(99)=198\).
Time = 3.64 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.39 \[ \int \frac {(c+d \sin (e+f x))^n}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {\sec (e+f x) \sqrt {1+\sin (e+f x)} (c+d \sin (e+f x))^n \left (\operatorname {AppellF1}\left (1,\frac {1}{2},-n,2,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \sqrt {2-2 \sin (e+f x)} (1+\sin (e+f x)) \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}-\frac {4 \operatorname {AppellF1}\left (-\frac {1}{2}-n,-\frac {1}{2},-n,\frac {1}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {-c+d}{d+d \sin (e+f x)}\right ) (1+\sin (e+f x)) \sqrt {1-\frac {2}{1+\sin (e+f x)}} \left (1+\frac {-1+\frac {c}{d}}{1+\sin (e+f x)}\right )^{-n}}{1+2 n}\right )}{4 \sqrt {3} f} \]
(Sec[e + f*x]*Sqrt[1 + Sin[e + f*x]]*(c + d*Sin[e + f*x])^n*((AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Sqrt[ 2 - 2*Sin[e + f*x]]*(1 + Sin[e + f*x]))/((c + d*Sin[e + f*x])/(c - d))^n - (4*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (-c + d)/( d + d*Sin[e + f*x])]*(1 + Sin[e + f*x])*Sqrt[1 - 2/(1 + Sin[e + f*x])])/(( 1 + 2*n)*(1 + (-1 + c/d)/(1 + Sin[e + f*x]))^n)))/(4*Sqrt[3]*f)
Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3267, 27, 154, 153}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(c+d \sin (e+f x))^n}{\sqrt {a \sin (e+f x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c+d \sin (e+f x))^n}{\sqrt {a \sin (e+f x)+a}}dx\) |
\(\Big \downarrow \) 3267 |
\(\displaystyle \frac {a^2 \cos (e+f x) \int \frac {(c+d \sin (e+f x))^n}{a (\sin (e+f x)+1) \sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \cos (e+f x) \int \frac {(c+d \sin (e+f x))^n}{(\sin (e+f x)+1) \sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 154 |
\(\displaystyle \frac {a \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} \int \frac {(c+d \sin (e+f x))^n}{(\sin (e+f x)+1) \sqrt {\frac {d}{c+d}-\frac {d \sin (e+f x)}{c+d}}}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 153 |
\(\displaystyle -\frac {a \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} \operatorname {AppellF1}\left (n+1,\frac {1}{2},1,n+2,\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d) (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}\) |
-((a*AppellF1[1 + n, 1/2, 1, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*S in[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)*f*(1 + n)*(a - a*Sin[e + f*x])*Sqrt[a + a*Sin[e + f*x]]))
3.7.65.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( b*c - a*d)], 0] && !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[b/(b*c - a*d)]^IntPart[n ]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d)), x]^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && !G tQ[Simplify[b/(b*c - a*d)], 0] && !SimplerQ[c + d*x, a + b*x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])) Subst[Int[(a + b*x)^(m - 1/2)*((c + d* x)^n/Sqrt[a - b*x]), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m , n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !IntegerQ[m]
\[\int \frac {\left (c +d \sin \left (f x +e \right )\right )^{n}}{\sqrt {a +a \sin \left (f x +e \right )}}d x\]
\[ \int \frac {(c+d \sin (e+f x))^n}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
\[ \int \frac {(c+d \sin (e+f x))^n}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int \frac {\left (c + d \sin {\left (e + f x \right )}\right )^{n}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
\[ \int \frac {(c+d \sin (e+f x))^n}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
\[ \int \frac {(c+d \sin (e+f x))^n}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^n}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]