Integrand size = 21, antiderivative size = 58 \[ \int \frac {\sin ^m(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},-m,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x)}{f \sqrt {1+\sin (e+f x)}} \] Output:
-AppellF1(1/2,-m,1,3/2,1-sin(f*x+e),1/2-1/2*sin(f*x+e))*cos(f*x+e)/f/(1+si n(f*x+e))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(225\) vs. \(2(58)=116\).
Time = 3.84 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.88 \[ \int \frac {\sin ^m(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx=\frac {\cos (e+f x) (-\sin (e+f x))^{-m} \sin ^m(e+f x) \sqrt {1+\sin (e+f x)} \left (1-\frac {1}{1+\sin (e+f x)}\right )^{-m} \left (4 \operatorname {AppellF1}\left (-\frac {1}{2}-m,-\frac {1}{2},-m,\frac {1}{2}-m,\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right ) (-\sin (e+f x))^m \sqrt {\frac {-1+\sin (e+f x)}{1+\sin (e+f x)}}-(1+2 m) \operatorname {AppellF1}\left (1,\frac {1}{2},-m,2,\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sqrt {2-2 \sin (e+f x)} \left (1-\frac {1}{1+\sin (e+f x)}\right )^m\right )}{4 f (1+2 m) (-1+\sin (e+f x))} \] Input:
Integrate[Sin[e + f*x]^m/Sqrt[1 + Sin[e + f*x]],x]
Output:
(Cos[e + f*x]*Sin[e + f*x]^m*Sqrt[1 + Sin[e + f*x]]*(4*AppellF1[-1/2 - m, -1/2, -m, 1/2 - m, 2/(1 + Sin[e + f*x]), (1 + Sin[e + f*x])^(-1)]*(-Sin[e + f*x])^m*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x])] - (1 + 2*m)*AppellF 1[1, 1/2, -m, 2, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sqrt[2 - 2*Sin[e + f*x]]*(1 - (1 + Sin[e + f*x])^(-1))^m))/(4*f*(1 + 2*m)*(-1 + Sin[e + f*x ])*(-Sin[e + f*x])^m*(1 - (1 + Sin[e + f*x])^(-1))^m)
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3264, 148, 333}
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.
\(\displaystyle \int \frac {\sin ^m(e+f x)}{\sqrt {\sin (e+f x)+1}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (e+f x)^m}{\sqrt {\sin (e+f x)+1}}dx\) |
\(\Big \downarrow \) 3264 |
\(\displaystyle -\frac {\cos (e+f x) \int \frac {\sin ^m(e+f x)}{\sqrt {1-\sin (e+f x)} (\sin (e+f x)+1)}d(1-\sin (e+f x))}{f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}\) |
\(\Big \downarrow \) 148 |
\(\displaystyle -\frac {2 \cos (e+f x) \int \frac {\sin ^m(e+f x)}{\sin (e+f x)+1}d\sqrt {1-\sin (e+f x)}}{f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle -\frac {\cos (e+f x) \operatorname {AppellF1}\left (\frac {1}{2},-m,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {\sin (e+f x)+1}}\) |
Input:
Int[Sin[e + f*x]^m/Sqrt[1 + Sin[e + f*x]],x]
Output:
-((AppellF1[1/2, -m, 1, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(d/b)^n*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])) Subst[Int[(a - x)^n*((2*a - x)^(m - 1 /2)/Sqrt[x]), x], x, a - b*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n} , x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]
\[\int \frac {\sin \left (f x +e \right )^{m}}{\sqrt {1+\sin \left (f x +e \right )}}d x\]
Input:
int(sin(f*x+e)^m/(1+sin(f*x+e))^(1/2),x)
Output:
int(sin(f*x+e)^m/(1+sin(f*x+e))^(1/2),x)
\[ \int \frac {\sin ^m(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{m}}{\sqrt {\sin \left (f x + e\right ) + 1}} \,d x } \] Input:
integrate(sin(f*x+e)^m/(1+sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
integral(sin(f*x + e)^m/sqrt(sin(f*x + e) + 1), x)
\[ \int \frac {\sin ^m(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx=\int \frac {\sin ^{m}{\left (e + f x \right )}}{\sqrt {\sin {\left (e + f x \right )} + 1}}\, dx \] Input:
integrate(sin(f*x+e)**m/(1+sin(f*x+e))**(1/2),x)
Output:
Integral(sin(e + f*x)**m/sqrt(sin(e + f*x) + 1), x)
\[ \int \frac {\sin ^m(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{m}}{\sqrt {\sin \left (f x + e\right ) + 1}} \,d x } \] Input:
integrate(sin(f*x+e)^m/(1+sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sin(f*x + e)^m/sqrt(sin(f*x + e) + 1), x)
\[ \int \frac {\sin ^m(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{m}}{\sqrt {\sin \left (f x + e\right ) + 1}} \,d x } \] Input:
integrate(sin(f*x+e)^m/(1+sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(sin(f*x + e)^m/sqrt(sin(f*x + e) + 1), x)
Timed out. \[ \int \frac {\sin ^m(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^m}{\sqrt {\sin \left (e+f\,x\right )+1}} \,d x \] Input:
int(sin(e + f*x)^m/(sin(e + f*x) + 1)^(1/2),x)
Output:
int(sin(e + f*x)^m/(sin(e + f*x) + 1)^(1/2), x)
\[ \int \frac {\sin ^m(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx=\int \frac {\sin \left (f x +e \right )^{m} \sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )+1}d x \] Input:
int(sin(f*x+e)^m/(1+sin(f*x+e))^(1/2),x)
Output:
int((sin(e + f*x)**m*sqrt(sin(e + f*x) + 1))/(sin(e + f*x) + 1),x)