Integrand size = 21, antiderivative size = 84 \[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=-\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},3,\frac {3}{2}+m,\frac {1}{2} (1+\csc (e+f x)),1+\csc (e+f x)\right ) \cot (e+f x) (a+a \csc (e+f x))^m}{f (1+2 m) \sqrt {1-\csc (e+f x)}} \]
-AppellF1(1/2+m,3,1/2,3/2+m,1+csc(f*x+e),1/2+1/2*csc(f*x+e))*cot(f*x+e)*(a +a*csc(f*x+e))^m*2^(1/2)/f/(1+2*m)/(1-csc(f*x+e))^(1/2)
\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx \]
Time = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 4315, 3042, 4314, 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.
\(\displaystyle \int \sin ^2(e+f x) (a \csc (e+f x)+a)^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \csc (e+f x)+a)^m}{\csc (e+f x)^2}dx\) |
\(\Big \downarrow \) 4315 |
\(\displaystyle (\csc (e+f x)+1)^{-m} (a \csc (e+f x)+a)^m \int (\csc (e+f x)+1)^m \sin ^2(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (\csc (e+f x)+1)^{-m} (a \csc (e+f x)+a)^m \int \frac {(\csc (e+f x)+1)^m}{\csc (e+f x)^2}dx\) |
\(\Big \downarrow \) 4314 |
\(\displaystyle \frac {\cot (e+f x) (\csc (e+f x)+1)^{-m-\frac {1}{2}} (a \csc (e+f x)+a)^m \int \frac {(\csc (e+f x)+1)^{m-\frac {1}{2}} \sin ^3(e+f x)}{\sqrt {1-\csc (e+f x)}}d\csc (e+f x)}{f \sqrt {1-\csc (e+f x)}}\) |
\(\Big \downarrow \) 153 |
\(\displaystyle -\frac {\sqrt {2} \cot (e+f x) (a \csc (e+f x)+a)^m \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},3,m+\frac {3}{2},\frac {1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right )}{f (2 m+1) \sqrt {1-\csc (e+f x)}}\) |
-((Sqrt[2]*AppellF1[1/2 + m, 1/2, 3, 3/2 + m, (1 + Csc[e + f*x])/2, 1 + Cs c[e + f*x]]*Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc [e + f*x]]))
3.1.35.3.1 Defintions of rubi rules used
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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^2*d*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x ]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(d*x)^(n - 1)*((a + b*x)^(m - 1/2 )/Sqrt[a - b*x]), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Csc[e + f*x])^FracPart[m ]/(1 + (b/a)*Csc[e + f*x])^FracPart[m]) Int[(1 + (b/a)*Csc[e + f*x])^m*(d *Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^ 2, 0] && !IntegerQ[m] && !GtQ[a, 0]
\[\int \left (a +a \csc \left (f x +e \right )\right )^{m} \sin \left (f x +e \right )^{2}d x\]
\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]
\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int \left (a \left (\csc {\left (e + f x \right )} + 1\right )\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx \]
\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]
\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^2\,{\left (a+\frac {a}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]