Integrand size = 21, antiderivative size = 85 \[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx=-\frac {2^{\frac {1}{2}+n} \operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{2}-n,\frac {3}{2},1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x) (1+\sin (c+d x))^{-\frac {1}{2}-n} (a+a \sin (c+d x))^n}{d} \]
-2^(1/2+n)*AppellF1(1/2,2,1/2-n,3/2,1-sin(d*x+c),1/2-1/2*sin(d*x+c))*cos(d *x+c)*(1+sin(d*x+c))^(-1/2-n)*(a+a*sin(d*x+c))^n/d
\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx=\int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx \]
Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3266, 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.
\(\displaystyle \int \csc ^2(c+d x) (a \sin (c+d x)+a)^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (c+d x)+a)^n}{\sin (c+d x)^2}dx\) |
\(\Big \downarrow \) 3266 |
\(\displaystyle (\sin (c+d x)+1)^{-n} (a \sin (c+d x)+a)^n \int \csc ^2(c+d x) (\sin (c+d x)+1)^ndx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (\sin (c+d x)+1)^{-n} (a \sin (c+d x)+a)^n \int \frac {(\sin (c+d x)+1)^n}{\sin (c+d x)^2}dx\) |
\(\Big \downarrow \) 3264 |
\(\displaystyle -\frac {\cos (c+d x) (\sin (c+d x)+1)^{-n-\frac {1}{2}} (a \sin (c+d x)+a)^n \int \frac {\csc ^2(c+d x) (\sin (c+d x)+1)^{n-\frac {1}{2}}}{\sqrt {1-\sin (c+d x)}}d(1-\sin (c+d x))}{d \sqrt {1-\sin (c+d x)}}\) |
\(\Big \downarrow \) 148 |
\(\displaystyle -\frac {2 \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac {1}{2}} (a \sin (c+d x)+a)^n \int \csc ^2(c+d x) (\sin (c+d x)+1)^{n-\frac {1}{2}}d\sqrt {1-\sin (c+d x)}}{d \sqrt {1-\sin (c+d x)}}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle -\frac {2^{n+\frac {1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac {1}{2}} (a \sin (c+d x)+a)^n \operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{2}-n,\frac {3}{2},1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right )}{d}\) |
-((2^(1/2 + n)*AppellF1[1/2, 2, 1/2 - n, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x] )^n)/d)
3.2.46.3.1 Defintions of rubi rules used
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[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Sin[e + f*x])^FracPart[m ]/(1 + (b/a)*Sin[e + f*x])^FracPart[m]) Int[(1 + (b/a)*Sin[e + f*x])^m*(d *Sin[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]
\[\int \left (\csc ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{n}d x\]
\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{2} \,d x } \]
\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{n} \csc ^{2}{\left (c + d x \right )}\, dx \]
\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{2} \,d x } \]
\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^n}{{\sin \left (c+d\,x\right )}^2} \,d x \]