Integrand size = 21, antiderivative size = 89 \[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx=\frac {2 \sqrt {2} \operatorname {AppellF1}\left (\frac {3}{2}+m,-\frac {1}{2},2,\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sec (e+f x) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{2+m}}{a^2 f (3+2 m)} \]
2*AppellF1(3/2+m,2,-1/2,5/2+m,1+sin(f*x+e),1/2+1/2*sin(f*x+e))*sec(f*x+e)* (a+a*sin(f*x+e))^(2+m)*2^(1/2)*(1-sin(f*x+e))^(1/2)/a^2/f/(3+2*m)
\[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx=\int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx \]
Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.30, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3198, 154, 27, 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.
\(\displaystyle \int \cot ^2(e+f x) (a \sin (e+f x)+a)^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^m}{\tan (e+f x)^2}dx\) |
\(\Big \downarrow \) 3198 |
\(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \int \frac {\csc ^2(e+f x) \sqrt {a-a \sin (e+f x)} (\sin (e+f x) a+a)^{m+\frac {1}{2}}}{a^2}d(a \sin (e+f x))}{a f}\) |
\(\Big \downarrow \) 154 |
\(\displaystyle \frac {\sqrt {2} \sec (e+f x) (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a} \int \frac {\csc ^2(e+f x) \sqrt {1-\sin (e+f x)} (\sin (e+f x) a+a)^{m+\frac {1}{2}}}{\sqrt {2} a^2}d(a \sin (e+f x))}{a f \sqrt {\frac {a-a \sin (e+f x)}{a}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sec (e+f x) (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a} \int \frac {\csc ^2(e+f x) \sqrt {1-\sin (e+f x)} (\sin (e+f x) a+a)^{m+\frac {1}{2}}}{a^2}d(a \sin (e+f x))}{a f \sqrt {\frac {a-a \sin (e+f x)}{a}}}\) |
\(\Big \downarrow \) 153 |
\(\displaystyle \frac {2 \sqrt {2} \sec (e+f x) (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{m+2} \operatorname {AppellF1}\left (m+\frac {3}{2},-\frac {1}{2},2,m+\frac {5}{2},\frac {\sin (e+f x) a+a}{2 a},\frac {\sin (e+f x) a+a}{a}\right )}{a^3 f (2 m+3) \sqrt {\frac {a-a \sin (e+f x)}{a}}}\) |
(2*Sqrt[2]*AppellF1[3/2 + m, -1/2, 2, 5/2 + m, (a + a*Sin[e + f*x])/(2*a), (a + a*Sin[e + f*x])/a]*Sec[e + f*x]*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(2 + m))/(a^3*f*(3 + 2*m)*Sqrt[(a - a*Sin[e + f*x])/a])
3.2.38.3.1 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_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]]*(Sqrt[a - b*Sin[e + f*x]]/(b* f*Cos[e + f*x])) Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/ 2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b ^2, 0] && !IntegerQ[m] && IntegerQ[p/2]
\[\int \left (\cot ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}d x\]
\[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{2} \,d x } \]
\[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cot ^{2}{\left (e + f x \right )}\, dx \]
\[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{2} \,d x } \]
\[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^m \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]