Integrand size = 23, antiderivative size = 222 \[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{27 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (3+3 \csc (e+f x))^2}+\frac {2 n \cos (e+f x) (d \csc (e+f x))^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2-n),-\frac {n}{2},\sin ^2(e+f x)\right )}{27 d^2 f \sqrt {\cos ^2(e+f x)}}-\frac {(1+2 n) \cos (e+f x) (d \csc (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-n),\frac {1-n}{2},\sin ^2(e+f x)\right )}{27 d f \sqrt {\cos ^2(e+f x)}} \]
-2/3*n*cot(f*x+e)*(d*csc(f*x+e))^(2+n)/a^2/d^2/f/(1+csc(f*x+e))+1/3*cot(f* x+e)*(d*csc(f*x+e))^(2+n)/d^2/f/(a+a*csc(f*x+e))^2+2/3*n*cos(f*x+e)*(d*csc (f*x+e))^(2+n)*hypergeom([1/2, -1-1/2*n],[-1/2*n],sin(f*x+e)^2)/a^2/d^2/f/ (cos(f*x+e)^2)^(1/2)-1/3*(1+2*n)*cos(f*x+e)*(d*csc(f*x+e))^(1+n)*hypergeom ([1/2, -1/2-1/2*n],[1/2-1/2*n],sin(f*x+e)^2)/a^2/d/f/(cos(f*x+e)^2)^(1/2)
\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx \]
Time = 1.08 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 3717, 3042, 4304, 25, 3042, 4508, 3042, 4274, 3042, 4259, 3042, 3122}
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.
| \(\displaystyle \int \frac {(d \csc (e+f x))^n}{(a \sin (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(d \csc (e+f x))^n}{(a \sin (e+f x)+a)^2}dx\) |
| \(\Big \downarrow \) 3717 |
| \(\displaystyle \frac {\int \frac {(d \csc (e+f x))^{n+2}}{(\csc (e+f x) a+a)^2}dx}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(d \csc (e+f x))^{n+2}}{(\csc (e+f x) a+a)^2}dx}{d^2}\) |
| \(\Big \downarrow \) 4304 |
| \(\displaystyle \frac {\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}-\frac {\int -\frac {(d \csc (e+f x))^{n+2} (a (1-n)+a (n+1) \csc (e+f x))}{\csc (e+f x) a+a}dx}{3 a^2}}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(d \csc (e+f x))^{n+2} (a (1-n)+a (n+1) \csc (e+f x))}{\csc (e+f x) a+a}dx}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(d \csc (e+f x))^{n+2} (a (1-n)+a (n+1) \csc (e+f x))}{\csc (e+f x) a+a}dx}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int (d \csc (e+f x))^{n+2} \left (a^2 (n+1) (2 n+1)-2 a^2 n (n+2) \csc (e+f x)\right )dx}{a^2}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{f (\csc (e+f x)+1)}}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int (d \csc (e+f x))^{n+2} \left (a^2 (n+1) (2 n+1)-2 a^2 n (n+2) \csc (e+f x)\right )dx}{a^2}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{f (\csc (e+f x)+1)}}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {a^2 (n+1) (2 n+1) \int (d \csc (e+f x))^{n+2}dx-\frac {2 a^2 n (n+2) \int (d \csc (e+f x))^{n+3}dx}{d}}{a^2}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{f (\csc (e+f x)+1)}}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^2 (n+1) (2 n+1) \int (d \csc (e+f x))^{n+2}dx-\frac {2 a^2 n (n+2) \int (d \csc (e+f x))^{n+3}dx}{d}}{a^2}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{f (\csc (e+f x)+1)}}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
| \(\Big \downarrow \) 4259 |
| \(\displaystyle \frac {\frac {\frac {a^2 (n+1) (2 n+1) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{-n-2}dx-\frac {2 a^2 n (n+2) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{-n-3}dx}{d}}{a^2}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{f (\csc (e+f x)+1)}}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^2 (n+1) (2 n+1) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{-n-2}dx-\frac {2 a^2 n (n+2) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{-n-3}dx}{d}}{a^2}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{f (\csc (e+f x)+1)}}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
| \(\Big \downarrow \) 3122 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 a^2 n \cos (e+f x) (d \csc (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-n-2),-\frac {n}{2},\sin ^2(e+f x)\right )}{f \sqrt {\cos ^2(e+f x)}}-\frac {a^2 d (2 n+1) \cos (e+f x) (d \csc (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-n-1),\frac {1-n}{2},\sin ^2(e+f x)\right )}{f \sqrt {\cos ^2(e+f x)}}}{a^2}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{f (\csc (e+f x)+1)}}{3 a^2}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 f (a \csc (e+f x)+a)^2}}{d^2}\) |
((Cot[e + f*x]*(d*Csc[e + f*x])^(2 + n))/(3*f*(a + a*Csc[e + f*x])^2) + (( -2*n*Cot[e + f*x]*(d*Csc[e + f*x])^(2 + n))/(f*(1 + Csc[e + f*x])) + ((2*a ^2*n*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*Hypergeometric2F1[1/2, (-2 - n) /2, -1/2*n, Sin[e + f*x]^2])/(f*Sqrt[Cos[e + f*x]^2]) - (a^2*d*(1 + 2*n)*C os[e + f*x]*(d*Csc[e + f*x])^(1 + n)*Hypergeometric2F1[1/2, (-1 - n)/2, (1 - n)/2, Sin[e + f*x]^2])/(f*Sqrt[Cos[e + f*x]^2]))/a^2)/(3*a^2))/d^2
3.9.18.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Csc[e + f*x])^(m - n*p )*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^(n - 1)*((Sin[c + d*x]/b)^(n - 1) Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-Cot[e + f*x])*(a + b*Csc[e + f*x])^m*((d*Csc [e + f*x])^n/(f*(2*m + 1))), x] + Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a*(2*m + n + 1) - b*(m + n + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ [m, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(2*m + 1))), x] - Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Cs c[e + f*x])^n*Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[ e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B , 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0]
\[\int \frac {\left (d \csc \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}d x\]
\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {\int \frac {\left (d \csc {\left (e + f x \right )}\right )^{n}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int \frac {{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]