\(\int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx\) [818]

Optimal result

Integrand size = 23, antiderivative size = 222 \[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{27 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (3+3 \csc (e+f x))^2}+\frac {2 n \cos (e+f x) (d \csc (e+f x))^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2-n),-\frac {n}{2},\sin ^2(e+f x)\right )}{27 d^2 f \sqrt {\cos ^2(e+f x)}}-\frac {(1+2 n) \cos (e+f x) (d \csc (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-n),\frac {1-n}{2},\sin ^2(e+f x)\right )}{27 d f \sqrt {\cos ^2(e+f x)}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3317, 3902, 4105, 3872, 3857, 2722} \[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {2 n \cos (e+f x) (d \csc (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-n-2),-\frac {n}{2},\sin ^2(e+f x)\right )}{3 a^2 d^2 f \sqrt {\cos ^2(e+f x)}}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{3 a^2 d^2 f (\csc (e+f x)+1)}-\frac {(2 n+1) \cos (e+f x) (d \csc (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-n-1),\frac {1-n}{2},\sin ^2(e+f x)\right )}{3 a^2 d f \sqrt {\cos ^2(e+f x)}}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 d^2 f (a \csc (e+f x)+a)^2} \]

[In]

Int[(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x])^2,x]

[Out]

(-2*n*Cot[e + f*x]*(d*Csc[e + f*x])^(2 + n))/(3*a^2*d^2*f*(1 + Csc[e + f*x])) + (Cot[e + f*x]*(d*Csc[e + f*x])
^(2 + n))/(3*d^2*f*(a + a*Csc[e + f*x])^2) + (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])/(3*a^2*d^2*f*Sqrt[Cos[e + f*x]^2]) - ((1 + 2*n)*Cos[e + f*x]*(d*Csc[e +
f*x])^(1 + n)*Hypergeometric2F1[1/2, (-1 - n)/2, (1 - n)/2, Sin[e + f*x]^2])/(3*a^2*d*f*Sqrt[Cos[e + f*x]^2])

Rule 2722

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]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 3317

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[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]

Rule 3857

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]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3902

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] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*C
sc[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])

Rule 4105

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] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(d \csc (e+f x))^{2+n}}{(a+a \csc (e+f x))^2} \, dx}{d^2} \\ & = \frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}-\frac {\int \frac {(d \csc (e+f x))^{2+n} (a (-1+n)-a (1+n) \csc (e+f x))}{a+a \csc (e+f x)} \, dx}{3 a^2 d^2} \\ & = -\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{3 a^2 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}-\frac {\int (d \csc (e+f x))^{2+n} \left (-a^2 (1+n) (1+2 n)+2 a^2 n (2+n) \csc (e+f x)\right ) \, dx}{3 a^4 d^2} \\ & = -\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{3 a^2 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}-\frac {(2 n (2+n)) \int (d \csc (e+f x))^{3+n} \, dx}{3 a^2 d^3}+\frac {((1+n) (1+2 n)) \int (d \csc (e+f x))^{2+n} \, dx}{3 a^2 d^2} \\ & = -\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{3 a^2 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}-\frac {\left (2 n (2+n) (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-3-n} \, dx}{3 a^2 d^3}+\frac {\left ((1+n) (1+2 n) (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-2-n} \, dx}{3 a^2 d^2} \\ & = -\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{3 a^2 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}+\frac {2 n \cot (e+f x) \csc (e+f x) (d \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2-n),-\frac {n}{2},\sin ^2(e+f x)\right )}{3 a^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 )}{3 a^2 d f \sqrt {\cos ^2(e+f x)}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx \]

[In]

Integrate[(d*Csc[e + f*x])^n/(3 + 3*Sin[e + f*x])^2,x]

[Out]

Integrate[(d*Csc[e + f*x])^n/(3 + 3*Sin[e + f*x])^2, x]

Maple [F]

\[\int \frac {\left (d \csc \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}d x\]

[In]

int((d*csc(f*x+e))^n/(a+a*sin(f*x+e))^2,x)

[Out]

int((d*csc(f*x+e))^n/(a+a*sin(f*x+e))^2,x)

Fricas [F]

\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*csc(f*x+e))^n/(a+a*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(-(d*csc(f*x + e))^n/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)

Sympy [F]

\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {\int \frac {\left (d \csc {\left (e + f x \right )}\right )^{n}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]

[In]

integrate((d*csc(f*x+e))**n/(a+a*sin(f*x+e))**2,x)

[Out]

Integral((d*csc(e + f*x))**n/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1), x)/a**2

Maxima [F]

\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*csc(f*x+e))^n/(a+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((d*csc(f*x + e))^n/(a*sin(f*x + e) + a)^2, x)

Giac [F]

\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*csc(f*x+e))^n/(a+a*sin(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*csc(f*x + e))^n/(a*sin(f*x + e) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int \frac {{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]

[In]

int((d/sin(e + f*x))^n/(a + a*sin(e + f*x))^2,x)

[Out]

int((d/sin(e + f*x))^n/(a + a*sin(e + f*x))^2, x)