3.818 \(\int \frac{(d \csc (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx\)

Optimal. Leaf size=231 \[ \frac{2 n \cos (e+f x) (d \csc (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-n-2);-\frac{n}{2};\sin ^2(e+f x)\right )}{3 a^2 d^2 f \sqrt{\cos ^2(e+f x)}}-\frac{2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{3 a^2 d^2 f (\csc (e+f x)+1)}-\frac{(2 n+1) \cos (e+f x) (d \csc (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\sin ^2(e+f x)\right )}{3 a^2 d f \sqrt{\cos ^2(e+f x)}}+\frac{\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 d^2 f (a \csc (e+f x)+a)^2} \]

[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, -n/2, 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])

________________________________________________________________________________________

Rubi [A]  time = 0.442028, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3238, 3817, 4020, 3787, 3772, 2643} \[ \frac{2 n \cos (e+f x) (d \csc (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-n-2);-\frac{n}{2};\sin ^2(e+f x)\right )}{3 a^2 d^2 f \sqrt{\cos ^2(e+f x)}}-\frac{2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{3 a^2 d^2 f (\csc (e+f x)+1)}-\frac{(2 n+1) \cos (e+f x) (d \csc (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\sin ^2(e+f x)\right )}{3 a^2 d f \sqrt{\cos ^2(e+f x)}}+\frac{\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 d^2 f (a \csc (e+f x)+a)^2} \]

Antiderivative was successfully verified.

[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, -n/2, 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 3238

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 3817

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*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])

Rule 4020

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]

Rule 3787

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 3772

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 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int \frac{(d \csc (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx &=\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 \, _2F_1\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} \, _2F_1\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]  time = 4.64156, size = 0, normalized size = 0. \[ \int \frac{(d \csc (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.776, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\csc \left ( fx+e \right ) \right ) ^{n}}{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (d \csc \left (f x + e\right )\right )^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)