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

Optimal. Leaf size=171 \[ \frac{d n \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{a f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\sin ^2(e+f x)\right )}{a f \sqrt{\cos ^2(e+f x)}}-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a \csc (e+f x)+a)} \]

[Out]

-((Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(a + a*Csc[e + f*x]))) + (d*n*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hy
pergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(a*f*(1 - n)*Sqrt[Cos[e + f*x]^2]) + (Cos[e + f*x]
*(d*Csc[e + f*x])^n*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Sin[e + f*x]^2])/(a*f*Sqrt[Cos[e + f*x]^2])

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Rubi [A]  time = 0.237542, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3238, 3820, 3787, 3772, 2643} \[ \frac{d n \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{a f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\sin ^2(e+f x)\right )}{a f \sqrt{\cos ^2(e+f x)}}-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a \csc (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(a + a*Csc[e + f*x]))) + (d*n*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hy
pergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(a*f*(1 - n)*Sqrt[Cos[e + f*x]^2]) + (Cos[e + f*x]
*(d*Csc[e + f*x])^n*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Sin[e + f*x]^2])/(a*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 3820

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[(b*d*Cot[e
+ f*x]*(d*Csc[e + f*x])^(n - 1))/(a*f*(a + b*Csc[e + f*x])), x] + Dist[(d*(n - 1))/(a*b), Int[(d*Csc[e + f*x]
)^(n - 1)*(a - b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 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)} \, dx &=\frac{\int \frac{(d \csc (e+f x))^{1+n}}{a+a \csc (e+f x)} \, dx}{d}\\ &=-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a+a \csc (e+f x))}+\frac{n \int (d \csc (e+f x))^n (a-a \csc (e+f x)) \, dx}{a^2}\\ &=-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a+a \csc (e+f x))}+\frac{n \int (d \csc (e+f x))^n \, dx}{a}-\frac{n \int (d \csc (e+f x))^{1+n} \, dx}{a d}\\ &=-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a+a \csc (e+f x))}+\frac{\left (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 )^{-n} \, dx}{a}-\frac{\left (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 )^{-1-n} \, dx}{a d}\\ &=-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a+a \csc (e+f x))}+\frac{\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\sin ^2(e+f x)\right )}{a f \sqrt{\cos ^2(e+f x)}}+\frac{n \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right ) \sin (e+f x)}{a f (1-n) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}

Mathematica [F]  time = 2.80116, size = 0, normalized size = 0. \[ \int \frac{(d \csc (e+f x))^n}{a+a \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.483, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\csc \left ( fx+e \right ) \right ) ^{n}}{a+a\sin \left ( fx+e \right ) }}\, 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)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \csc \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a}\,{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)),x, algorithm="maxima")

[Out]

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

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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 \sin \left (f x + e\right ) + a}, 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)),x, algorithm="fricas")

[Out]

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

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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{\left (e + f x \right )} + 1}\, dx}{a} \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)),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \csc \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a}\,{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)),x, algorithm="giac")

[Out]

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