3.816 \(\int (d \csc (e+f x))^n (a+a \sin (e+f x)) \, dx\)

Optimal. Leaf size=149 \[ \frac{a d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n) \sqrt{\cos ^2(e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.149725, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3238, 3787, 3772, 2643} \[ \frac{a d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n) \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(
1 - n)*Sqrt[Cos[e + f*x]^2]) + (a*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2,
(4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*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 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 (d \csc (e+f x))^n (a+a \sin (e+f x)) \, dx &=d \int (d \csc (e+f x))^{-1+n} (a+a \csc (e+f x)) \, dx\\ &=a \int (d \csc (e+f x))^n \, dx+(a d) \int (d \csc (e+f x))^{-1+n} \, dx\\ &=\left (a (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+\left (a d (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\\ &=\frac{a \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)}{f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{a \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right ) \sin ^2(e+f x)}{f (2-n) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}

Mathematica [C]  time = 1.64396, size = 280, normalized size = 1.88 \[ \frac{a 2^{n-1} \left (-1+e^{2 i (e+f x)}\right ) e^{-i (e+f n x)} \left (\frac{i e^{i (e+f x)}}{-1+e^{2 i (e+f x)}}\right )^n (\csc (e+f x)+1) \left (e^{i e} (n-1) \left (n e^{i (e+f (n+1) x)} \, _2F_1\left (1,\frac{3-n}{2};\frac{n+3}{2};e^{2 i (e+f x)}\right )+2 i (n+1) e^{i f n x} \, _2F_1\left (1,1-\frac{n}{2};\frac{n+2}{2};e^{2 i (e+f x)}\right )\right )-n (n+1) e^{i f (n-1) x} \, _2F_1\left (1,\frac{1-n}{2};\frac{n+1}{2};e^{2 i (e+f x)}\right )\right ) \csc ^{-n-1}(e+f x) (d \csc (e+f x))^n}{f (n-1) n (n+1) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 1.033, size = 0, normalized size = 0. \begin{align*} \int \left ( d\csc \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \right ) \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{\left (a \sin \left (f x + e\right ) + a\right )} \left (d \csc \left (f x + e\right )\right )^{n}\,{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((a*sin(f*x + e) + a)*(d*csc(f*x + e))^n, x)

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

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

[Out]

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

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