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

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

[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]) + (b*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 = 0.148498, 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} \[ \frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[e + f*x])^n*(a + b*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]) + (b*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+b \sin (e+f x)) \, dx &=d \int (d \csc (e+f x))^{-1+n} (b+a \csc (e+f x)) \, dx\\ &=a \int (d \csc (e+f x))^n \, dx+(b 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 (b 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{b \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 [A]  time = 0.246177, size = 105, normalized size = 0.7 \[ -\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n-1} \left (a \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(e+f x)\right )+b \sqrt{\sin ^2(e+f x)} \csc (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{3}{2};\cos ^2(e+f x)\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.145, size = 0, normalized size = 0. \begin{align*} \int \left ( d\csc \left ( fx+e \right ) \right ) ^{n} \left ( a+b\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+b*sin(f*x+e)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \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+b*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((b*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 (b \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+b*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((b*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*} \int \left (d \csc{\left (e + f x \right )}\right )^{n} \left (a + b \sin{\left (e + f x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \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+b*sin(f*x+e)),x, algorithm="giac")

[Out]

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