3.33 \(\int (a+a \csc (e+f x))^m \, dx\)

Optimal. Leaf size=84 \[ -\frac{\sqrt{2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left (m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right )}{f (2 m+1) \sqrt{1-\csc (e+f x)}} \]

[Out]

-((Sqrt[2]*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1 + Csc[e + f*x])/2, 1 + Csc[e + f*x]]*Cot[e + f*x]*(a + a*Csc[
e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc[e + f*x]]))

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Rubi [A]  time = 0.0574375, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3779, 3778, 136} \[ -\frac{\sqrt{2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left (m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right )}{f (2 m+1) \sqrt{1-\csc (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Csc[e + f*x])^m,x]

[Out]

-((Sqrt[2]*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1 + Csc[e + f*x])/2, 1 + Csc[e + f*x]]*Cot[e + f*x]*(a + a*Csc[
e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc[e + f*x]]))

Rule 3779

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[(a^IntPart[n]*(a + b*Csc[c + d*x])^FracPart
[n])/(1 + (b*Csc[c + d*x])/a)^FracPart[n], Int[(1 + (b*Csc[c + d*x])/a)^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rule 3778

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[(a^n*Cot[c + d*x])/(d*Sqrt[1 + Csc[c + d*x]
]*Sqrt[1 - Csc[c + d*x]]), Subst[Int[(1 + (b*x)/a)^(n - 1/2)/(x*Sqrt[1 - (b*x)/a]), x], x, Csc[c + d*x]], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rule 136

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*e - a*
f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps

\begin{align*} \int (a+a \csc (e+f x))^m \, dx &=\left ((1+\csc (e+f x))^{-m} (a+a \csc (e+f x))^m\right ) \int (1+\csc (e+f x))^m \, dx\\ &=\frac{\left (\cot (e+f x) (1+\csc (e+f x))^{-\frac{1}{2}-m} (a+a \csc (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{(1+x)^{-\frac{1}{2}+m}}{\sqrt{1-x} x} \, dx,x,\csc (e+f x)\right )}{f \sqrt{1-\csc (e+f x)}}\\ &=-\frac{\sqrt{2} F_1\left (\frac{1}{2}+m;\frac{1}{2},1;\frac{3}{2}+m;\frac{1}{2} (1+\csc (e+f x)),1+\csc (e+f x)\right ) \cot (e+f x) (a+a \csc (e+f x))^m}{f (1+2 m) \sqrt{1-\csc (e+f x)}}\\ \end{align*}

Mathematica [F]  time = 0.594863, size = 0, normalized size = 0. \[ \int (a+a \csc (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + a*Csc[e + f*x])^m,x]

[Out]

Integrate[(a + a*Csc[e + f*x])^m, x]

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Maple [F]  time = 0.287, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\csc \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*csc(f*x+e))^m,x)

[Out]

int((a+a*csc(f*x+e))^m,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((a*csc(f*x + e) + a)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((a*csc(f*x + e) + a)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(f*x+e))**m,x)

[Out]

Integral((a*csc(e + f*x) + a)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((a*csc(f*x + e) + a)^m, x)