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

Optimal. Leaf size=104 \[ -\frac{\sqrt{2} \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{f \sqrt{\csc (e+f x)+1}} \]

[Out]

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

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Rubi [A]  time = 0.0738236, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3834, 139, 138} \[ -\frac{\sqrt{2} \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{f \sqrt{\csc (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m,x]

[Out]

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

Rule 3834

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[Cot[e + f*x]/(f*Sqr
t[1 + Csc[e + f*x]]*Sqrt[1 - Csc[e + f*x]]), Subst[Int[(a + b*x)^m/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Csc[e + f
*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&  !IntegerQ[2*m]

Rule 139

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

Rule 138

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

Rubi steps

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

Mathematica [F]  time = 1.9129, size = 0, normalized size = 0. \[ \int \csc (e+f x) (a+b \csc (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Csc[e + f*x]*(a + b*Csc[e + f*x])^m,x]

[Out]

Integrate[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x]

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Maple [F]  time = 0.511, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( fx+e \right ) \left ( a+b\csc \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*csc(e + f*x))**m*csc(e + f*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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