3.54 \(\int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx\)

Optimal. Leaf size=274 \[ -\frac{\sqrt{2} \left (a^2+b^2 (m+1)\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt{\csc (e+f x)+1}}+\frac{\sqrt{2} a (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt{\csc (e+f x)+1}}-\frac{\cot (e+f x) (a+b \csc (e+f x))^{m+1}}{b f (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.34266, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3840, 4007, 3834, 139, 138} \[ -\frac{\sqrt{2} \left (a^2+b^2 (m+1)\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt{\csc (e+f x)+1}}+\frac{\sqrt{2} a (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac{a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt{\csc (e+f x)+1}}-\frac{\cot (e+f x) (a+b \csc (e+f x))^{m+1}}{b f (m+2)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^3*(a + b*Csc[e + f*x])^m,x]

[Out]

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

Rule 3840

Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(Cot[e + f*x]*(a
+ b*Csc[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*(b
*(m + 1) - a*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4007

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))
, x_Symbol] :> Dist[(A*b - a*B)/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] + Dist[B/b, Int[Csc[e + f*x
]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^
2, 0]

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

Mathematica [F]  time = 4.181, size = 0, normalized size = 0. \[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Csc[e + f*x]^3*(a + b*Csc[e + f*x])^m,x]

[Out]

Integrate[Csc[e + f*x]^3*(a + b*Csc[e + f*x])^m, x]

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Maple [F]  time = 0.309, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( fx+e \right ) \right ) ^{3} \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)^3*(a+b*csc(f*x+e))^m,x)

[Out]

int(csc(f*x+e)^3*(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 )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3*(a+b*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e) + a)^m*csc(f*x + e)^3, 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 )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3*(a+b*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e) + a)^m*csc(f*x + e)^3, 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 ^{3}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**3*(a+b*csc(f*x+e))**m,x)

[Out]

Integral((a + b*csc(e + f*x))**m*csc(e + f*x)**3, 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 )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3*(a+b*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e) + a)^m*csc(f*x + e)^3, x)