3.31 \(\int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx\)

Optimal. Leaf size=109 \[ -\frac{2^{m+\frac{1}{2}} m \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac{1}{2}} (a \csc (e+f x)+a)^m \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2}-m,\frac{3}{2},\frac{1}{2} (1-\csc (e+f x))\right )}{f (m+1)}-\frac{\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)} \]

[Out]

-((Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*m*Cot[e + f*x]*(1 + Csc[e + f*x])^(-1/2 -
m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Csc[e + f*x])/2])/(f*(1 + m))

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Rubi [A]  time = 0.103176, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3798, 3828, 3827, 69} \[ -\frac{2^{m+\frac{1}{2}} m \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac{1}{2}} (a \csc (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x))\right )}{f (m+1)}-\frac{\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^2*(a + a*Csc[e + f*x])^m,x]

[Out]

-((Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*m*Cot[e + f*x]*(1 + Csc[e + f*x])^(-1/2 -
m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Csc[e + f*x])/2])/(f*(1 + m))

Rule 3798

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

Rule 3828

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[(a^In
tPart[m]*(a + b*Csc[e + f*x])^FracPart[m])/(1 + (b*Csc[e + f*x])/a)^FracPart[m], Int[(1 + (b*Csc[e + f*x])/a)^
m*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] &&  !GtQ
[a, 0]

Rule 3827

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[(a^2*
d*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[((d*x)^(n - 1)*(a + b*x)^(m -
1/2))/Sqrt[a - b*x], x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !In
tegerQ[m] && GtQ[a, 0]

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx &=-\frac{\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}+\frac{m \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx}{1+m}\\ &=-\frac{\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}+\frac{\left (m (1+\csc (e+f x))^{-m} (a+a \csc (e+f x))^m\right ) \int \csc (e+f x) (1+\csc (e+f x))^m \, dx}{1+m}\\ &=-\frac{\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}+\frac{\left (m \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}} \, dx,x,\csc (e+f x)\right )}{f (1+m) \sqrt{1-\csc (e+f x)}}\\ &=-\frac{\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}-\frac{2^{\frac{1}{2}+m} m \cot (e+f x) (1+\csc (e+f x))^{-\frac{1}{2}-m} (a+a \csc (e+f x))^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x))\right )}{f (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.431278, size = 131, normalized size = 1.2 \[ -\frac{\cot \left (\frac{1}{2} (e+f x)\right ) \left (\tan \left (\frac{1}{2} (e+f x)\right )+1\right )^{-2 m} (a (\csc (e+f x)+1))^m \left (-2 \text{Hypergeometric2F1}\left (-2 m-1,-m-1,-m,-\tan \left (\frac{1}{2} (e+f x)\right )\right )+2 \text{Hypergeometric2F1}\left (-m-1,-2 m,-m,-\tan \left (\frac{1}{2} (e+f x)\right )\right )+\text{Hypergeometric2F1}\left (-m-1,-2 (m+1),-m,-\tan \left (\frac{1}{2} (e+f x)\right )\right )\right )}{2 f (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^2*(a + a*Csc[e + f*x])^m,x]

[Out]

-(Cot[(e + f*x)/2]*(a*(1 + Csc[e + f*x]))^m*(-2*Hypergeometric2F1[-1 - 2*m, -1 - m, -m, -Tan[(e + f*x)/2]] + 2
*Hypergeometric2F1[-1 - m, -2*m, -m, -Tan[(e + f*x)/2]] + Hypergeometric2F1[-1 - m, -2*(1 + m), -m, -Tan[(e +
f*x)/2]]))/(2*f*(1 + m)*(1 + Tan[(e + f*x)/2])^(2*m))

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Maple [F]  time = 0.448, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( fx+e \right ) \right ) ^{2} \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(csc(f*x+e)^2*(a+a*csc(f*x+e))^m,x)

[Out]

int(csc(f*x+e)^2*(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} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2*(a+a*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((a*csc(f*x + e) + a)^m*csc(f*x + e)^2, 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} \csc \left (f x + e\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2*(a+a*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((a*csc(f*x + e) + a)^m*csc(f*x + e)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\csc{\left (e + f x \right )} + 1\right )\right )^{m} \csc ^{2}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**2*(a+a*csc(f*x+e))**m,x)

[Out]

Integral((a*(csc(e + f*x) + 1))**m*csc(e + f*x)**2, 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} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2*(a+a*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((a*csc(f*x + e) + a)^m*csc(f*x + e)^2, x)