\(\int \csc (e+f x) (a+a \csc (e+f x))^m \, dx\) [32]

Optimal result

Integrand size = 19, antiderivative size = 74 \[ \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx=-\frac {2^{\frac {1}{2}+m} \cot (e+f x) (1+\csc (e+f x))^{-\frac {1}{2}-m} (a+a \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x))\right )}{f} \]

[Out]

-2^(1/2+m)*cot(f*x+e)*(1+csc(f*x+e))^(-1/2-m)*(a+a*csc(f*x+e))^m*hypergeom([1/2, 1/2-m],[3/2],1/2-1/2*csc(f*x+
e))/f

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3913, 3912, 71} \[ \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx=-\frac {2^{m+\frac {1}{2}} \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac {1}{2}} (a \csc (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x))\right )}{f} \]

[In]

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

[Out]

-((2^(1/2 + 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)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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]))

Rule 3912

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 3913

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[a^Int
Part[m]*((a + b*Csc[e + f*x])^FracPart[m]/(1 + (b/a)*Csc[e + f*x])^FracPart[m]), Int[(1 + (b/a)*Csc[e + f*x])^
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]

Rubi steps \begin{align*} \text {integral}& = \left ((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 \\ & = \frac {\left (\cot (e+f x) (1+\csc (e+f x))^{-\frac {1}{2}-m} (a+a \csc (e+f x))^m\right ) \text {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m}}{\sqrt {1-x}} \, dx,x,\csc (e+f x)\right )}{f \sqrt {1-\csc (e+f x)}} \\ & = -\frac {2^{\frac {1}{2}+m} \cot (e+f x) (1+\csc (e+f x))^{-\frac {1}{2}-m} (a+a \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x))\right )}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81 \[ \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx=-\frac {(a (1+\csc (e+f x)))^m \operatorname {Hypergeometric2F1}\left (-2 m,-m,1-m,-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )^{-2 m}}{f m} \]

[In]

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

[Out]

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

Maple [F]

\[\int \csc \left (f x +e \right ) \left (a +a \csc \left (f x +e \right )\right )^{m}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx=\int \left (a \left (\csc {\left (e + f x \right )} + 1\right )\right )^{m} \csc {\left (e + f x \right )}\, dx \]

[In]

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

[Out]

Integral((a*(csc(e + f*x) + 1))**m*csc(e + f*x), x)

Maxima [F]

\[ \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx=\int \frac {{\left (a+\frac {a}{\sin \left (e+f\,x\right )}\right )}^m}{\sin \left (e+f\,x\right )} \,d x \]

[In]

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

[Out]

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