\(\int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx\) [35]

Optimal result

Integrand size = 21, antiderivative size = 84 \[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=-\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},3,\frac {3}{2}+m,\frac {1}{2} (1+\csc (e+f x)),1+\csc (e+f x)\right ) \cot (e+f x) (a+a \csc (e+f x))^m}{f (1+2 m) \sqrt {1-\csc (e+f x)}} \]

[Out]

-AppellF1(1/2+m,3,1/2,3/2+m,1+csc(f*x+e),1/2+1/2*csc(f*x+e))*cot(f*x+e)*(a+a*csc(f*x+e))^m*2^(1/2)/f/(1+2*m)/(
1-csc(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3913, 3912, 141} \[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=-\frac {\sqrt {2} \cot (e+f x) (a \csc (e+f x)+a)^m \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},3,m+\frac {3}{2},\frac {1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right )}{f (2 m+1) \sqrt {1-\csc (e+f x)}} \]

[In]

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

[Out]

-((Sqrt[2]*AppellF1[1/2 + m, 1/2, 3, 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 141

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)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(
b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], 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])

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 (1+\csc (e+f x))^m \sin ^2(e+f x) \, 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} x^3} \, dx,x,\csc (e+f x)\right )}{f \sqrt {1-\csc (e+f x)}} \\ & = -\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},3,\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]

\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (a +a \csc \left (f x +e \right )\right )^{m} \sin \left (f x +e \right )^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

integral(-(cos(f*x + e)^2 - 1)*(a*csc(f*x + e) + a)^m, x)

Sympy [F]

\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int \left (a \left (\csc {\left (e + f x \right )} + 1\right )\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx \]

[In]

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

[Out]

Integral((a*(csc(e + f*x) + 1))**m*sin(e + f*x)**2, x)

Maxima [F]

\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^2\,{\left (a+\frac {a}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

int(sin(e + f*x)^2*(a + a/sin(e + f*x))^m,x)

[Out]

int(sin(e + f*x)^2*(a + a/sin(e + f*x))^m, x)