3.34 \(\int (a+a \csc (e+f x))^m \sin (e+f x) \, dx\)

Optimal. Leaf size=83 \[ \frac{\sqrt{2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left (m+\frac{1}{2};\frac{1}{2},2;m+\frac{3}{2};\frac{1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right )}{f (2 m+1) \sqrt{1-\csc (e+f x)}} \]

[Out]

(Sqrt[2]*AppellF1[1/2 + m, 1/2, 2, 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]])

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Rubi [A]  time = 0.0895867, antiderivative size = 83, 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 = {3828, 3827, 136} \[ \frac{\sqrt{2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left (m+\frac{1}{2};\frac{1}{2},2;m+\frac{3}{2};\frac{1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right )}{f (2 m+1) \sqrt{1-\csc (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*AppellF1[1/2 + m, 1/2, 2, 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 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 136

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

Rubi steps

\begin{align*} \int (a+a \csc (e+f x))^m \sin (e+f x) \, dx &=\left ((1+\csc (e+f x))^{-m} (a+a \csc (e+f x))^m\right ) \int (1+\csc (e+f x))^m \sin (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 ) \operatorname{Subst}\left (\int \frac{(1+x)^{-\frac{1}{2}+m}}{\sqrt{1-x} x^2} \, dx,x,\csc (e+f x)\right )}{f \sqrt{1-\csc (e+f x)}}\\ &=\frac{\sqrt{2} F_1\left (\frac{1}{2}+m;\frac{1}{2},2;\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]  time = 3.64719, size = 0, normalized size = 0. \[ \int (a+a \csc (e+f x))^m \sin (e+f x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.678, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\csc \left ( fx+e \right ) \right ) ^{m}\sin \left ( fx+e \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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