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

Optimal. Leaf size=220 \[ \frac {\sqrt {2} a \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right )}{b f \sqrt {\csc (e+f x)+1}}-\frac {\sqrt {2} (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m-1;\frac {3}{2};\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right )}{b f \sqrt {\csc (e+f x)+1}} \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3838, 3834, 139, 138} \[ \frac {\sqrt {2} a \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right )}{b f \sqrt {\csc (e+f x)+1}}-\frac {\sqrt {2} (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m-1;\frac {3}{2};\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right )}{b f \sqrt {\csc (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*(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*f*Sqrt[1 + Csc[e + f*x]]*((a + b*Csc[e + f*x])/(a + b))^m)) + (Sqrt[2]*a*Ap
pellF1[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*f*Sqrt[1 + Csc[e + f*x]]*((a + b*Csc[e + f*x])/(a + b))^m)

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])

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 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 3838

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

Rubi steps

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

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Mathematica [F]  time = 2.77, size = 0, normalized size = 0.00 \[ \int \csc ^2(e+f x) (a+b \csc (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.32, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{2}\left (f x +e \right )\right ) \left (a +b \csc \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{\sin \left (e+f\,x\right )}\right )}^m}{{\sin \left (e+f\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \csc {\left (e + f x \right )}\right )^{m} \csc ^{2}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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