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

Optimal. Leaf size=109 \[ -\frac {2^{m+\frac {1}{2}} m \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac {1}{2}} (a \csc (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\csc (e+f x))\right )}{f (m+1)}-\frac {\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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 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]))

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 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 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]

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*}

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Mathematica [A]  time = 1.35, size = 126, normalized size = 1.16 \[ -\frac {\tan \left (\frac {1}{2} (e+f x)\right ) \left (\tan \left (\frac {1}{2} (e+f x)\right )+1\right )^{-2 m} (a (\csc (e+f x)+1))^m \left ((m+1) \, _2F_1\left (1-m,-2 m;2-m;-\tan \left (\frac {1}{2} (e+f x)\right )\right )+(m-1) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (-m-1,-2 m;-m;-\tan \left (\frac {1}{2} (e+f x)\right )\right )\right )}{2 f (m-1) (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 2.00, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{2}\left (f x +e \right )\right ) \left (a +a \csc \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2*(a+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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {a}{\sin \left (e+f\,x\right )}\right )}^m}{{\sin \left (e+f\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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