3.717 \(\int (a+b \cot (x))^n \csc ^2(x) \, dx\)

Optimal. Leaf size=20 \[ -\frac {(a+b \cot (x))^{n+1}}{b (n+1)} \]

[Out]

-(a+b*cot(x))^(1+n)/b/(1+n)

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 32} \[ -\frac {(a+b \cot (x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[x])^n*Csc[x]^2,x]

[Out]

-((a + b*Cot[x])^(1 + n)/(b*(1 + n)))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 19, normalized size = 0.95 \[ -\frac {(a+b \cot (x))^{n+1}}{b n+b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cot[x])^n*Csc[x]^2,x]

[Out]

-((a + b*Cot[x])^(1 + n)/(b + b*n))

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fricas [A]  time = 0.65, size = 38, normalized size = 1.90 \[ -\frac {{\left (b \cos \relax (x) + a \sin \relax (x)\right )} \left (\frac {b \cos \relax (x) + a \sin \relax (x)}{\sin \relax (x)}\right )^{n}}{{\left (b n + b\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(x))^n*csc(x)^2,x, algorithm="fricas")

[Out]

-(b*cos(x) + a*sin(x))*((b*cos(x) + a*sin(x))/sin(x))^n/((b*n + b)*sin(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(x))^n*csc(x)^2,x, algorithm="giac")

[Out]

integrate((b*cot(x) + a)^n*csc(x)^2, x)

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maple [A]  time = 0.08, size = 21, normalized size = 1.05 \[ -\frac {\left (a +b \cot \relax (x )\right )^{n +1}}{b \left (n +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cot(x))^n*csc(x)^2,x)

[Out]

-(a+b*cot(x))^(n+1)/b/(n+1)

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maxima [A]  time = 0.32, size = 20, normalized size = 1.00 \[ -\frac {{\left (b \cot \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(x))^n*csc(x)^2,x, algorithm="maxima")

[Out]

-(b*cot(x) + a)^(n + 1)/(b*(n + 1))

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mupad [B]  time = 3.19, size = 43, normalized size = 2.15 \[ \left \{\begin {array}{cl} -\frac {\ln \left (a+\frac {b}{\mathrm {tan}\relax (x)}\right )}{b} & \text {\ if\ \ }n=-1\\ -\frac {{\left (a+\frac {b}{\mathrm {tan}\relax (x)}\right )}^{n+1}}{b\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cot(x))^n/sin(x)^2,x)

[Out]

piecewise(n == -1, -log(a + b/tan(x))/b, n ~= -1, -(a + b/tan(x))^(n + 1)/(b*(n + 1)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(x))**n*csc(x)**2,x)

[Out]

Integral((a + b*cot(x))**n*csc(x)**2, x)

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