3.22 \(\int \frac {\csc ^2(x)}{(a+b \cot (x))^2} \, dx\)

Optimal. Leaf size=12 \[ \frac {1}{b (a+b \cot (x))} \]

[Out]

1/b/(a+b*cot(x))

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Rubi [A]  time = 0.04, antiderivative size = 12, 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} \[ \frac {1}{b (a+b \cot (x))} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2/(a + b*Cot[x])^2,x]

[Out]

1/(b*(a + b*Cot[x]))

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

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Mathematica [A]  time = 0.03, size = 17, normalized size = 1.42 \[ \frac {\sin (x)}{b (a \sin (x)+b \cos (x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/(b*(b*Cos[x] + a*Sin[x]))

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fricas [B]  time = 0.71, size = 39, normalized size = 3.25 \[ -\frac {a \cos \relax (x) - b \sin \relax (x)}{{\left (a^{2} b + b^{3}\right )} \cos \relax (x) + {\left (a^{3} + a b^{2}\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*cos(x) - b*sin(x))/((a^2*b + b^3)*cos(x) + (a^3 + a*b^2)*sin(x))

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giac [A]  time = 0.60, size = 13, normalized size = 1.08 \[ -\frac {1}{{\left (a \tan \relax (x) + b\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((a*tan(x) + b)*a)

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maple [A]  time = 0.20, size = 13, normalized size = 1.08 \[ \frac {1}{b \left (a +b \cot \relax (x )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b/(a+b*cot(x))

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maxima [A]  time = 0.75, size = 12, normalized size = 1.00 \[ \frac {1}{{\left (b \cot \relax (x) + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((b*cot(x) + a)*b)

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mupad [B]  time = 0.19, size = 14, normalized size = 1.17 \[ -\frac {1}{\mathrm {tan}\relax (x)\,a^2+b\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^2*(a + b*cot(x))^2),x)

[Out]

-1/(a^2*tan(x) + a*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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