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 = 0.0400216, antiderivative size = 12, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0275174, 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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Maple [A]  time = 0.05, size = 13, normalized size = 1.1 \begin{align*}{\frac{1}{b \left ( a+b\cot \left ( x \right ) \right ) }} \end{align*}

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 = 1.18473, size = 16, normalized size = 1.33 \begin{align*} \frac{1}{{\left (b \cot \left (x\right ) + a\right )} b} \end{align*}

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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Fricas [B]  time = 1.59851, size = 95, normalized size = 7.92 \begin{align*} -\frac{a \cos \left (x\right ) - b \sin \left (x\right )}{{\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) +{\left (a^{3} + a b^{2}\right )} \sin \left (x\right )} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (x \right )}}{\left (a + b \cot{\left (x \right )}\right )^{2}}\, dx \end{align*}

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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Giac [A]  time = 1.27932, size = 18, normalized size = 1.5 \begin{align*} -\frac{1}{{\left (a \tan \left (x\right ) + b\right )} a} \end{align*}

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)