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

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

[Out]

-(Log[a + b*Cot[x]]/b)

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Rubi [A]  time = 0.0411302, 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, 31} \[ -\frac{\log (a+b \cot (x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2/(a + b*Cot[x]),x]

[Out]

-(Log[a + b*Cot[x]]/b)

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0586226, size = 20, normalized size = 1.67 \[ \frac{\log (\sin (x))-\log (a \sin (x)+b \cos (x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2/(a + b*Cot[x]),x]

[Out]

(Log[Sin[x]] - Log[b*Cos[x] + a*Sin[x]])/b

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Maple [A]  time = 0.025, size = 13, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( a+b\cot \left ( x \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(a+b*cot(x))/b

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Maxima [A]  time = 0.960791, size = 16, normalized size = 1.33 \begin{align*} -\frac{\log \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)),x, algorithm="maxima")

[Out]

-log(b*cot(x) + a)/b

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Fricas [B]  time = 2.267, size = 123, normalized size = 10.25 \begin{align*} -\frac{\log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right ) - \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2) - log(-1/4*cos(x)^2 + 1/4))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)**2/(a + b*cot(x)), x)

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Giac [A]  time = 1.12864, size = 30, normalized size = 2.5 \begin{align*} -\frac{\log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{b} + \frac{\log \left ({\left | \tan \left (x\right ) \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(a*tan(x) + b))/b + log(abs(tan(x)))/b