3.139 \(\int \cot (a+b x) \csc (a+b x) \, dx\)

Optimal. Leaf size=11 \[ -\frac{\csc (a+b x)}{b} \]

[Out]

-(Csc[a + b*x]/b)

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Rubi [A]  time = 0.0093198, antiderivative size = 11, 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 = {2606, 8} \[ -\frac{\csc (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]*Csc[a + b*x],x]

[Out]

-(Csc[a + b*x]/b)

Rule 2606

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0083335, size = 11, normalized size = 1. \[ -\frac{\csc (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]*Csc[a + b*x],x]

[Out]

-(Csc[a + b*x]/b)

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Maple [A]  time = 0.002, size = 14, normalized size = 1.3 \begin{align*} -{\frac{1}{b\sin \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)/sin(b*x+a)^2,x)

[Out]

-1/b/sin(b*x+a)

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Maxima [A]  time = 0.994688, size = 18, normalized size = 1.64 \begin{align*} -\frac{1}{b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*sin(b*x + a))

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Fricas [A]  time = 1.81136, size = 28, normalized size = 2.55 \begin{align*} -\frac{1}{b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*sin(b*x + a))

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Sympy [A]  time = 0.684761, size = 20, normalized size = 1.82 \begin{align*} \begin{cases} - \frac{1}{b \sin{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos{\left (a \right )}}{\sin ^{2}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+a)**2,x)

[Out]

Piecewise((-1/(b*sin(a + b*x)), Ne(b, 0)), (x*cos(a)/sin(a)**2, True))

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Giac [A]  time = 1.15179, size = 18, normalized size = 1.64 \begin{align*} -\frac{1}{b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*sin(b*x + a))