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

Optimal. Leaf size=15 \[ -\frac{\csc ^2(a+b x)}{2 b} \]

[Out]

-Csc[a + b*x]^2/(2*b)

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Rubi [A]  time = 0.0178605, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2606, 30} \[ -\frac{\csc ^2(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]*Csc[a + b*x]^2,x]

[Out]

-Csc[a + b*x]^2/(2*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0121944, size = 15, normalized size = 1. \[ -\frac{\csc ^2(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Csc[a + b*x]^2/(2*b)

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Maple [A]  time = 0.004, size = 14, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)/sin(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.961035, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{2 \, b \sin \left (b x + a\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2/(b*sin(b*x + a)^2)

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Fricas [A]  time = 2.48954, size = 38, normalized size = 2.53 \begin{align*} \frac{1}{2 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+a)^3,x, algorithm="fricas")

[Out]

1/2/(b*cos(b*x + a)^2 - b)

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Sympy [A]  time = 0.976703, size = 24, normalized size = 1.6 \begin{align*} \begin{cases} - \frac{1}{2 b \sin ^{2}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos{\left (a \right )}}{\sin ^{3}{\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)**3,x)

[Out]

Piecewise((-1/(2*b*sin(a + b*x)**2), Ne(b, 0)), (x*cos(a)/sin(a)**3, True))

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Giac [A]  time = 1.13086, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{2 \, b \sin \left (b x + a\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+a)^3,x, algorithm="giac")

[Out]

-1/2/(b*sin(b*x + a)^2)