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

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

[Out]

-(Cot[a + b*x]/b)

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Rubi [A]  time = 0.0083176, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767, 8} \[ -\frac{\cot (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2,x]

[Out]

-(Cot[a + b*x]/b)

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0082027, size = 11, normalized size = 1. \[ -\frac{\cot (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2,x]

[Out]

-(Cot[a + b*x]/b)

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Maple [A]  time = 0.011, size = 12, normalized size = 1.1 \begin{align*} -{\frac{\cot \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cot(b*x+a)/b

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Maxima [A]  time = 1.01663, size = 18, normalized size = 1.64 \begin{align*} -\frac{1}{b \tan \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*tan(b*x + a))

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Fricas [A]  time = 0.448256, size = 43, normalized size = 3.91 \begin{align*} -\frac{\cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(b*x + a)/(b*sin(b*x + a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(a + b*x)**2, x)

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Giac [A]  time = 1.28183, size = 18, normalized size = 1.64 \begin{align*} -\frac{1}{b \tan \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*tan(b*x + a))