3.4 \(\int (a+b \csc ^2(c+d x)) \, dx\)

Optimal. Leaf size=16 \[ a x-\frac{b \cot (c+d x)}{d} \]

[Out]

a*x - (b*Cot[c + d*x])/d

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Rubi [A]  time = 0.0124905, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3767, 8} \[ a x-\frac{b \cot (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Csc[c + d*x]^2,x]

[Out]

a*x - (b*Cot[c + d*x])/d

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

Mathematica [A]  time = 0.0082534, size = 16, normalized size = 1. \[ a x-\frac{b \cot (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Csc[c + d*x]^2,x]

[Out]

a*x - (b*Cot[c + d*x])/d

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Maple [A]  time = 0.013, size = 17, normalized size = 1.1 \begin{align*} ax-{\frac{b\cot \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x-b*cot(d*x+c)/d

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Maxima [A]  time = 0.976349, size = 24, normalized size = 1.5 \begin{align*} a x - \frac{b}{d \tan \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x - b/(d*tan(d*x + c))

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Fricas [A]  time = 0.468637, size = 76, normalized size = 4.75 \begin{align*} \frac{a d x \sin \left (d x + c\right ) - b \cos \left (d x + c\right )}{d \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*d*x*sin(d*x + c) - b*cos(d*x + c))/(d*sin(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*csc(c + d*x)**2, x)

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Giac [A]  time = 1.37975, size = 24, normalized size = 1.5 \begin{align*} a x - \frac{b}{d \tan \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x - b/(d*tan(d*x + c))