3.6 \(\int \frac{1}{a+a \csc (c+d x)} \, dx\)

Optimal. Leaf size=28 \[ \frac{\cot (c+d x)}{d (a \csc (c+d x)+a)}+\frac{x}{a} \]

[Out]

x/a + Cot[c + d*x]/(d*(a + a*Csc[c + d*x]))

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Rubi [A]  time = 0.0126391, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3777, 8} \[ \frac{\cot (c+d x)}{d (a \csc (c+d x)+a)}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Csc[c + d*x])^(-1),x]

[Out]

x/a + Cot[c + d*x]/(d*(a + a*Csc[c + d*x]))

Rule 3777

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(Cot[c + d*x]*(a + b*Csc[c + d*x])^n)/(d*(
2*n + 1)), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]
), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{1}{a+a \csc (c+d x)} \, dx &=\frac{\cot (c+d x)}{d (a+a \csc (c+d x))}+\frac{\int a \, dx}{a^2}\\ &=\frac{x}{a}+\frac{\cot (c+d x)}{d (a+a \csc (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0894926, size = 47, normalized size = 1.68 \[ \frac{-\frac{2 \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+c+d x}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Csc[c + d*x])^(-1),x]

[Out]

(c + d*x - (2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(a*d)

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Maple [A]  time = 0.042, size = 41, normalized size = 1.5 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{ad}}+2\,{\frac{1}{ad \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*csc(d*x+c)),x)

[Out]

2/a/d*arctan(tan(1/2*d*x+1/2*c))+2/a/d/(tan(1/2*d*x+1/2*c)+1)

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Maxima [A]  time = 1.47103, size = 68, normalized size = 2.43 \begin{align*} \frac{2 \,{\left (\frac{\arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{1}{a + \frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(d*x+c)),x, algorithm="maxima")

[Out]

2*(arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 1/(a + a*sin(d*x + c)/(cos(d*x + c) + 1)))/d

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Fricas [A]  time = 0.467268, size = 142, normalized size = 5.07 \begin{align*} \frac{d x +{\left (d x + 1\right )} \cos \left (d x + c\right ) +{\left (d x - 1\right )} \sin \left (d x + c\right ) + 1}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(d*x+c)),x, algorithm="fricas")

[Out]

(d*x + (d*x + 1)*cos(d*x + c) + (d*x - 1)*sin(d*x + c) + 1)/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\csc{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(d*x+c)),x)

[Out]

Integral(1/(csc(c + d*x) + 1), x)/a

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Giac [A]  time = 1.2772, size = 43, normalized size = 1.54 \begin{align*} \frac{\frac{d x + c}{a} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(d*x+c)),x, algorithm="giac")

[Out]

((d*x + c)/a + 2/(a*(tan(1/2*d*x + 1/2*c) + 1)))/d