3.11 \(\int \frac{1}{(a+a \csc (c+d x))^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{4 \cot (c+d x)}{3 a^2 d (\csc (c+d x)+1)}+\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a \csc (c+d x)+a)^2} \]

[Out]

x/a^2 + (4*Cot[c + d*x])/(3*a^2*d*(1 + Csc[c + d*x])) + Cot[c + d*x]/(3*d*(a + a*Csc[c + d*x])^2)

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Rubi [A]  time = 0.066061, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3777, 3919, 3794} \[ \frac{4 \cot (c+d x)}{3 a^2 d (\csc (c+d x)+1)}+\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a \csc (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Csc[c + d*x])^(-2),x]

[Out]

x/a^2 + (4*Cot[c + d*x])/(3*a^2*d*(1 + Csc[c + d*x])) + Cot[c + d*x]/(3*d*(a + a*Csc[c + d*x])^2)

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 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+a \csc (c+d x))^2} \, dx &=\frac{\cot (c+d x)}{3 d (a+a \csc (c+d x))^2}-\frac{\int \frac{-3 a+a \csc (c+d x)}{a+a \csc (c+d x)} \, dx}{3 a^2}\\ &=\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a+a \csc (c+d x))^2}-\frac{4 \int \frac{\csc (c+d x)}{a+a \csc (c+d x)} \, dx}{3 a}\\ &=\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a+a \csc (c+d x))^2}+\frac{4 \cot (c+d x)}{3 d \left (a^2+a^2 \csc (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.324959, size = 108, normalized size = 1.89 \[ \frac{3 (3 c+3 d x-4) \cos \left (\frac{1}{2} (c+d x)\right )+(-3 c-3 d x+10) \cos \left (\frac{3}{2} (c+d x)\right )+6 \sin \left (\frac{1}{2} (c+d x)\right ) ((c+d x) \cos (c+d x)+2 c+2 d x-3)}{6 a^2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Csc[c + d*x])^(-2),x]

[Out]

(3*(-4 + 3*c + 3*d*x)*Cos[(c + d*x)/2] + (10 - 3*c - 3*d*x)*Cos[(3*(c + d*x))/2] + 6*(-3 + 2*c + 2*d*x + (c +
d*x)*Cos[c + d*x])*Sin[(c + d*x)/2])/(6*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)

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Maple [A]  time = 0.063, size = 83, normalized size = 1.5 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}}-{\frac{4}{3\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+2\,{\frac{1}{d{a}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+2\,{\frac{1}{d{a}^{2} \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))^2,x)

[Out]

2/d/a^2*arctan(tan(1/2*d*x+1/2*c))-4/3/d/a^2/(tan(1/2*d*x+1/2*c)+1)^3+2/d/a^2/(tan(1/2*d*x+1/2*c)+1)^2+2/d/a^2
/(tan(1/2*d*x+1/2*c)+1)

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Maxima [B]  time = 1.47041, size = 192, normalized size = 3.37 \begin{align*} \frac{2 \,{\left (\frac{\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac{3 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac{3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(d*x+c))^2,x, algorithm="maxima")

[Out]

2/3*((9*sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 4)/(a^2 + 3*a^2*sin(d*x + c)
/(cos(d*x + c) + 1) + 3*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3) + 3
*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

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Fricas [B]  time = 0.480864, size = 306, normalized size = 5.37 \begin{align*} \frac{{\left (3 \, d x - 5\right )} \cos \left (d x + c\right )^{2} - 6 \, d x -{\left (3 \, d x + 4\right )} \cos \left (d x + c\right ) -{\left (6 \, d x +{\left (3 \, d x + 5\right )} \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) + 1}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d -{\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(d*x+c))^2,x, algorithm="fricas")

[Out]

1/3*((3*d*x - 5)*cos(d*x + c)^2 - 6*d*x - (3*d*x + 4)*cos(d*x + c) - (6*d*x + (3*d*x + 5)*cos(d*x + c) + 1)*si
n(d*x + c) + 1)/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x
+ c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\csc ^{2}{\left (c + d x \right )} + 2 \csc{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(d*x+c))**2,x)

[Out]

Integral(1/(csc(c + d*x)**2 + 2*csc(c + d*x) + 1), x)/a**2

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Giac [A]  time = 1.33023, size = 81, normalized size = 1.42 \begin{align*} \frac{\frac{3 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*csc(d*x+c))^2,x, algorithm="giac")

[Out]

1/3*(3*(d*x + c)/a^2 + 2*(3*tan(1/2*d*x + 1/2*c)^2 + 9*tan(1/2*d*x + 1/2*c) + 4)/(a^2*(tan(1/2*d*x + 1/2*c) +
1)^3))/d