\(\int \frac {1}{(a+a \csc (c+d x))^2} \, dx\) [11]

Optimal result

Integrand size = 12, antiderivative size = 57 \[ \int \frac {1}{(a+a \csc (c+d x))^2} \, dx=\frac {x}{a^2}+\frac {4 \cot (c+d x)}{3 a^2 d (1+\csc (c+d x))}+\frac {\cot (c+d x)}{3 d (a+a \csc (c+d x))^2} \]

[Out]

x/a^2+4/3*cot(d*x+c)/a^2/d/(1+csc(d*x+c))+1/3*cot(d*x+c)/d/(a+a*csc(d*x+c))^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3862, 4004, 3879} \[ \int \frac {1}{(a+a \csc (c+d x))^2} \, dx=\frac {4 \cot (c+d x)}{3 a^2 d (\csc (c+d x)+1)}+\frac {x}{a^2}+\frac {\cot (c+d x)}{3 d (a \csc (c+d x)+a)^2} \]

[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 3862

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 3879

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]

Rule 4004

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.60 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.89 \[ \int \frac {1}{(a+a \csc (c+d x))^2} \, dx=\frac {3 (-4+3 c+3 d x) \cos \left (\frac {1}{2} (c+d x)\right )+(10-3 c-3 d x) \cos \left (\frac {3}{2} (c+d x)\right )+6 (-3+2 c+2 d x+(c+d x) \cos (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right )}{6 a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]

[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)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {x}{a^{2}}+\frac {6 i {\mathrm e}^{i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}-\frac {10}{3}}{d \,a^{2} \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )^{3}}\)	\(54\)
derivativedivides	\(\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4}}{a^{2} d}\)	\(67\)
default	\(\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4}}{a^{2} d}\)	\(67\)
parallelrisch	\(\frac {\left (3 d x -8\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (9 d x -18\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (9 d x -6\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3 d x}{3 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\)	\(79\)
norman	\(\frac {\frac {x}{a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d a}+\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {2}{3 a d}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d a}}{a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\)	\(118\)

[In]

int(1/(a+a*csc(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

x/a^2+2/3*(9*I*exp(I*(d*x+c))+6*exp(2*I*(d*x+c))-5)/d/a^2/(I+exp(I*(d*x+c)))^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (53) = 106\).

Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.18 \[ \int \frac {1}{(a+a \csc (c+d x))^2} \, dx=\frac {{\left (3 \, d x - 5\right )} \cos \left (d x + c\right )^{2} - 6 \, d x - {\left (3 \, d x + 4\right )} \cos \left (d x + c\right ) - {\left (6 \, d x + {\left (3 \, d x + 5\right )} \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) + 1}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d - {\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \]

[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))

Sympy [F]

\[ \int \frac {1}{(a+a \csc (c+d x))^2} \, dx=\frac {\int \frac {1}{\csc ^{2}{\left (c + d x \right )} + 2 \csc {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (53) = 106\).

Time = 0.35 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.49 \[ \int \frac {1}{(a+a \csc (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(a+a \csc (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{3 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 18.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(a+a \csc (c+d x))^2} \, dx=\frac {x}{a^2}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {8}{3}}{a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3} \]

[In]

int(1/(a + a/sin(c + d*x))^2,x)

[Out]

x/a^2 + (6*tan(c/2 + (d*x)/2) + 2*tan(c/2 + (d*x)/2)^2 + 8/3)/(a^2*d*(tan(c/2 + (d*x)/2) + 1)^3)