### 3.30 $$\int \frac{1}{-5-3 \sin (c+d x)} \, dx$$

Optimal. Leaf size=31 $-\frac{\tan ^{-1}\left (\frac{\cos (c+d x)}{\sin (c+d x)+3}\right )}{2 d}-\frac{x}{4}$

[Out]

-x/4 - ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])]/(2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0117896, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {2658} $-\frac{\tan ^{-1}\left (\frac{\cos (c+d x)}{\sin (c+d x)+3}\right )}{2 d}-\frac{x}{4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(-5 - 3*Sin[c + d*x])^(-1),x]

[Out]

-x/4 - ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])]/(2*d)

Rule 2658

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, -Simp[x/q, x] - Sim
p[(2*ArcTan[(b*Cos[c + d*x])/(a - q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && NegQ[a]

Rubi steps

\begin{align*} \int \frac{1}{-5-3 \sin (c+d x)} \, dx &=-\frac{x}{4}-\frac{\tan ^{-1}\left (\frac{\cos (c+d x)}{3+\sin (c+d x)}\right )}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0241396, size = 56, normalized size = 1.81 $-\frac{\tan ^{-1}\left (\frac{2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}\right )}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-5 - 3*Sin[c + d*x])^(-1),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 20, normalized size = 0.7 \begin{align*} -{\frac{1}{2\,d}\arctan \left ({\frac{5}{4}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{3}{4}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-5-3*sin(d*x+c)),x)

[Out]

-1/2/d*arctan(5/4*tan(1/2*d*x+1/2*c)+3/4)

________________________________________________________________________________________

Maxima [A]  time = 1.43388, size = 35, normalized size = 1.13 \begin{align*} -\frac{\arctan \left (\frac{5 \, \sin \left (d x + c\right )}{4 \,{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{3}{4}\right )}{2 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*arctan(5/4*sin(d*x + c)/(cos(d*x + c) + 1) + 3/4)/d

________________________________________________________________________________________

Fricas [A]  time = 1.94536, size = 73, normalized size = 2.35 \begin{align*} -\frac{\arctan \left (\frac{5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right )}{4 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/4*arctan(1/4*(5*sin(d*x + c) + 3)/cos(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 0.943528, size = 49, normalized size = 1.58 \begin{align*} \begin{cases} - \frac{\operatorname{atan}{\left (\frac{5 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{4} + \frac{3}{4} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor }{2 d} & \text{for}\: d \neq 0 \\\frac{x}{- 3 \sin{\left (c \right )} - 5} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*sin(d*x+c)),x)

[Out]

Piecewise((-(atan(5*tan(c/2 + d*x/2)/4 + 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))/(2*d), Ne(d, 0)), (x/(-3*si
n(c) - 5), True))

________________________________________________________________________________________

Giac [A]  time = 1.12618, size = 66, normalized size = 2.13 \begin{align*} -\frac{d x + c + 2 \, \arctan \left (-\frac{3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{4 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*sin(d*x+c)),x, algorithm="giac")

[Out]

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