### 3.31 $$\int \frac{1}{(-5-3 \sin (c+d x))^2} \, dx$$

Optimal. Leaf size=56 $\frac{3 \cos (c+d x)}{16 d (3 \sin (c+d x)+5)}+\frac{5 \tan ^{-1}\left (\frac{\cos (c+d x)}{\sin (c+d x)+3}\right )}{32 d}+\frac{5 x}{64}$

[Out]

(5*x)/64 + (5*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(32*d) + (3*Cos[c + d*x])/(16*d*(5 + 3*Sin[c + d*x]))

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Rubi [A]  time = 0.0307029, antiderivative size = 56, 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 = {2664, 12, 2658} $\frac{3 \cos (c+d x)}{16 d (3 \sin (c+d x)+5)}+\frac{5 \tan ^{-1}\left (\frac{\cos (c+d x)}{\sin (c+d x)+3}\right )}{32 d}+\frac{5 x}{64}$

Antiderivative was successfully veriﬁed.

[In]

Int[(-5 - 3*Sin[c + d*x])^(-2),x]

[Out]

(5*x)/64 + (5*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(32*d) + (3*Cos[c + d*x])/(16*d*(5 + 3*Sin[c + d*x]))

Rule 2664

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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))^2} \, dx &=\frac{3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}-\frac{1}{16} \int \frac{5}{-5-3 \sin (c+d x)} \, dx\\ &=\frac{3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}-\frac{5}{16} \int \frac{1}{-5-3 \sin (c+d x)} \, dx\\ &=\frac{5 x}{64}+\frac{5 \tan ^{-1}\left (\frac{\cos (c+d x)}{3+\sin (c+d x)}\right )}{32 d}+\frac{3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0991729, size = 91, normalized size = 1.62 $\frac{\frac{6 (-3 \sin (c+d x)+5 \cos (c+d x)-5)}{3 \sin (c+d x)+5}+25 \tan ^{-1}\left (\frac{2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}\right )}{160 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-5 - 3*Sin[c + d*x])^(-2),x]

[Out]

(25*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])] + (6*(-5 + 5*Cos[c
+ d*x] - 3*Sin[c + d*x]))/(5 + 3*Sin[c + d*x]))/(160*d)

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Maple [A]  time = 0.03, size = 92, normalized size = 1.6 \begin{align*}{\frac{9}{200\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+{\frac{6}{5}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+1 \right ) ^{-1}}+{\frac{3}{40\,d} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+{\frac{6}{5}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+1 \right ) ^{-1}}+{\frac{5}{32\,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))^2,x)

[Out]

9/200/d/(tan(1/2*d*x+1/2*c)^2+6/5*tan(1/2*d*x+1/2*c)+1)*tan(1/2*d*x+1/2*c)+3/40/d/(tan(1/2*d*x+1/2*c)^2+6/5*ta
n(1/2*d*x+1/2*c)+1)+5/32/d*arctan(5/4*tan(1/2*d*x+1/2*c)+3/4)

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Maxima [A]  time = 1.44171, size = 126, normalized size = 2.25 \begin{align*} \frac{\frac{12 \,{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 5\right )}}{\frac{6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5} + 25 \, \arctan \left (\frac{5 \, \sin \left (d x + c\right )}{4 \,{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{3}{4}\right )}{160 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/160*(12*(3*sin(d*x + c)/(cos(d*x + c) + 1) + 5)/(6*sin(d*x + c)/(cos(d*x + c) + 1) + 5*sin(d*x + c)^2/(cos(d
*x + c) + 1)^2 + 5) + 25*arctan(5/4*sin(d*x + c)/(cos(d*x + c) + 1) + 3/4))/d

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Fricas [A]  time = 1.93719, size = 162, normalized size = 2.89 \begin{align*} \frac{5 \,{\left (3 \, \sin \left (d x + c\right ) + 5\right )} \arctan \left (\frac{5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right ) + 12 \, \cos \left (d x + c\right )}{64 \,{\left (3 \, d \sin \left (d x + c\right ) + 5 \, d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/64*(5*(3*sin(d*x + c) + 5)*arctan(1/4*(5*sin(d*x + c) + 3)/cos(d*x + c)) + 12*cos(d*x + c))/(3*d*sin(d*x + c
) + 5*d)

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Sympy [A]  time = 3.07124, size = 389, normalized size = 6.95 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*sin(d*x+c))**2,x)

[Out]

Piecewise((x/(-5 + 3*sin(2*atan(3/5 - 4*I/5)))**2, Eq(c, -d*x - 2*atan(3/5 - 4*I/5))), (x/(-5 + 3*sin(2*atan(3
/5 + 4*I/5)))**2, Eq(c, -d*x - 2*atan(3/5 + 4*I/5))), (x/(-3*sin(c) - 5)**2, Eq(d, 0)), (125*(atan(5*tan(c/2 +
d*x/2)/4 + 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**2/(800*d*tan(c/2 + d*x/2)**2 + 960*d*t
an(c/2 + d*x/2) + 800*d) + 150*(atan(5*tan(c/2 + d*x/2)/4 + 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2
+ d*x/2)/(800*d*tan(c/2 + d*x/2)**2 + 960*d*tan(c/2 + d*x/2) + 800*d) + 125*(atan(5*tan(c/2 + d*x/2)/4 + 3/4)
+ pi*floor((c/2 + d*x/2 - pi/2)/pi))/(800*d*tan(c/2 + d*x/2)**2 + 960*d*tan(c/2 + d*x/2) + 800*d) + 36*tan(c/2
+ d*x/2)/(800*d*tan(c/2 + d*x/2)**2 + 960*d*tan(c/2 + d*x/2) + 800*d) + 60/(800*d*tan(c/2 + d*x/2)**2 + 960*d
*tan(c/2 + d*x/2) + 800*d), True))

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Giac [A]  time = 1.18743, size = 128, normalized size = 2.29 \begin{align*} \frac{25 \, d x + 25 \, c + \frac{24 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5\right )}}{5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5} + 50 \, \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 )}{320 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/320*(25*d*x + 25*c + 24*(3*tan(1/2*d*x + 1/2*c) + 5)/(5*tan(1/2*d*x + 1/2*c)^2 + 6*tan(1/2*d*x + 1/2*c) + 5)
+ 50*arctan(-(3*cos(d*x + c) + sin(d*x + c) + 3)/(cos(d*x + c) - 3*sin(d*x + c) - 9)))/d