### 3.28 $$\int \frac{1}{(-5+3 \sin (c+d x))^3} \, dx$$

Optimal. Leaf size=83 $\frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}+\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}+\frac{59 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{1024 d}-\frac{59 x}{2048}$

[Out]

(-59*x)/2048 + (59*ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])])/(1024*d) + (3*Cos[c + d*x])/(32*d*(5 - 3*Sin[c + d
*x])^2) + (45*Cos[c + d*x])/(512*d*(5 - 3*Sin[c + d*x]))

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Rubi [A]  time = 0.0631747, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {2664, 2754, 12, 2658} $\frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}+\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}+\frac{59 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{1024 d}-\frac{59 x}{2048}$

Antiderivative was successfully veriﬁed.

[In]

Int[(-5 + 3*Sin[c + d*x])^(-3),x]

[Out]

(-59*x)/2048 + (59*ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])])/(1024*d) + (3*Cos[c + d*x])/(32*d*(5 - 3*Sin[c + d
*x])^2) + (45*Cos[c + d*x])/(512*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 2754

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

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))^3} \, dx &=\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}-\frac{1}{32} \int \frac{10+3 \sin (c+d x)}{(-5+3 \sin (c+d x))^2} \, dx\\ &=\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}+\frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}+\frac{1}{512} \int \frac{59}{-5+3 \sin (c+d x)} \, dx\\ &=\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}+\frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}+\frac{59}{512} \int \frac{1}{-5+3 \sin (c+d x)} \, dx\\ &=-\frac{59 x}{2048}+\frac{59 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{1024 d}+\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}+\frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.278905, size = 113, normalized size = 1.36 $\frac{\frac{546 \cos (c+d x)+9 (60 \sin (c+d x)-15 \sin (2 (c+d x))+9 \cos (2 (c+d x))-59)}{(5-3 \sin (c+d x))^2}+59 \tan ^{-1}\left (\frac{2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}\right )}{1024 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(59*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])] + (546*Cos[c + d*x
] + 9*(-59 + 9*Cos[2*(c + d*x)] + 60*Sin[c + d*x] - 15*Sin[2*(c + d*x)]))/(5 - 3*Sin[c + d*x])^2)/(1024*d)

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Maple [B]  time = 0.036, size = 184, normalized size = 2.2 \begin{align*} -{\frac{963}{1280\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-2}}+{\frac{11739}{6400\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-2}}-{\frac{2313}{1280\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-2}}+{\frac{273}{256\,d} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-2}}-{\frac{59}{1024\,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))^3,x)

[Out]

-963/1280/d/(5*tan(1/2*d*x+1/2*c)^2-6*tan(1/2*d*x+1/2*c)+5)^2*tan(1/2*d*x+1/2*c)^3+11739/6400/d/(5*tan(1/2*d*x
+1/2*c)^2-6*tan(1/2*d*x+1/2*c)+5)^2*tan(1/2*d*x+1/2*c)^2-2313/1280/d/(5*tan(1/2*d*x+1/2*c)^2-6*tan(1/2*d*x+1/2
*c)+5)^2*tan(1/2*d*x+1/2*c)+273/256/d/(5*tan(1/2*d*x+1/2*c)^2-6*tan(1/2*d*x+1/2*c)+5)^2-59/1024/d*arctan(5/4*t
an(1/2*d*x+1/2*c)-3/4)

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Maxima [B]  time = 1.46139, size = 234, normalized size = 2.82 \begin{align*} \frac{\frac{12 \,{\left (\frac{3855 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3913 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1605 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 2275\right )}}{\frac{60 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{86 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{60 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{25 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 25} - 1475 \, \arctan \left (\frac{5 \, \sin \left (d x + c\right )}{4 \,{\left (\cos \left (d x + c\right ) + 1\right )}} - \frac{3}{4}\right )}{25600 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25600*(12*(3855*sin(d*x + c)/(cos(d*x + c) + 1) - 3913*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1605*sin(d*x +
c)^3/(cos(d*x + c) + 1)^3 - 2275)/(60*sin(d*x + c)/(cos(d*x + c) + 1) - 86*sin(d*x + c)^2/(cos(d*x + c) + 1)^2
+ 60*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 25*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 25) - 1475*arctan(5/4*sin
(d*x + c)/(cos(d*x + c) + 1) - 3/4))/d

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Fricas [A]  time = 2.00305, size = 273, normalized size = 3.29 \begin{align*} -\frac{59 \,{\left (9 \, \cos \left (d x + c\right )^{2} + 30 \, \sin \left (d x + c\right ) - 34\right )} \arctan \left (\frac{5 \, \sin \left (d x + c\right ) - 3}{4 \, \cos \left (d x + c\right )}\right ) - 540 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 1092 \, \cos \left (d x + c\right )}{2048 \,{\left (9 \, d \cos \left (d x + c\right )^{2} + 30 \, d \sin \left (d x + c\right ) - 34 \, d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2048*(59*(9*cos(d*x + c)^2 + 30*sin(d*x + c) - 34)*arctan(1/4*(5*sin(d*x + c) - 3)/cos(d*x + c)) - 540*cos(
d*x + c)*sin(d*x + c) + 1092*cos(d*x + c))/(9*d*cos(d*x + c)^2 + 30*d*sin(d*x + c) - 34*d)

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Sympy [A]  time = 8.9146, size = 915, normalized size = 11.02 \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))**3,x)

[Out]

Piecewise((x/(-5 + 3*sin(2*atan(3/5 - 4*I/5)))**3, Eq(c, -d*x + 2*atan(3/5 - 4*I/5))), (x/(-5 + 3*sin(2*atan(3
/5 + 4*I/5)))**3, Eq(c, -d*x + 2*atan(3/5 + 4*I/5))), (x/(3*sin(c) - 5)**3, Eq(d, 0)), (-1475*(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)**4/(25600*d*tan(c/2 + d*x/2)**4 - 6144
0*d*tan(c/2 + d*x/2)**3 + 88064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/2 + d*x/2) + 25600*d) + 3540*(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)**3/(25600*d*tan(c/2 + d*x/2)**4 -
61440*d*tan(c/2 + d*x/2)**3 + 88064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/2 + d*x/2) + 25600*d) - 5074*(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/(25600*d*tan(c/2 + d*x/2)
**4 - 61440*d*tan(c/2 + d*x/2)**3 + 88064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/2 + d*x/2) + 25600*d) + 3540*(
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)/(25600*d*tan(c/2 + d*x/
2)**4 - 61440*d*tan(c/2 + d*x/2)**3 + 88064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/2 + d*x/2) + 25600*d) - 1475
*(atan(5*tan(c/2 + d*x/2)/4 - 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))/(25600*d*tan(c/2 + d*x/2)**4 - 61440*d
*tan(c/2 + d*x/2)**3 + 88064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/2 + d*x/2) + 25600*d) - 321*tan(c/2 + d*x/2
)**4/(25600*d*tan(c/2 + d*x/2)**4 - 61440*d*tan(c/2 + d*x/2)**3 + 88064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/
2 + d*x/2) + 25600*d) + 774*tan(c/2 + d*x/2)**2/(25600*d*tan(c/2 + d*x/2)**4 - 61440*d*tan(c/2 + d*x/2)**3 + 8
8064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/2 + d*x/2) + 25600*d) - 1080*tan(c/2 + d*x/2)/(25600*d*tan(c/2 + d*
x/2)**4 - 61440*d*tan(c/2 + d*x/2)**3 + 88064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/2 + d*x/2) + 25600*d) + 77
1/(25600*d*tan(c/2 + d*x/2)**4 - 61440*d*tan(c/2 + d*x/2)**3 + 88064*d*tan(c/2 + d*x/2)**2 - 61440*d*tan(c/2 +
d*x/2) + 25600*d), True))

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Giac [A]  time = 1.22392, size = 165, normalized size = 1.99 \begin{align*} -\frac{1475 \, d x + 1475 \, c + \frac{24 \,{\left (1605 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3913 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3855 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2275\right )}}{{\left (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\right )}^{2}} + 2950 \, \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 )}{51200 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/51200*(1475*d*x + 1475*c + 24*(1605*tan(1/2*d*x + 1/2*c)^3 - 3913*tan(1/2*d*x + 1/2*c)^2 + 3855*tan(1/2*d*x
+ 1/2*c) - 2275)/(5*tan(1/2*d*x + 1/2*c)^2 - 6*tan(1/2*d*x + 1/2*c) + 5)^2 + 2950*arctan((3*cos(d*x + c) - si
n(d*x + c) + 3)/(cos(d*x + c) + 3*sin(d*x + c) - 9)))/d