3.32 \(\int \frac{1}{(-5-3 \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=81 \[ \frac{45 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)}+\frac{3 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}+\frac{59 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{1024 d}-\frac{59 x}{2048} \]

[Out]

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

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Rubi [A]  time = 0.0611682, antiderivative size = 81, 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} \[ \frac{45 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)}+\frac{3 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}+\frac{59 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{1024 d}-\frac{59 x}{2048} \]

Antiderivative was successfully verified.

[In]

Int[(-5 - 3*Cos[c + d*x])^(-3),x]

[Out]

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

Mathematica [A]  time = 0.121325, size = 65, normalized size = 0.8 \[ \frac{546 \sin (c+d x)+135 \sin (2 (c+d x))+59 (3 \cos (c+d x)+5)^2 \tan ^{-1}\left (2 \cot \left (\frac{1}{2} (c+d x)\right )\right )}{1024 d (3 \cos (c+d x)+5)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(59*ArcTan[2*Cot[(c + d*x)/2]]*(5 + 3*Cos[c + d*x])^2 + 546*Sin[c + d*x] + 135*Sin[2*(c + d*x)])/(1024*d*(5 +
3*Cos[c + d*x])^2)

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Maple [A]  time = 0.037, size = 79, normalized size = 1. \begin{align*}{\frac{69}{512\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-2}}+{\frac{51}{128\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-2}}-{\frac{59}{1024\,d}\arctan \left ({\frac{1}{2}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

69/512/d/(tan(1/2*d*x+1/2*c)^2+4)^2*tan(1/2*d*x+1/2*c)^3+51/128/d/(tan(1/2*d*x+1/2*c)^2+4)^2*tan(1/2*d*x+1/2*c
)-59/1024/d*arctan(1/2*tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 2.01823, size = 150, normalized size = 1.85 \begin{align*} \frac{\frac{6 \,{\left (\frac{68 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{23 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{\frac{8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 16} - 59 \, \arctan \left (\frac{\sin \left (d x + c\right )}{2 \,{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{1024 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024*(6*(68*sin(d*x + c)/(cos(d*x + c) + 1) + 23*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(8*sin(d*x + c)^2/(cos
(d*x + c) + 1)^2 + sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 16) - 59*arctan(1/2*sin(d*x + c)/(cos(d*x + c) + 1)))
/d

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Fricas [A]  time = 1.6588, size = 257, normalized size = 3.17 \begin{align*} \frac{59 \,{\left (9 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 25\right )} \arctan \left (\frac{5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right ) + 12 \,{\left (45 \, \cos \left (d x + c\right ) + 91\right )} \sin \left (d x + c\right )}{2048 \,{\left (9 \, d \cos \left (d x + c\right )^{2} + 30 \, d \cos \left (d x + c\right ) + 25 \, d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2048*(59*(9*cos(d*x + c)^2 + 30*cos(d*x + c) + 25)*arctan(1/4*(5*cos(d*x + c) + 3)/sin(d*x + c)) + 12*(45*co
s(d*x + c) + 91)*sin(d*x + c))/(9*d*cos(d*x + c)^2 + 30*d*cos(d*x + c) + 25*d)

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Sympy [A]  time = 6.4919, size = 362, normalized size = 4.47 \begin{align*} \begin{cases} \frac{x}{\left (-5 - 3 \cosh{\left (2 \operatorname{atanh}{\left (2 \right )} \right )}\right )^{3}} & \text{for}\: c = - d x - 2 i \operatorname{atanh}{\left (2 \right )} \vee c = - d x + 2 i \operatorname{atanh}{\left (2 \right )} \\\frac{x}{\left (- 3 \cos{\left (c \right )} - 5\right )^{3}} & \text{for}\: d = 0 \\- \frac{59 \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} - \frac{472 \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} - \frac{944 \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} + \frac{138 \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} + \frac{408 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{1024 d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 8192 d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 16384 d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(-5 - 3*cosh(2*atanh(2)))**3, Eq(c, -d*x - 2*I*atanh(2)) | Eq(c, -d*x + 2*I*atanh(2))), (x/(-3*co
s(c) - 5)**3, Eq(d, 0)), (-59*(atan(tan(c/2 + d*x/2)/2) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)*
*4/(1024*d*tan(c/2 + d*x/2)**4 + 8192*d*tan(c/2 + d*x/2)**2 + 16384*d) - 472*(atan(tan(c/2 + d*x/2)/2) + pi*fl
oor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**2/(1024*d*tan(c/2 + d*x/2)**4 + 8192*d*tan(c/2 + d*x/2)**2 + 1
6384*d) - 944*(atan(tan(c/2 + d*x/2)/2) + pi*floor((c/2 + d*x/2 - pi/2)/pi))/(1024*d*tan(c/2 + d*x/2)**4 + 819
2*d*tan(c/2 + d*x/2)**2 + 16384*d) + 138*tan(c/2 + d*x/2)**3/(1024*d*tan(c/2 + d*x/2)**4 + 8192*d*tan(c/2 + d*
x/2)**2 + 16384*d) + 408*tan(c/2 + d*x/2)/(1024*d*tan(c/2 + d*x/2)**4 + 8192*d*tan(c/2 + d*x/2)**2 + 16384*d),
True))

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Giac [A]  time = 1.31137, size = 101, normalized size = 1.25 \begin{align*} -\frac{59 \, d x + 59 \, c - \frac{12 \,{\left (23 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 68 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4\right )}^{2}} - 118 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{2048 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2048*(59*d*x + 59*c - 12*(23*tan(1/2*d*x + 1/2*c)^3 + 68*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 4)
^2 - 118*arctan(sin(d*x + c)/(cos(d*x + c) + 3)))/d