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

Optimal. Leaf size=113 \[ \frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}-\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}-\frac{43 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]

[Out]

(43*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(2048*d) - (43*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])/(20
48*d) - (5*Sin[c + d*x])/(32*d*(3 - 5*Cos[c + d*x])^2) + (45*Sin[c + d*x])/(512*d*(3 - 5*Cos[c + d*x]))

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Rubi [A]  time = 0.0724024, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2664, 2754, 12, 2659, 207} \[ \frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}-\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}-\frac{43 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 5*Cos[c + d*x])^(-3),x]

[Out]

(43*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(2048*d) - (43*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])/(20
48*d) - (5*Sin[c + d*x])/(32*d*(3 - 5*Cos[c + d*x])^2) + (45*Sin[c + d*x])/(512*d*(3 - 5*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 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(3-5 \cos (c+d x))^3} \, dx &=-\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{1}{32} \int \frac{-6-5 \cos (c+d x)}{(3-5 \cos (c+d x))^2} \, dx\\ &=-\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{1}{512} \int \frac{43}{3-5 \cos (c+d x)} \, dx\\ &=-\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{43}{512} \int \frac{1}{3-5 \cos (c+d x)} \, dx\\ &=-\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{43 \operatorname{Subst}\left (\int \frac{1}{-2+8 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{256 d}\\ &=\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}-\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}-\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.144925, size = 211, normalized size = 1.87 \[ -\frac{45 \sin \left (\frac{1}{2} (c+d x)\right )}{1024 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}-\frac{45 \sin \left (\frac{1}{2} (c+d x)\right )}{1024 d \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}-\frac{5}{512 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{5}{512 d \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{43 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}-\frac{43 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(43*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(2048*d) - (43*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])/(20
48*d) - 5/(512*d*(Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2])^2) - (45*Sin[(c + d*x)/2])/(1024*d*(Cos[(c + d*x)/2]
- 2*Sin[(c + d*x)/2])) + 5/(512*d*(Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2])^2) - (45*Sin[(c + d*x)/2])/(1024*d*(
Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]))

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Maple [A]  time = 0.041, size = 120, normalized size = 1.1 \begin{align*} -{\frac{25}{2048\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-2}}+{\frac{35}{2048\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}}+{\frac{43}{2048\,d}\ln \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }+{\frac{25}{2048\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-2}}+{\frac{35}{2048\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}}-{\frac{43}{2048\,d}\ln \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/2048/d/(2*tan(1/2*d*x+1/2*c)-1)^2+35/2048/d/(2*tan(1/2*d*x+1/2*c)-1)+43/2048/d*ln(2*tan(1/2*d*x+1/2*c)-1)+
25/2048/d/(1+2*tan(1/2*d*x+1/2*c))^2+35/2048/d/(1+2*tan(1/2*d*x+1/2*c))-43/2048/d*ln(1+2*tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.58646, size = 185, normalized size = 1.64 \begin{align*} \frac{\frac{20 \,{\left (\frac{17 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{28 \, \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{16 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1} - 43 \, \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 43 \, \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{2048 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2048*(20*(17*sin(d*x + c)/(cos(d*x + c) + 1) - 28*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(8*sin(d*x + c)^2/(co
s(d*x + c) + 1)^2 - 16*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 1) - 43*log(2*sin(d*x + c)/(cos(d*x + c) + 1) + 1
) + 43*log(2*sin(d*x + c)/(cos(d*x + c) + 1) - 1))/d

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Fricas [A]  time = 1.69143, size = 383, normalized size = 3.39 \begin{align*} -\frac{43 \,{\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) - 43 \,{\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) + 40 \,{\left (45 \, \cos \left (d x + c\right ) - 11\right )} \sin \left (d x + c\right )}{4096 \,{\left (25 \, d \cos \left (d x + c\right )^{2} - 30 \, d \cos \left (d x + c\right ) + 9 \, d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4096*(43*(25*cos(d*x + c)^2 - 30*cos(d*x + c) + 9)*log(-3/2*cos(d*x + c) + 2*sin(d*x + c) + 5/2) - 43*(25*c
os(d*x + c)^2 - 30*cos(d*x + c) + 9)*log(-3/2*cos(d*x + c) - 2*sin(d*x + c) + 5/2) + 40*(45*cos(d*x + c) - 11)
*sin(d*x + c))/(25*d*cos(d*x + c)^2 - 30*d*cos(d*x + c) + 9*d)

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Sympy [A]  time = 5.18501, size = 490, normalized size = 4.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(3 - 5*cos(2*atan(1/2)))**3, Eq(c, -d*x - 2*atan(1/2)) | Eq(c, -d*x + 2*atan(1/2))), (x/(3 - 5*co
s(c))**3, Eq(d, 0)), (688*log(tan(c/2 + d*x/2) - 1/2)*tan(c/2 + d*x/2)**4/(32768*d*tan(c/2 + d*x/2)**4 - 16384
*d*tan(c/2 + d*x/2)**2 + 2048*d) - 344*log(tan(c/2 + d*x/2) - 1/2)*tan(c/2 + d*x/2)**2/(32768*d*tan(c/2 + d*x/
2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) + 43*log(tan(c/2 + d*x/2) - 1/2)/(32768*d*tan(c/2 + d*x/2)**4 -
16384*d*tan(c/2 + d*x/2)**2 + 2048*d) - 688*log(tan(c/2 + d*x/2) + 1/2)*tan(c/2 + d*x/2)**4/(32768*d*tan(c/2 +
d*x/2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) + 344*log(tan(c/2 + d*x/2) + 1/2)*tan(c/2 + d*x/2)**2/(3276
8*d*tan(c/2 + d*x/2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) - 43*log(tan(c/2 + d*x/2) + 1/2)/(32768*d*tan(
c/2 + d*x/2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) + 560*tan(c/2 + d*x/2)**3/(32768*d*tan(c/2 + d*x/2)**4
- 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) - 340*tan(c/2 + d*x/2)/(32768*d*tan(c/2 + d*x/2)**4 - 16384*d*tan(c/2
+ d*x/2)**2 + 2048*d), True))

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Giac [A]  time = 1.21798, size = 113, normalized size = 1. \begin{align*} \frac{\frac{20 \,{\left (28 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 17 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}} - 43 \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 43 \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{2048 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2048*(20*(28*tan(1/2*d*x + 1/2*c)^3 - 17*tan(1/2*d*x + 1/2*c))/(4*tan(1/2*d*x + 1/2*c)^2 - 1)^2 - 43*log(abs
(2*tan(1/2*d*x + 1/2*c) + 1)) + 43*log(abs(2*tan(1/2*d*x + 1/2*c) - 1)))/d