3.41 \(\int \frac{1}{(3-5 \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=138 \[ -\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}-\frac{279 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}+\frac{279 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d} \]

[Out]

(-279*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(32768*d) + (279*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])
/(32768*d) - (5*Sin[c + d*x])/(48*d*(3 - 5*Cos[c + d*x])^3) + (25*Sin[c + d*x])/(512*d*(3 - 5*Cos[c + d*x])^2)
- (995*Sin[c + d*x])/(24576*d*(3 - 5*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.113442, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}-\frac{279 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}+\frac{279 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 5*Cos[c + d*x])^(-4),x]

[Out]

(-279*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(32768*d) + (279*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])
/(32768*d) - (5*Sin[c + d*x])/(48*d*(3 - 5*Cos[c + d*x])^3) + (25*Sin[c + d*x])/(512*d*(3 - 5*Cos[c + d*x])^2)
- (995*Sin[c + d*x])/(24576*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))^4} \, dx &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{1}{48} \int \frac{-9-10 \cos (c+d x)}{(3-5 \cos (c+d x))^3} \, dx\\ &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}+\frac{\int \frac{154+75 \cos (c+d x)}{(3-5 \cos (c+d x))^2} \, dx}{1536}\\ &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac{\int -\frac{837}{3-5 \cos (c+d x)} \, dx}{24576}\\ &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}-\frac{279 \int \frac{1}{3-5 \cos (c+d x)} \, dx}{8192}\\ &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}-\frac{279 \operatorname{Subst}\left (\int \frac{1}{-2+8 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{4096 d}\\ &=-\frac{279 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}+\frac{279 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}\\ \end{align*}

Mathematica [B]  time = 0.231309, size = 288, normalized size = 2.09 \[ \frac{226140 \sin (c+d x)-190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))-104625 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )+467046 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )-765855 \cos (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+104625 \cos (3 (c+d x)) \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-467046 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{393216 d (5 \cos (c+d x)-3)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 5*Cos[c + d*x])^(-4),x]

[Out]

(467046*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - 104625*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - 2*Sin[(c +
d*x)/2]] - 765855*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 2*Sin[(c
+ d*x)/2]]) + 376650*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 2*S
in[(c + d*x)/2]]) - 467046*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]] + 104625*Cos[3*(c + d*x)]*Log[Cos[(c + d
*x)/2] + 2*Sin[(c + d*x)/2]] + 226140*Sin[c + d*x] - 190800*Sin[2*(c + d*x)] + 99500*Sin[3*(c + d*x)])/(393216
*d*(-3 + 5*Cos[c + d*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 160, normalized size = 1.2 \begin{align*} -{\frac{125}{49152\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-3}}+{\frac{25}{8192\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-2}}-{\frac{295}{32768\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}}-{\frac{279}{32768\,d}\ln \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }-{\frac{125}{49152\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-3}}-{\frac{25}{8192\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-2}}-{\frac{295}{32768\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}}+{\frac{279}{32768\,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))^4,x)

[Out]

-125/49152/d/(2*tan(1/2*d*x+1/2*c)-1)^3+25/8192/d/(2*tan(1/2*d*x+1/2*c)-1)^2-295/32768/d/(2*tan(1/2*d*x+1/2*c)
-1)-279/32768/d*ln(2*tan(1/2*d*x+1/2*c)-1)-125/49152/d/(1+2*tan(1/2*d*x+1/2*c))^3-25/8192/d/(1+2*tan(1/2*d*x+1
/2*c))^2-295/32768/d/(1+2*tan(1/2*d*x+1/2*c))+279/32768/d*ln(1+2*tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [A]  time = 1.84675, size = 239, normalized size = 1.73 \begin{align*} -\frac{\frac{20 \,{\left (\frac{447 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1696 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{2832 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac{12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{64 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1} - 837 \, \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 837 \, \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{98304 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-5*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/98304*(20*(447*sin(d*x + c)/(cos(d*x + c) + 1) - 1696*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2832*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5)/(12*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 48*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 +
64*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 1) - 837*log(2*sin(d*x + c)/(cos(d*x + c) + 1) + 1) + 837*log(2*sin(d
*x + c)/(cos(d*x + c) + 1) - 1))/d

________________________________________________________________________________________

Fricas [A]  time = 1.68231, size = 522, normalized size = 3.78 \begin{align*} \frac{837 \,{\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\right )} \log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) - 837 \,{\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\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 (4975 \, \cos \left (d x + c\right )^{2} - 4770 \, \cos \left (d x + c\right ) + 1583\right )} \sin \left (d x + c\right )}{196608 \,{\left (125 \, d \cos \left (d x + c\right )^{3} - 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) - 27 \, d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-5*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

1/196608*(837*(125*cos(d*x + c)^3 - 225*cos(d*x + c)^2 + 135*cos(d*x + c) - 27)*log(-3/2*cos(d*x + c) + 2*sin(
d*x + c) + 5/2) - 837*(125*cos(d*x + c)^3 - 225*cos(d*x + c)^2 + 135*cos(d*x + c) - 27)*log(-3/2*cos(d*x + c)
- 2*sin(d*x + c) + 5/2) + 40*(4975*cos(d*x + c)^2 - 4770*cos(d*x + c) + 1583)*sin(d*x + c))/(125*d*cos(d*x + c
)^3 - 225*d*cos(d*x + c)^2 + 135*d*cos(d*x + c) - 27*d)

________________________________________________________________________________________

Sympy [A]  time = 14.0992, size = 831, normalized size = 6.02 \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))**4,x)

[Out]

Piecewise((x/(3 - 5*cos(2*atan(1/2)))**4, Eq(c, -d*x - 2*atan(1/2)) | Eq(c, -d*x + 2*atan(1/2))), (x/(3 - 5*co
s(c))**4, Eq(d, 0)), (-53568*log(tan(c/2 + d*x/2) - 1/2)*tan(c/2 + d*x/2)**6/(6291456*d*tan(c/2 + d*x/2)**6 -
4718592*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)**2 - 98304*d) + 40176*log(tan(c/2 + d*x/2) - 1/2)*t
an(c/2 + d*x/2)**4/(6291456*d*tan(c/2 + d*x/2)**6 - 4718592*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)
**2 - 98304*d) - 10044*log(tan(c/2 + d*x/2) - 1/2)*tan(c/2 + d*x/2)**2/(6291456*d*tan(c/2 + d*x/2)**6 - 471859
2*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)**2 - 98304*d) + 837*log(tan(c/2 + d*x/2) - 1/2)/(6291456*
d*tan(c/2 + d*x/2)**6 - 4718592*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)**2 - 98304*d) + 53568*log(t
an(c/2 + d*x/2) + 1/2)*tan(c/2 + d*x/2)**6/(6291456*d*tan(c/2 + d*x/2)**6 - 4718592*d*tan(c/2 + d*x/2)**4 + 11
79648*d*tan(c/2 + d*x/2)**2 - 98304*d) - 40176*log(tan(c/2 + d*x/2) + 1/2)*tan(c/2 + d*x/2)**4/(6291456*d*tan(
c/2 + d*x/2)**6 - 4718592*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)**2 - 98304*d) + 10044*log(tan(c/2
+ d*x/2) + 1/2)*tan(c/2 + d*x/2)**2/(6291456*d*tan(c/2 + d*x/2)**6 - 4718592*d*tan(c/2 + d*x/2)**4 + 1179648*
d*tan(c/2 + d*x/2)**2 - 98304*d) - 837*log(tan(c/2 + d*x/2) + 1/2)/(6291456*d*tan(c/2 + d*x/2)**6 - 4718592*d*
tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)**2 - 98304*d) - 56640*tan(c/2 + d*x/2)**5/(6291456*d*tan(c/2
+ d*x/2)**6 - 4718592*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)**2 - 98304*d) + 33920*tan(c/2 + d*x/2
)**3/(6291456*d*tan(c/2 + d*x/2)**6 - 4718592*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)**2 - 98304*d)
- 8940*tan(c/2 + d*x/2)/(6291456*d*tan(c/2 + d*x/2)**6 - 4718592*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 +
d*x/2)**2 - 98304*d), True))

________________________________________________________________________________________

Giac [A]  time = 1.15496, size = 131, normalized size = 0.95 \begin{align*} -\frac{\frac{20 \,{\left (2832 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1696 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 447 \, \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 )}^{3}} - 837 \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 837 \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{98304 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-5*cos(d*x+c))^4,x, algorithm="giac")

[Out]

-1/98304*(20*(2832*tan(1/2*d*x + 1/2*c)^5 - 1696*tan(1/2*d*x + 1/2*c)^3 + 447*tan(1/2*d*x + 1/2*c))/(4*tan(1/2
*d*x + 1/2*c)^2 - 1)^3 - 837*log(abs(2*tan(1/2*d*x + 1/2*c) + 1)) + 837*log(abs(2*tan(1/2*d*x + 1/2*c) - 1)))/
d