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

Optimal. Leaf size=140 \[ \frac{995 \sin (c+d x)}{24576 d (5 \cos (c+d x)+3)}-\frac{25 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)^2}+\frac{5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}+\frac{279 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}-\frac{279 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d} \]

[Out]

(279*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(32768*d) - (279*Log[2*Cos[(c + d*x)/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]))

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Rubi [A]  time = 0.113443, antiderivative size = 140, 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, 206} \[ \frac{995 \sin (c+d x)}{24576 d (5 \cos (c+d x)+3)}-\frac{25 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)^2}+\frac{5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}+\frac{279 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}-\frac{279 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Cos[c + d*x])^(-4),x]

[Out]

(279*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(32768*d) - (279*Log[2*Cos[(c + d*x)/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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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}{8-2 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{4096 d}\\ &=\frac{279 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}-\frac{279 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )+\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 = 0.245671, size = 296, normalized size = 2.11 \[ \frac{226140 \sin (c+d x)+190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))+104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+467046 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+765855 \cos (c+d x) \left (\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-104625 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )-467046 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{393216 d (5 \cos (c+d x)+3)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(467046*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 104625*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] - Sin[(c +
d*x)/2]] + 765855*Cos[c + d*x]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2]]) + 376650*Cos[2*(c + d*x)]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + S
in[(c + d*x)/2]]) - 467046*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 104625*Cos[3*(c + d*x)]*Log[2*Cos[(c +
d*x)/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)

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Maple [A]  time = 0.04, size = 144, normalized size = 1. \begin{align*} -{\frac{125}{6144\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-3}}+{\frac{175}{4096\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-2}}-{\frac{745}{16384\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-1}}-{\frac{279}{32768\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }-{\frac{125}{6144\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-3}}-{\frac{175}{4096\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-2}}-{\frac{745}{16384\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-1}}+{\frac{279}{32768\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \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]

-125/6144/d/(tan(1/2*d*x+1/2*c)+2)^3+175/4096/d/(tan(1/2*d*x+1/2*c)+2)^2-745/16384/d/(tan(1/2*d*x+1/2*c)+2)-27
9/32768/d*ln(tan(1/2*d*x+1/2*c)+2)-125/6144/d/(tan(1/2*d*x+1/2*c)-2)^3-175/4096/d/(tan(1/2*d*x+1/2*c)-2)^2-745
/16384/d/(tan(1/2*d*x+1/2*c)-2)+279/32768/d*ln(tan(1/2*d*x+1/2*c)-2)

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Maxima [A]  time = 1.41692, size = 235, normalized size = 1.68 \begin{align*} -\frac{\frac{20 \,{\left (\frac{2832 \, \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{447 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac{48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 64} + 837 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 837 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\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*(2832*sin(d*x + c)/(cos(d*x + c) + 1) - 1696*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 447*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5)/(48*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 12*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 +
sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 64) + 837*log(sin(d*x + c)/(cos(d*x + c) + 1) + 2) - 837*log(sin(d*x + c
)/(cos(d*x + c) + 1) - 2))/d

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Fricas [A]  time = 1.66475, size = 521, normalized size = 3.72 \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)

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Sympy [A]  time = 13.6307, size = 813, normalized size = 5.81 \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/(5*cos(2*atan(2)) + 3)**4, Eq(c, -d*x - 2*atan(2)) | Eq(c, -d*x + 2*atan(2))), (x/(5*cos(c) + 3)*
*4, Eq(d, 0)), (837*log(tan(c/2 + d*x/2) - 2)*tan(c/2 + d*x/2)**6/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan
(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 10044*log(tan(c/2 + d*x/2) - 2)*tan(c/2 + d*x/
2)**4/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d
) + 40176*log(tan(c/2 + d*x/2) - 2)*tan(c/2 + d*x/2)**2/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x
/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 53568*log(tan(c/2 + d*x/2) - 2)/(98304*d*tan(c/2 + d*x/2
)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 837*log(tan(c/2 + d*x/2) +
2)*tan(c/2 + d*x/2)**6/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x
/2)**2 - 6291456*d) + 10044*log(tan(c/2 + d*x/2) + 2)*tan(c/2 + d*x/2)**4/(98304*d*tan(c/2 + d*x/2)**6 - 11796
48*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 40176*log(tan(c/2 + d*x/2) + 2)*tan(c/
2 + d*x/2)**2/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6
291456*d) + 53568*log(tan(c/2 + d*x/2) + 2)/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 471
8592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 8940*tan(c/2 + d*x/2)**5/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*ta
n(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) + 33920*tan(c/2 + d*x/2)**3/(98304*d*tan(c/2 +
d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 56640*tan(c/2 + d*x/2
)/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d), T
rue))

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Giac [A]  time = 1.18952, size = 123, normalized size = 0.88 \begin{align*} -\frac{\frac{20 \,{\left (447 \, \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} + 2832 \, \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 )}^{3}} + 837 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \right |}\right ) - 837 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \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*(447*tan(1/2*d*x + 1/2*c)^5 - 1696*tan(1/2*d*x + 1/2*c)^3 + 2832*tan(1/2*d*x + 1/2*c))/(tan(1/2*d
*x + 1/2*c)^2 - 4)^3 + 837*log(abs(tan(1/2*d*x + 1/2*c) + 2)) - 837*log(abs(tan(1/2*d*x + 1/2*c) - 2)))/d