Integrand size = 12, antiderivative size = 140 \[ \int \frac {1}{(3+5 \cos (c+d x))^4} \, dx=\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {995 \sin (c+d x)}{24576 d (3+5 \cos (c+d x))} \] Output:
279/32768*ln(2*cos(1/2*d*x+1/2*c)-sin(1/2*d*x+1/2*c))/d-279/32768*ln(2*cos (1/2*d*x+1/2*c)+sin(1/2*d*x+1/2*c))/d+5/48*sin(d*x+c)/d/(3+5*cos(d*x+c))^3 -25/512*sin(d*x+c)/d/(3+5*cos(d*x+c))^2+995/24576*sin(d*x+c)/d/(3+5*cos(d* x+c))
Leaf count is larger than twice the leaf count of optimal. \(296\) vs. \(2(140)=280\).
Time = 0.33 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.11 \[ \int \frac {1}{(3+5 \cos (c+d x))^4} \, dx=\frac {467046 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+765855 \cos (c+d x) \left (\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-467046 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+226140 \sin (c+d x)+190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))}{393216 d (3+5 \cos (c+d x))^3} \] Input:
Integrate[(3 + 5*Cos[c + d*x])^(-4),x]
Output:
(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] + Sin[(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)
Time = 0.52 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.77, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3143, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3138, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(5 \cos (c+d x)+3)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (5 \sin \left (c+d x+\frac {\pi }{2}\right )+3\right )^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle \frac {1}{48} \int -\frac {9-10 \cos (c+d x)}{(5 \cos (c+d x)+3)^3}dx+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}-\frac {1}{48} \int \frac {9-10 \cos (c+d x)}{(5 \cos (c+d x)+3)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}-\frac {1}{48} \int \frac {9-10 \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (5 \sin \left (c+d x+\frac {\pi }{2}\right )+3\right )^3}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{48} \left (-\frac {1}{32} \int -\frac {154-75 \cos (c+d x)}{(5 \cos (c+d x)+3)^2}dx-\frac {75 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}\right )+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {154-75 \cos (c+d x)}{(5 \cos (c+d x)+3)^2}dx-\frac {75 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}\right )+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {154-75 \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (5 \sin \left (c+d x+\frac {\pi }{2}\right )+3\right )^2}dx-\frac {75 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}\right )+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {1}{16} \int -\frac {837}{5 \cos (c+d x)+3}dx+\frac {995 \sin (c+d x)}{16 d (5 \cos (c+d x)+3)}\right )-\frac {75 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}\right )+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \sin (c+d x)}{16 d (5 \cos (c+d x)+3)}-\frac {837}{16} \int \frac {1}{5 \cos (c+d x)+3}dx\right )-\frac {75 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}\right )+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \sin (c+d x)}{16 d (5 \cos (c+d x)+3)}-\frac {837}{16} \int \frac {1}{5 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx\right )-\frac {75 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}\right )+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \sin (c+d x)}{16 d (5 \cos (c+d x)+3)}-\frac {837 \int \frac {1}{8-2 \tan ^2\left (\frac {1}{2} (c+d x)\right )}d\tan \left (\frac {1}{2} (c+d x)\right )}{8 d}\right )-\frac {75 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}\right )+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \sin (c+d x)}{16 d (5 \cos (c+d x)+3)}-\frac {837 \text {arctanh}\left (\frac {1}{2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{32 d}\right )-\frac {75 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}\right )+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}\) |
Input:
Int[(3 + 5*Cos[c + d*x])^(-4),x]
Output:
(5*Sin[c + d*x])/(48*d*(3 + 5*Cos[c + d*x])^3) + ((-75*Sin[c + d*x])/(32*d *(3 + 5*Cos[c + d*x])^2) + ((-837*ArcTanh[Tan[(c + d*x)/2]/2])/(32*d) + (9 95*Sin[c + d*x])/(16*d*(3 + 5*Cos[c + d*x])))/32)/48
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[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]
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] + Simp [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] && IntegerQ[2*n]
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] + Simp[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]
Time = 0.77 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71
| method | result | size |
| norman | \(\frac {-\frac {295 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512 d}+\frac {265 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{768 d}-\frac {745 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8192 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-4\right )^{3}}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768 d}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768 d}\) | \(99\) |
| derivativedivides | \(\frac {-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}}{d}\) | \(124\) |
| default | \(\frac {-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}}{d}\) | \(124\) |
| risch | \(\frac {i \left (20925 \,{\mathrm e}^{5 i \left (d x +c \right )}+62775 \,{\mathrm e}^{4 i \left (d x +c \right )}+111042 \,{\mathrm e}^{3 i \left (d x +c \right )}+119310 \,{\mathrm e}^{2 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}+24875\right )}{12288 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{3}}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}+\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}\) | \(129\) |
| parallelrisch | \(\frac {\left (765855 \cos \left (d x +c \right )+376650 \cos \left (2 d x +2 c \right )+104625 \cos \left (3 d x +3 c \right )+467046\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )+\left (-765855 \cos \left (d x +c \right )-376650 \cos \left (2 d x +2 c \right )-104625 \cos \left (3 d x +3 c \right )-467046\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )+226140 \sin \left (d x +c \right )+190800 \sin \left (2 d x +2 c \right )+99500 \sin \left (3 d x +3 c \right )}{98304 d \left (558+125 \cos \left (3 d x +3 c \right )+915 \cos \left (d x +c \right )+450 \cos \left (2 d x +2 c \right )\right )}\) | \(161\) |
Input:
int(1/(3+5*cos(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
(-295/512/d*tan(1/2*d*x+1/2*c)+265/768/d*tan(1/2*d*x+1/2*c)^3-745/8192/d*t an(1/2*d*x+1/2*c)^5)/(tan(1/2*d*x+1/2*c)^2-4)^3+279/32768/d*ln(tan(1/2*d*x +1/2*c)-2)-279/32768/d*ln(tan(1/2*d*x+1/2*c)+2)
Time = 0.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(3+5 \cos (c+d x))^4} \, dx=-\frac {837 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 837 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 40 \, {\left (4975 \, \cos \left (d x + c\right )^{2} + 4770 \, \cos \left (d x + c\right ) + 1583\right )} \sin \left (d x + c\right )}{196608 \, {\left (125 \, d \cos \left (d x + c\right )^{3} + 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) + 27 \, d\right )}} \] Input:
integrate(1/(3+5*cos(d*x+c))^4,x, algorithm="fricas")
Output:
-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)
Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (126) = 252\).
Time = 2.36 (sec) , antiderivative size = 813, normalized size of antiderivative = 5.81 \[ \int \frac {1}{(3+5 \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(3+5*cos(d*x+c))**4,x)
Output:
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(ta n(c/2 + d*x/2) - 2)*tan(c/2 + d*x/2)**4/(98304*d*tan(c/2 + d*x/2)**6 - 117 9648*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 + 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 - 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)/(983 04*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) - 8940*tan(c/2 + d*x/2)**5/(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)*...
Time = 0.04 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(3+5 \cos (c+d x))^4} \, dx=-\frac {\frac {20 \, {\left (\frac {2832 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1696 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {447 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 64} + 837 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 837 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{98304 \, d} \] Input:
integrate(1/(3+5*cos(d*x+c))^4,x, algorithm="maxima")
Output:
-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 + si n(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
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(3+5 \cos (c+d x))^4} \, dx=-\frac {\frac {20 \, {\left (447 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1696 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{3}} + 837 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) - 837 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{98304 \, d} \] Input:
integrate(1/(3+5*cos(d*x+c))^4,x, algorithm="giac")
Output:
-1/98304*(20*(447*tan(1/2*d*x + 1/2*c)^5 - 1696*tan(1/2*d*x + 1/2*c)^3 + 2 832*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
Time = 41.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(3+5 \cos (c+d x))^4} \, dx=-\frac {279\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{16384\,d}-\frac {\frac {745\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192}-\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{768}+\frac {295\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-64\right )} \] Input:
int(1/(5*cos(c + d*x) + 3)^4,x)
Output:
- (279*atanh(tan(c/2 + (d*x)/2)/2))/(16384*d) - ((295*tan(c/2 + (d*x)/2))/ 512 - (265*tan(c/2 + (d*x)/2)^3)/768 + (745*tan(c/2 + (d*x)/2)^5)/8192)/(d *(48*tan(c/2 + (d*x)/2)^2 - 12*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)^6 - 64))
Time = 0.17 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.74 \[ \int \frac {1}{(3+5 \cos (c+d x))^4} \, dx=\frac {104625 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right ) \sin \left (d x +c \right )^{2}-217620 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )-104625 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right ) \sin \left (d x +c \right )^{2}+217620 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )-95400 \cos \left (d x +c \right ) \sin \left (d x +c \right )+188325 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right ) \sin \left (d x +c \right )^{2}-210924 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )-188325 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right ) \sin \left (d x +c \right )^{2}+210924 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )+99500 \sin \left (d x +c \right )^{3}-131160 \sin \left (d x +c \right )}{98304 d \left (125 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-260 \cos \left (d x +c \right )+225 \sin \left (d x +c \right )^{2}-252\right )} \] Input:
int(1/(3+5*cos(d*x+c))^4,x)
Output:
(104625*cos(c + d*x)*log(tan((c + d*x)/2) - 2)*sin(c + d*x)**2 - 217620*co s(c + d*x)*log(tan((c + d*x)/2) - 2) - 104625*cos(c + d*x)*log(tan((c + d* x)/2) + 2)*sin(c + d*x)**2 + 217620*cos(c + d*x)*log(tan((c + d*x)/2) + 2) - 95400*cos(c + d*x)*sin(c + d*x) + 188325*log(tan((c + d*x)/2) - 2)*sin( c + d*x)**2 - 210924*log(tan((c + d*x)/2) - 2) - 188325*log(tan((c + d*x)/ 2) + 2)*sin(c + d*x)**2 + 210924*log(tan((c + d*x)/2) + 2) + 99500*sin(c + d*x)**3 - 131160*sin(c + d*x))/(98304*d*(125*cos(c + d*x)*sin(c + d*x)**2 - 260*cos(c + d*x) + 225*sin(c + d*x)**2 - 252))