Integrand size = 12, antiderivative size = 138 \[ \int \frac {1}{(-3+5 \cos (c+d x))^4} \, dx=-\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))} \]
-279/32768*ln(cos(1/2*d*x+1/2*c)-2*sin(1/2*d*x+1/2*c))/d+279/32768*ln(cos( 1/2*d*x+1/2*c)+2*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. \(288\) vs. \(2(138)=276\).
Time = 0.01 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.09 \[ \int \frac {1}{(-3+5 \cos (c+d x))^4} \, dx=\frac {467046 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-104625 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-765855 \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-467046 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )+104625 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )+226140 \sin (c+d x)-190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))}{393216 d (-3+5 \cos (c+d x))^3} \]
(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[C os[(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*Sin[(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)
Time = 0.49 (sec) , antiderivative size = 106, 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, 220}
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 |
\(\displaystyle \frac {1}{48} \int -\frac {10 \cos (c+d x)+9}{(3-5 \cos (c+d x))^3}dx-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{48} \int \frac {10 \cos (c+d x)+9}{(3-5 \cos (c+d x))^3}dx-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{48} \int \frac {10 \sin \left (c+d x+\frac {\pi }{2}\right )+9}{\left (3-5 \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 3233 |
\(\displaystyle \frac {1}{48} \left (\frac {75 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac {1}{32} \int -\frac {75 \cos (c+d x)+154}{(3-5 \cos (c+d x))^2}dx\right )-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {75 \cos (c+d x)+154}{(3-5 \cos (c+d x))^2}dx+\frac {75 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}\right )-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {75 \sin \left (c+d x+\frac {\pi }{2}\right )+154}{\left (3-5 \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx+\frac {75 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}\right )-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 3233 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {1}{16} \int -\frac {837}{3-5 \cos (c+d x)}dx-\frac {995 \sin (c+d x)}{16 d (3-5 \cos (c+d x))}\right )+\frac {75 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}\right )-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (-\frac {837}{16} \int \frac {1}{3-5 \cos (c+d x)}dx-\frac {995 \sin (c+d x)}{16 d (3-5 \cos (c+d x))}\right )+\frac {75 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}\right )-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (-\frac {837}{16} \int \frac {1}{3-5 \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {995 \sin (c+d x)}{16 d (3-5 \cos (c+d x))}\right )+\frac {75 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}\right )-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (-\frac {837 \int \frac {1}{8 \tan ^2\left (\frac {1}{2} (c+d x)\right )-2}d\tan \left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {995 \sin (c+d x)}{16 d (3-5 \cos (c+d x))}\right )+\frac {75 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}\right )-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {837 \text {arctanh}\left (2 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{32 d}-\frac {995 \sin (c+d x)}{16 d (3-5 \cos (c+d x))}\right )+\frac {75 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}\right )-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}\) |
(-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[2*Tan[(c + d*x)/2]])/(32*d) - (99 5*Sin[c + d*x])/(16*d*(3 - 5*Cos[c + d*x])))/32)/48
3.1.45.3.1 Defintions of rubi rules used
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[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 = 0.71 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76
method | result | size |
norman | \(\frac {-\frac {745 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192 d}+\frac {265 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768 d}-\frac {295 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512 d}}{{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}^{3}}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768 d}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768 d}\) | \(105\) |
risch | \(-\frac {i \left (20925 \,{\mathrm e}^{5 i \left (d x +c \right )}-62775 \,{\mathrm e}^{4 i \left (d x +c \right )}+111042 \,{\mathrm e}^{3 i \left (d x +c \right )}-119310 \,{\mathrm e}^{2 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}-24875\right )}{12288 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{3}}+\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}\) | \(129\) |
derivativedivides | \(\frac {-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768}-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}}{d}\) | \(140\) |
default | \(\frac {-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768}-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}}{d}\) | \(140\) |
parallelrisch | \(\frac {\left (-765855 \cos \left (d x +c \right )+376650 \cos \left (2 d x +2 c \right )-104625 \cos \left (3 d x +3 c \right )+467046\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )+\left (765855 \cos \left (d x +c \right )-376650 \cos \left (2 d x +2 c \right )+104625 \cos \left (3 d x +3 c \right )-467046\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{2}\right )+226140 \sin \left (d x +c \right )-190800 \sin \left (2 d x +2 c \right )+99500 \sin \left (3 d x +3 c \right )}{98304 d \left (-558+125 \cos \left (3 d x +3 c \right )-450 \cos \left (2 d x +2 c \right )+915 \cos \left (d x +c \right )\right )}\) | \(161\) |
(-745/8192/d*tan(1/2*d*x+1/2*c)+265/768/d*tan(1/2*d*x+1/2*c)^3-295/512/d*t an(1/2*d*x+1/2*c)^5)/(4*tan(1/2*d*x+1/2*c)^2-1)^3-279/32768/d*ln(2*tan(1/2 *d*x+1/2*c)-1)+279/32768/d*ln(2*tan(1/2*d*x+1/2*c)+1)
Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(-3+5 \cos (c+d x))^4} \, dx=\frac {837 \, {\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\right )} \log \left (-\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 837 \, {\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\right )} \log \left (-\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 40 \, {\left (4975 \, \cos \left (d x + c\right )^{2} - 4770 \, \cos \left (d x + c\right ) + 1583\right )} \sin \left (d x + c\right )}{196608 \, {\left (125 \, d \cos \left (d x + c\right )^{3} - 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) - 27 \, d\right )}} \]
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. 831 vs. \(2 (126) = 252\).
Time = 2.48 (sec) , antiderivative size = 831, normalized size of antiderivative = 6.02 \[ \int \frac {1}{(-3+5 \cos (c+d x))^4} \, dx=\text {Too large to display} \]
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/(5*cos(c) - 3)**4, Eq(d, 0)), (-53568*log(2*ta n(c/2 + d*x/2) - 1)*tan(c/2 + d*x/2)**6/(6291456*d*tan(c/2 + d*x/2)**6 - 4 718592*d*tan(c/2 + d*x/2)**4 + 1179648*d*tan(c/2 + d*x/2)**2 - 98304*d) + 40176*log(2*tan(c/2 + d*x/2) - 1)*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(2*tan(c/2 + d*x/2) - 1)*tan(c/2 + d*x/2)**2/(62914 56*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(2*tan(c/2 + d*x/2) - 1)/(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(2*tan(c/2 + d*x/2) + 1)*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(2*tan(c/2 + d*x/2) + 1)*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(2*tan(c/2 + d*x/ 2) + 1)*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(2*ta n(c/2 + d*x/2) + 1)/(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 + ...
Time = 0.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(-3+5 \cos (c+d x))^4} \, dx=-\frac {\frac {20 \, {\left (\frac {447 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1696 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2832 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {64 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1} - 837 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 837 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{98304 \, d} \]
-1/98304*(20*(447*sin(d*x + c)/(cos(d*x + c) + 1) - 1696*sin(d*x + c)^3/(c os(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
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(-3+5 \cos (c+d x))^4} \, dx=-\frac {\frac {20 \, {\left (2832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1696 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 447 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} - 837 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 837 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{98304 \, d} \]
-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
Time = 0.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(-3+5 \cos (c+d x))^4} \, dx=\frac {279\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16384\,d}-\frac {\frac {295\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32768}-\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{49152}+\frac {745\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{524288}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16}-\frac {1}{64}\right )} \]