Integrand size = 21, antiderivative size = 185 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {1}{128 a^3 d (1-\cos (c+d x))^2}+\frac {5}{64 a^3 d (1-\cos (c+d x))}+\frac {1}{40 a^3 d (1+\cos (c+d x))^5}-\frac {13}{64 a^3 d (1+\cos (c+d x))^4}+\frac {35}{48 a^3 d (1+\cos (c+d x))^3}-\frac {99}{64 a^3 d (1+\cos (c+d x))^2}+\frac {303}{128 a^3 d (1+\cos (c+d x))}+\frac {37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac {219 \log (1+\cos (c+d x))}{256 a^3 d} \]
-1/128/a^3/d/(1-cos(d*x+c))^2+5/64/a^3/d/(1-cos(d*x+c))+1/40/a^3/d/(1+cos( d*x+c))^5-13/64/a^3/d/(1+cos(d*x+c))^4+35/48/a^3/d/(1+cos(d*x+c))^3-99/64/ a^3/d/(1+cos(d*x+c))^2+303/128/a^3/d/(1+cos(d*x+c))+37/256*ln(1-cos(d*x+c) )/a^3/d+219/256*ln(1+cos(d*x+c))/a^3/d
Time = 0.89 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\left (12-195 \cos ^2\left (\frac {1}{2} (c+d x)\right )+1400 \cos ^4\left (\frac {1}{2} (c+d x)\right )+60 \cos ^8\left (\frac {1}{2} (c+d x)\right ) \left (303+10 \cot ^2\left (\frac {1}{2} (c+d x)\right )\right )-30 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (198+\cot ^4\left (\frac {1}{2} (c+d x)\right )\right )+120 \cos ^{10}\left (\frac {1}{2} (c+d x)\right ) \left (219 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+37 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x)}{1920 a^3 d (1+\sec (c+d x))^3} \]
((12 - 195*Cos[(c + d*x)/2]^2 + 1400*Cos[(c + d*x)/2]^4 + 60*Cos[(c + d*x) /2]^8*(303 + 10*Cot[(c + d*x)/2]^2) - 30*Cos[(c + d*x)/2]^6*(198 + Cot[(c + d*x)/2]^4) + 120*Cos[(c + d*x)/2]^10*(219*Log[Cos[(c + d*x)/2]] + 37*Log [Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^4*Sec[c + d*x]^3)/(1920*a^3*d*(1 + S ec[c + d*x])^3)
Time = 0.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.75, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4367, 27, 99, 2009}
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.
| \(\displaystyle \int \frac {\cot ^5(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^5 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {a^6 \int \frac {\cos ^8(c+d x)}{a^9 (1-\cos (c+d x))^3 (\cos (c+d x)+1)^6}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cos ^8(c+d x)}{(1-\cos (c+d x))^3 (\cos (c+d x)+1)^6}d\cos (c+d x)}{a^3 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (-\frac {219}{256 (\cos (c+d x)+1)}+\frac {303}{128 (\cos (c+d x)+1)^2}-\frac {99}{32 (\cos (c+d x)+1)^3}+\frac {35}{16 (\cos (c+d x)+1)^4}-\frac {13}{16 (\cos (c+d x)+1)^5}+\frac {1}{8 (\cos (c+d x)+1)^6}-\frac {37}{256 (\cos (c+d x)-1)}-\frac {5}{64 (\cos (c+d x)-1)^2}-\frac {1}{64 (\cos (c+d x)-1)^3}\right )d\cos (c+d x)}{a^3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {5}{64 (1-\cos (c+d x))}-\frac {303}{128 (\cos (c+d x)+1)}+\frac {1}{128 (1-\cos (c+d x))^2}+\frac {99}{64 (\cos (c+d x)+1)^2}-\frac {35}{48 (\cos (c+d x)+1)^3}+\frac {13}{64 (\cos (c+d x)+1)^4}-\frac {1}{40 (\cos (c+d x)+1)^5}-\frac {37}{256} \log (1-\cos (c+d x))-\frac {219}{256} \log (\cos (c+d x)+1)}{a^3 d}\) |
-((1/(128*(1 - Cos[c + d*x])^2) - 5/(64*(1 - Cos[c + d*x])) - 1/(40*(1 + C os[c + d*x])^5) + 13/(64*(1 + Cos[c + d*x])^4) - 35/(48*(1 + Cos[c + d*x]) ^3) + 99/(64*(1 + Cos[c + d*x])^2) - 303/(128*(1 + Cos[c + d*x])) - (37*Lo g[1 - Cos[c + d*x]])/256 - (219*Log[1 + Cos[c + d*x]])/256)/(a^3*d))
3.1.94.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[1/(a^(m - n - 1)*b^n*d) Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Time = 0.74 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.62
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{128 \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {5}{64 \left (\cos \left (d x +c \right )-1\right )}+\frac {37 \ln \left (\cos \left (d x +c \right )-1\right )}{256}+\frac {1}{40 \left (\cos \left (d x +c \right )+1\right )^{5}}-\frac {13}{64 \left (\cos \left (d x +c \right )+1\right )^{4}}+\frac {35}{48 \left (\cos \left (d x +c \right )+1\right )^{3}}-\frac {99}{64 \left (\cos \left (d x +c \right )+1\right )^{2}}+\frac {303}{128 \left (\cos \left (d x +c \right )+1\right )}+\frac {219 \ln \left (\cos \left (d x +c \right )+1\right )}{256}}{d \,a^{3}}\) | \(115\) |
| default | \(\frac {-\frac {1}{128 \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {5}{64 \left (\cos \left (d x +c \right )-1\right )}+\frac {37 \ln \left (\cos \left (d x +c \right )-1\right )}{256}+\frac {1}{40 \left (\cos \left (d x +c \right )+1\right )^{5}}-\frac {13}{64 \left (\cos \left (d x +c \right )+1\right )^{4}}+\frac {35}{48 \left (\cos \left (d x +c \right )+1\right )^{3}}-\frac {99}{64 \left (\cos \left (d x +c \right )+1\right )^{2}}+\frac {303}{128 \left (\cos \left (d x +c \right )+1\right )}+\frac {219 \ln \left (\cos \left (d x +c \right )+1\right )}{256}}{d \,a^{3}}\) | \(115\) |
| risch | \(-\frac {i x}{a^{3}}-\frac {2 i c}{a^{3} d}+\frac {4395 \,{\mathrm e}^{13 i \left (d x +c \right )}+11010 \,{\mathrm e}^{12 i \left (d x +c \right )}-1390 \,{\mathrm e}^{11 i \left (d x +c \right )}-47190 \,{\mathrm e}^{10 i \left (d x +c \right )}-50987 \,{\mathrm e}^{9 i \left (d x +c \right )}+25428 \,{\mathrm e}^{8 i \left (d x +c \right )}+86748 \,{\mathrm e}^{7 i \left (d x +c \right )}+25428 \,{\mathrm e}^{6 i \left (d x +c \right )}-50987 \,{\mathrm e}^{5 i \left (d x +c \right )}-47190 \,{\mathrm e}^{4 i \left (d x +c \right )}-1390 \,{\mathrm e}^{3 i \left (d x +c \right )}+11010 \,{\mathrm e}^{2 i \left (d x +c \right )}+4395 \,{\mathrm e}^{i \left (d x +c \right )}}{960 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4}}+\frac {219 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 a^{3} d}+\frac {37 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 a^{3} d}\) | \(237\) |
1/d/a^3*(-1/128/(cos(d*x+c)-1)^2-5/64/(cos(d*x+c)-1)+37/256*ln(cos(d*x+c)- 1)+1/40/(cos(d*x+c)+1)^5-13/64/(cos(d*x+c)+1)^4+35/48/(cos(d*x+c)+1)^3-99/ 64/(cos(d*x+c)+1)^2+303/128/(cos(d*x+c)+1)+219/256*ln(cos(d*x+c)+1))
Time = 0.29 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.71 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {8790 \, \cos \left (d x + c\right )^{6} + 11010 \, \cos \left (d x + c\right )^{5} - 13880 \, \cos \left (d x + c\right )^{4} - 25560 \, \cos \left (d x + c\right )^{3} - 734 \, \cos \left (d x + c\right )^{2} + 3285 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 555 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 13878 \, \cos \left (d x + c\right ) + 5536}{3840 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
1/3840*(8790*cos(d*x + c)^6 + 11010*cos(d*x + c)^5 - 13880*cos(d*x + c)^4 - 25560*cos(d*x + c)^3 - 734*cos(d*x + c)^2 + 3285*(cos(d*x + c)^7 + 3*cos (d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d *x + c)^2 + 3*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 555*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2 ) + 13878*cos(d*x + c) + 5536)/(a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c )^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c) ^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)
\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Time = 0.23 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (4395 \, \cos \left (d x + c\right )^{6} + 5505 \, \cos \left (d x + c\right )^{5} - 6940 \, \cos \left (d x + c\right )^{4} - 12780 \, \cos \left (d x + c\right )^{3} - 367 \, \cos \left (d x + c\right )^{2} + 6939 \, \cos \left (d x + c\right ) + 2768\right )}}{a^{3} \cos \left (d x + c\right )^{7} + 3 \, a^{3} \cos \left (d x + c\right )^{6} + a^{3} \cos \left (d x + c\right )^{5} - 5 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}} + \frac {3285 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {555 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{3840 \, d} \]
1/3840*(2*(4395*cos(d*x + c)^6 + 5505*cos(d*x + c)^5 - 6940*cos(d*x + c)^4 - 12780*cos(d*x + c)^3 - 367*cos(d*x + c)^2 + 6939*cos(d*x + c) + 2768)/( a^3*cos(d*x + c)^7 + 3*a^3*cos(d*x + c)^6 + a^3*cos(d*x + c)^5 - 5*a^3*cos (d*x + c)^4 - 5*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2 + 3*a^3*cos(d*x + c) + a^3) + 3285*log(cos(d*x + c) + 1)/a^3 + 555*log(cos(d*x + c) - 1)/a^3 )/d
Time = 0.49 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.41 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {30 \, {\left (\frac {18 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {111 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {2220 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {15360 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\frac {9780 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2790 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {740 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {135 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {12 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{15}}}{15360 \, d} \]
-1/15360*(30*(18*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 111*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 1)*(cos(d*x + c) + 1)^2/(a^3*(cos(d*x + c) - 1)^2) - 2220*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^3 + 153 60*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^3 + (9780*a^12*( cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2790*a^12*(cos(d*x + c) - 1)^2/(cos (d*x + c) + 1)^2 + 740*a^12*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 13 5*a^12*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 12*a^12*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5)/a^15)/d
Time = 14.54 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.92 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{256\,a^3\,d}-\frac {93\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,a^3\,d}+\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{768\,a^3\,d}-\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,a^3\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{1280\,a^3\,d}+\frac {37\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a^3\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^3\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {1}{4}\right )}{128\,a^3\,d} \]
(163*tan(c/2 + (d*x)/2)^2)/(256*a^3*d) - (93*tan(c/2 + (d*x)/2)^4)/(512*a^ 3*d) + (37*tan(c/2 + (d*x)/2)^6)/(768*a^3*d) - (9*tan(c/2 + (d*x)/2)^8)/(1 024*a^3*d) + tan(c/2 + (d*x)/2)^10/(1280*a^3*d) + (37*log(tan(c/2 + (d*x)/ 2)))/(128*a^3*d) - log(tan(c/2 + (d*x)/2)^2 + 1)/(a^3*d) + (cot(c/2 + (d*x )/2)^4*((9*tan(c/2 + (d*x)/2)^2)/2 - 1/4))/(128*a^3*d)