Integrand size = 21, antiderivative size = 128 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{128 a^3 d}-\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {a^2}{40 d (a+a \cos (c+d x))^5}+\frac {3 a}{64 d (a+a \cos (c+d x))^4}-\frac {1}{64 a d (a+a \cos (c+d x))^2}-\frac {3}{128 d \left (a^3+a^3 \cos (c+d x)\right )} \]
3/128*arctanh(cos(d*x+c))/a^3/d-1/128/a/d/(a-a*cos(d*x+c))^2-1/40*a^2/d/(a +a*cos(d*x+c))^5+3/64*a/d/(a+a*cos(d*x+c))^4-1/64/a/d/(a+a*cos(d*x+c))^2-3 /128/d/(a^3+a^3*cos(d*x+c))
Time = 3.78 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\left (4-15 \cos ^2\left (\frac {1}{2} (c+d x)\right )+60 \cos ^8\left (\frac {1}{2} (c+d x)\right )+10 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (2+\cot ^4\left (\frac {1}{2} (c+d x)\right )\right )-120 \cos ^{10}\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x)}{640 a^3 d (1+\sec (c+d x))^3} \]
-1/640*((4 - 15*Cos[(c + d*x)/2]^2 + 60*Cos[(c + d*x)/2]^8 + 10*Cos[(c + d *x)/2]^6*(2 + Cot[(c + d*x)/2]^4) - 120*Cos[(c + d*x)/2]^10*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^4*Sec[c + d*x]^3)/(a^3 *d*(1 + Sec[c + d*x])^3)
Time = 0.47 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 25, 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.
| \(\displaystyle \int \frac {\csc ^5(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(a (-\cos (c+d x))-a)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(\cos (c+d x) a+a)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(a \cos (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^5 \left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int -\frac {\cos ^3(c+d x)}{(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^6}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^5 \int \frac {\cos ^3(c+d x)}{(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^6}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \int \frac {a^3 \cos ^3(c+d x)}{(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^6}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^2 \int \left (-\frac {1}{8 (\cos (c+d x) a+a)^6}+\frac {3}{16 (\cos (c+d x) a+a)^5 a}+\frac {1}{64 (a-a \cos (c+d x))^3 a^3}-\frac {1}{32 (\cos (c+d x) a+a)^3 a^3}-\frac {3}{128 \left (a^2-a^2 \cos ^2(c+d x)\right ) a^4}-\frac {3}{128 (\cos (c+d x) a+a)^2 a^4}\right )d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \left (-\frac {3 \text {arctanh}(\cos (c+d x))}{128 a^5}+\frac {3}{128 a^4 (a \cos (c+d x)+a)}+\frac {1}{128 a^3 (a-a \cos (c+d x))^2}+\frac {1}{64 a^3 (a \cos (c+d x)+a)^2}-\frac {3}{64 a (a \cos (c+d x)+a)^4}+\frac {1}{40 (a \cos (c+d x)+a)^5}\right )}{d}\) |
-((a^2*((-3*ArcTanh[Cos[c + d*x]])/(128*a^5) + 1/(128*a^3*(a - a*Cos[c + d *x])^2) + 1/(40*(a + a*Cos[c + d*x])^5) - 3/(64*a*(a + a*Cos[c + d*x])^4) + 1/(64*a^3*(a + a*Cos[c + d*x])^2) + 3/(128*a^4*(a + a*Cos[c + d*x]))))/d )
3.1.99.3.1 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[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.99 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{40 \left (\cos \left (d x +c \right )+1\right )^{5}}+\frac {3}{64 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {1}{64 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {3}{128 \left (\cos \left (d x +c \right )+1\right )}+\frac {3 \ln \left (\cos \left (d x +c \right )+1\right )}{256}-\frac {1}{128 \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {3 \ln \left (\cos \left (d x +c \right )-1\right )}{256}}{d \,a^{3}}\) | \(91\) |
| default | \(\frac {-\frac {1}{40 \left (\cos \left (d x +c \right )+1\right )^{5}}+\frac {3}{64 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {1}{64 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {3}{128 \left (\cos \left (d x +c \right )+1\right )}+\frac {3 \ln \left (\cos \left (d x +c \right )+1\right )}{256}-\frac {1}{128 \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {3 \ln \left (\cos \left (d x +c \right )-1\right )}{256}}{d \,a^{3}}\) | \(91\) |
| parallelrisch | \(\frac {-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-10 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-60 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-20 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5120 a^{3} d}\) | \(113\) |
| norman | \(\frac {-\frac {1}{512 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{1024 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{1280 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{256 d a}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{256 d a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{512 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{256 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{2}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a^{3} d}\) | \(158\) |
| risch | \(-\frac {15 \,{\mathrm e}^{13 i \left (d x +c \right )}+90 \,{\mathrm e}^{12 i \left (d x +c \right )}+170 \,{\mathrm e}^{11 i \left (d x +c \right )}-30 \,{\mathrm e}^{10 i \left (d x +c \right )}+1521 \,{\mathrm e}^{9 i \left (d x +c \right )}+1476 \,{\mathrm e}^{8 i \left (d x +c \right )}+3756 \,{\mathrm e}^{7 i \left (d x +c \right )}+1476 \,{\mathrm e}^{6 i \left (d x +c \right )}+1521 \,{\mathrm e}^{5 i \left (d x +c \right )}-30 \,{\mathrm e}^{4 i \left (d x +c \right )}+170 \,{\mathrm e}^{3 i \left (d x +c \right )}+90 \,{\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}}{320 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 a^{3} d}\) | \(220\) |
1/d/a^3*(-1/40/(cos(d*x+c)+1)^5+3/64/(cos(d*x+c)+1)^4-1/64/(cos(d*x+c)+1)^ 2-3/128/(cos(d*x+c)+1)+3/256*ln(cos(d*x+c)+1)-1/128/(cos(d*x+c)-1)^2-3/256 *ln(cos(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (117) = 234\).
Time = 0.27 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.48 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {30 \, \cos \left (d x + c\right )^{6} + 90 \, \cos \left (d x + c\right )^{5} + 40 \, \cos \left (d x + c\right )^{4} - 120 \, \cos \left (d x + c\right )^{3} + 122 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 126 \, \cos \left (d x + c\right ) + 32}{1280 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
-1/1280*(30*cos(d*x + c)^6 + 90*cos(d*x + c)^5 + 40*cos(d*x + c)^4 - 120*c os(d*x + c)^3 + 122*cos(d*x + c)^2 - 15*(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) + 15*(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 + co s(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 126*cos( d*x + c) + 32)/(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)
\[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\csc ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Time = 0.20 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.47 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{6} + 45 \, \cos \left (d x + c\right )^{5} + 20 \, \cos \left (d x + c\right )^{4} - 60 \, \cos \left (d x + c\right )^{3} + 61 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right ) + 16\right )}}{a^{3} \cos \left (d x + c\right )^{7} + 3 \, a^{3} \cos \left (d x + c\right )^{6} + a^{3} \cos \left (d x + c\right )^{5} - 5 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}} - \frac {15 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {15 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{1280 \, d} \]
-1/1280*(2*(15*cos(d*x + c)^6 + 45*cos(d*x + c)^5 + 20*cos(d*x + c)^4 - 60 *cos(d*x + c)^3 + 61*cos(d*x + c)^2 + 63*cos(d*x + c) + 16)/(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) - 1 5*log(cos(d*x + c) + 1)/a^3 + 15*log(cos(d*x + c) - 1)/a^3)/d
Time = 0.45 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.81 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {10 \, {\left (\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {60 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {\frac {60 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {30 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {20 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{15}}}{5120 \, d} \]
1/5120*(10*(2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*(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) - 60*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^3 + (60*a^12*( cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 30*a^12*(cos(d*x + c) - 1)^2/(cos(d *x + c) + 1)^2 - 20*a^12*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 5*a^1 2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 4*a^12*(cos(d*x + c) - 1)^5/ (cos(d*x + c) + 1)^5)/a^15)/d
Time = 13.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.35 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3\,\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{128\,a^3\,d}-\frac {\frac {3\,{\cos \left (c+d\,x\right )}^6}{128}+\frac {9\,{\cos \left (c+d\,x\right )}^5}{128}+\frac {{\cos \left (c+d\,x\right )}^4}{32}-\frac {3\,{\cos \left (c+d\,x\right )}^3}{32}+\frac {61\,{\cos \left (c+d\,x\right )}^2}{640}+\frac {63\,\cos \left (c+d\,x\right )}{640}+\frac {1}{40}}{d\,\left (a^3\,{\cos \left (c+d\,x\right )}^7+3\,a^3\,{\cos \left (c+d\,x\right )}^6+a^3\,{\cos \left (c+d\,x\right )}^5-5\,a^3\,{\cos \left (c+d\,x\right )}^4-5\,a^3\,{\cos \left (c+d\,x\right )}^3+a^3\,{\cos \left (c+d\,x\right )}^2+3\,a^3\,\cos \left (c+d\,x\right )+a^3\right )} \]
(3*atanh(cos(c + d*x)))/(128*a^3*d) - ((63*cos(c + d*x))/640 + (61*cos(c + d*x)^2)/640 - (3*cos(c + d*x)^3)/32 + cos(c + d*x)^4/32 + (9*cos(c + d*x) ^5)/128 + (3*cos(c + d*x)^6)/128 + 1/40)/(d*(3*a^3*cos(c + d*x) + a^3 + a^ 3*cos(c + d*x)^2 - 5*a^3*cos(c + d*x)^3 - 5*a^3*cos(c + d*x)^4 + a^3*cos(c + d*x)^5 + 3*a^3*cos(c + d*x)^6 + a^3*cos(c + d*x)^7))