Integrand size = 21, antiderivative size = 146 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{64 a^2 d}-\frac {1}{64 d (a-a \cos (c+d x))^2}+\frac {a^2}{32 d (a+a \cos (c+d x))^4}-\frac {a}{48 d (a+a \cos (c+d x))^3}-\frac {1}{32 d (a+a \cos (c+d x))^2}-\frac {1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac {1}{32 d \left (a^2+a^2 \cos (c+d x)\right )} \]
1/64*arctanh(cos(d*x+c))/a^2/d-1/64/d/(a-a*cos(d*x+c))^2+1/32*a^2/d/(a+a*c os(d*x+c))^4-1/48*a/d/(a+a*cos(d*x+c))^3-1/32/d/(a+a*cos(d*x+c))^2-1/64/d/ (a^2-a^2*cos(d*x+c))-1/32/d/(a^2+a^2*cos(d*x+c))
Time = 0.62 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.04 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (12 \csc ^2\left (\frac {1}{2} (c+d x)\right )+6 \csc ^4\left (\frac {1}{2} (c+d x)\right )+24 \left (-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+24 \sec ^2\left (\frac {1}{2} (c+d x)\right )+12 \sec ^4\left (\frac {1}{2} (c+d x)\right )+4 \sec ^6\left (\frac {1}{2} (c+d x)\right )-3 \sec ^8\left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x)}{384 a^2 d (1+\sec (c+d x))^2} \]
-1/384*(Cos[(c + d*x)/2]^4*(12*Csc[(c + d*x)/2]^2 + 6*Csc[(c + d*x)/2]^4 + 24*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]) + 24*Sec[(c + d*x)/2] ^2 + 12*Sec[(c + d*x)/2]^4 + 4*Sec[(c + d*x)/2]^6 - 3*Sec[(c + d*x)/2]^8)* Sec[c + d*x]^2)/(a^2*d*(1 + Sec[c + d*x])^2)
Time = 0.48 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4360, 3042, 3315, 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.
| \(\displaystyle \int \frac {\csc ^5(c+d x)}{(a \sec (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{(a (-\cos (c+d x))-a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^5 \left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int \frac {\cos ^2(c+d x)}{(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^5}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^3 \int \frac {a^2 \cos ^2(c+d x)}{(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^5}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^3 \int \left (\frac {1}{8 (\cos (c+d x) a+a)^5 a}-\frac {1}{16 (\cos (c+d x) a+a)^4 a^2}+\frac {1}{32 (a-a \cos (c+d x))^3 a^3}-\frac {1}{16 (\cos (c+d x) a+a)^3 a^3}-\frac {1}{64 \left (a^2-a^2 \cos ^2(c+d x)\right ) a^4}+\frac {1}{64 (a-a \cos (c+d x))^2 a^4}-\frac {1}{32 (\cos (c+d x) a+a)^2 a^4}\right )d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \left (\frac {\text {arctanh}(\cos (c+d x))}{64 a^5}-\frac {1}{64 a^4 (a-a \cos (c+d x))}-\frac {1}{32 a^4 (a \cos (c+d x)+a)}-\frac {1}{64 a^3 (a-a \cos (c+d x))^2}-\frac {1}{32 a^3 (a \cos (c+d x)+a)^2}-\frac {1}{48 a^2 (a \cos (c+d x)+a)^3}+\frac {1}{32 a (a \cos (c+d x)+a)^4}\right )}{d}\) |
(a^3*(ArcTanh[Cos[c + d*x]]/(64*a^5) - 1/(64*a^3*(a - a*Cos[c + d*x])^2) - 1/(64*a^4*(a - a*Cos[c + d*x])) + 1/(32*a*(a + a*Cos[c + d*x])^4) - 1/(48 *a^2*(a + a*Cos[c + d*x])^3) - 1/(32*a^3*(a + a*Cos[c + d*x])^2) - 1/(32*a ^4*(a + a*Cos[c + d*x]))))/d
3.1.82.3.1 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 = 0.61 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68
| method | result | size |
| parallelrisch | \(\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-6 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-24 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1536 a^{2} d}\) | \(100\) |
| derivativedivides | \(\frac {-\frac {1}{64 \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {1}{64 \cos \left (d x +c \right )-64}-\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{128}+\frac {1}{32 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {1}{48 \left (\cos \left (d x +c \right )+1\right )^{3}}-\frac {1}{32 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {1}{32 \left (\cos \left (d x +c \right )+1\right )}+\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{128}}{d \,a^{2}}\) | \(103\) |
| default | \(\frac {-\frac {1}{64 \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {1}{64 \cos \left (d x +c \right )-64}-\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{128}+\frac {1}{32 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {1}{48 \left (\cos \left (d x +c \right )+1\right )^{3}}-\frac {1}{32 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {1}{32 \left (\cos \left (d x +c \right )+1\right )}+\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{128}}{d \,a^{2}}\) | \(103\) |
| norman | \(\frac {-\frac {1}{256 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{512 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{256 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{192 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2} d}\) | \(139\) |
| risch | \(-\frac {3 \,{\mathrm e}^{11 i \left (d x +c \right )}+12 \,{\mathrm e}^{10 i \left (d x +c \right )}+7 \,{\mathrm e}^{9 i \left (d x +c \right )}-32 \,{\mathrm e}^{8 i \left (d x +c \right )}+566 \,{\mathrm e}^{7 i \left (d x +c \right )}+424 \,{\mathrm e}^{6 i \left (d x +c \right )}+566 \,{\mathrm e}^{5 i \left (d x +c \right )}-32 \,{\mathrm e}^{4 i \left (d x +c \right )}+7 \,{\mathrm e}^{3 i \left (d x +c \right )}+12 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}}{96 a^{2} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 a^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 a^{2} d}\) | \(198\) |
1/1536*(3*tan(1/2*d*x+1/2*c)^8+8*tan(1/2*d*x+1/2*c)^6-6*cot(1/2*d*x+1/2*c) ^4-6*tan(1/2*d*x+1/2*c)^4-24*cot(1/2*d*x+1/2*c)^2-48*tan(1/2*d*x+1/2*c)^2- 24*ln(tan(1/2*d*x+1/2*c)))/a^2/d
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
Time = 0.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.94 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {6 \, \cos \left (d x + c\right )^{5} + 12 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - 20 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 70 \, \cos \left (d x + c\right ) + 32}{384 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
-1/384*(6*cos(d*x + c)^5 + 12*cos(d*x + c)^4 - 4*cos(d*x + c)^3 - 20*cos(d *x + c)^2 - 3*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos( d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1 /2) + 3*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 70*cos(d*x + c) + 32)/(a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^ 2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2 *d*cos(d*x + c) + a^2*d)
\[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\csc ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Time = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 6 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 10 \, \cos \left (d x + c\right )^{2} + 35 \, \cos \left (d x + c\right ) + 16\right )}}{a^{2} \cos \left (d x + c\right )^{6} + 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}} - \frac {3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{384 \, d} \]
-1/384*(2*(3*cos(d*x + c)^5 + 6*cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 10*cos (d*x + c)^2 + 35*cos(d*x + c) + 16)/(a^2*cos(d*x + c)^6 + 2*a^2*cos(d*x + c)^5 - a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^3 - a^2*cos(d*x + c)^2 + 2* a^2*cos(d*x + c) + a^2) - 3*log(cos(d*x + c) + 1)/a^2 + 3*log(cos(d*x + c) - 1)/a^2)/d
Time = 0.38 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.42 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {6 \, {\left (\frac {4 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {12 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac {\frac {48 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {6 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{8}}}{1536 \, d} \]
1/1536*(6*(4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*(cos(d*x + c) - 1)^ 2/(cos(d*x + c) + 1)^2 - 1)*(cos(d*x + c) + 1)^2/(a^2*(cos(d*x + c) - 1)^2 ) - 12*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^2 + (48*a^6*(co s(d*x + c) - 1)/(cos(d*x + c) + 1) - 6*a^6*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 8*a^6*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 3*a^6*(cos( d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)/a^8)/d
Time = 13.64 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.04 \[ \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{64\,a^2\,d}-\frac {\frac {{\cos \left (c+d\,x\right )}^5}{64}+\frac {{\cos \left (c+d\,x\right )}^4}{32}-\frac {{\cos \left (c+d\,x\right )}^3}{96}-\frac {5\,{\cos \left (c+d\,x\right )}^2}{96}+\frac {35\,\cos \left (c+d\,x\right )}{192}+\frac {1}{12}}{d\,\left (a^2\,{\cos \left (c+d\,x\right )}^6+2\,a^2\,{\cos \left (c+d\,x\right )}^5-a^2\,{\cos \left (c+d\,x\right )}^4-4\,a^2\,{\cos \left (c+d\,x\right )}^3-a^2\,{\cos \left (c+d\,x\right )}^2+2\,a^2\,\cos \left (c+d\,x\right )+a^2\right )} \]