Integrand size = 19, antiderivative size = 131 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{d}-\frac {3 a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \]
a*arctanh(sin(d*x+c))/d-a*cot(d*x+c)/d-a*cot(d*x+c)^3/d-3/5*a*cot(d*x+c)^5 /d-1/7*a*cot(d*x+c)^7/d-a*csc(d*x+c)/d-1/3*a*csc(d*x+c)^3/d-1/5*a*csc(d*x+ c)^5/d-1/7*a*csc(d*x+c)^7/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {16 a \cot (c+d x)}{35 d}-\frac {8 a \cot (c+d x) \csc ^2(c+d x)}{35 d}-\frac {6 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac {a \cot (c+d x) \csc ^6(c+d x)}{7 d}-\frac {a \csc ^7(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},\sin ^2(c+d x)\right )}{7 d} \]
(-16*a*Cot[c + d*x])/(35*d) - (8*a*Cot[c + d*x]*Csc[c + d*x]^2)/(35*d) - ( 6*a*Cot[c + d*x]*Csc[c + d*x]^4)/(35*d) - (a*Cot[c + d*x]*Csc[c + d*x]^6)/ (7*d) - (a*Csc[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, Sin[c + d*x]^2] )/(7*d)
Time = 0.49 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.79, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {3042, 4360, 25, 25, 3042, 25, 3317, 25, 3042, 3101, 25, 254, 2009, 4254, 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.
| \(\displaystyle \int \csc ^8(c+d x) (a \sec (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\left (\csc ^8(c+d x) \sec (c+d x) (a (-\cos (c+d x))-a)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\left ((\cos (c+d x) a+a) \csc ^8(c+d x) \sec (c+d x)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^8(c+d x) \sec (c+d x) (a \cos (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {a-a \sin \left (c+d x-\frac {\pi }{2}\right )}{\sin \left (c+d x-\frac {\pi }{2}\right ) \cos \left (c+d x-\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^8 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \csc ^8(c+d x)dx-a \int -\csc ^8(c+d x) \sec (c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a \int \csc ^8(c+d x)dx+a \int \csc ^8(c+d x) \sec (c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \csc (c+d x)^8dx+a \int \csc (c+d x)^8 \sec (c+d x)dx\) |
| \(\Big \downarrow \) 3101 |
| \(\displaystyle a \int \csc (c+d x)^8dx-\frac {a \int -\frac {\csc ^8(c+d x)}{1-\csc ^2(c+d x)}d\csc (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \int \frac {\csc ^8(c+d x)}{1-\csc ^2(c+d x)}d\csc (c+d x)}{d}+a \int \csc (c+d x)^8dx\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {a \int \left (-\csc ^6(c+d x)-\csc ^4(c+d x)-\csc ^2(c+d x)+\frac {1}{1-\csc ^2(c+d x)}-1\right )d\csc (c+d x)}{d}+a \int \csc (c+d x)^8dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \csc (c+d x)^8dx-\frac {a \left (-\text {arctanh}(\csc (c+d x))+\frac {1}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)+\csc (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {a \int \left (\cot ^6(c+d x)+3 \cot ^4(c+d x)+3 \cot ^2(c+d x)+1\right )d\cot (c+d x)}{d}-\frac {a \left (-\text {arctanh}(\csc (c+d x))+\frac {1}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)+\csc (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (-\text {arctanh}(\csc (c+d x))+\frac {1}{7} \csc ^7(c+d x)+\frac {1}{5} \csc ^5(c+d x)+\frac {1}{3} \csc ^3(c+d x)+\csc (c+d x)\right )}{d}-\frac {a \left (\frac {1}{7} \cot ^7(c+d x)+\frac {3}{5} \cot ^5(c+d x)+\cot ^3(c+d x)+\cot (c+d x)\right )}{d}\) |
-((a*(Cot[c + d*x] + Cot[c + d*x]^3 + (3*Cot[c + d*x]^5)/5 + Cot[c + d*x]^ 7/7))/d) - (a*(-ArcTanh[Csc[c + d*x]] + Csc[c + d*x] + Csc[c + d*x]^3/3 + Csc[c + d*x]^5/5 + Csc[c + d*x]^7/7))/d
3.1.17.3.1 Defintions of rubi rules used
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_S ymbol] :> Simp[-(f*a^n)^(-1) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/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 = 1.00 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {16}{35}-\frac {\csc \left (d x +c \right )^{6}}{7}-\frac {6 \csc \left (d x +c \right )^{4}}{35}-\frac {8 \csc \left (d x +c \right )^{2}}{35}\right ) \cot \left (d x +c \right )}{d}\) | \(103\) |
| default | \(\frac {a \left (-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {16}{35}-\frac {\csc \left (d x +c \right )^{6}}{7}-\frac {6 \csc \left (d x +c \right )^{4}}{35}-\frac {8 \csc \left (d x +c \right )^{2}}{35}\right ) \cot \left (d x +c \right )}{d}\) | \(103\) |
| parallelrisch | \(-\frac {\left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\frac {56 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {203 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+448 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+203 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+448 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-448 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right ) a}{448 d}\) | \(121\) |
| norman | \(\frac {-\frac {a}{448 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{40 d}-\frac {29 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{192 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {29 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{24 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{320 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(158\) |
| risch | \(-\frac {2 i a \left (105 \,{\mathrm e}^{11 i \left (d x +c \right )}-210 \,{\mathrm e}^{10 i \left (d x +c \right )}-455 \,{\mathrm e}^{9 i \left (d x +c \right )}+1120 \,{\mathrm e}^{8 i \left (d x +c \right )}+686 \,{\mathrm e}^{7 i \left (d x +c \right )}-2492 \,{\mathrm e}^{6 i \left (d x +c \right )}-274 \,{\mathrm e}^{5 i \left (d x +c \right )}+1360 \,{\mathrm e}^{4 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}-402 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+48\right )}{105 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(195\) |
1/d*(a*(-1/7/sin(d*x+c)^7-1/5/sin(d*x+c)^5-1/3/sin(d*x+c)^3-1/sin(d*x+c)+l n(sec(d*x+c)+tan(d*x+c)))+a*(-16/35-1/7*csc(d*x+c)^6-6/35*csc(d*x+c)^4-8/3 5*csc(d*x+c)^2)*cot(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (121) = 242\).
Time = 0.27 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.15 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {96 \, a \cos \left (d x + c\right )^{6} + 114 \, a \cos \left (d x + c\right )^{5} - 450 \, a \cos \left (d x + c\right )^{4} - 250 \, a \cos \left (d x + c\right )^{3} + 670 \, a \cos \left (d x + c\right )^{2} - 105 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 142 \, a \cos \left (d x + c\right ) - 352 \, a}{210 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \]
-1/210*(96*a*cos(d*x + c)^6 + 114*a*cos(d*x + c)^5 - 450*a*cos(d*x + c)^4 - 250*a*cos(d*x + c)^3 + 670*a*cos(d*x + c)^2 - 105*(a*cos(d*x + c)^5 - a* cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a)*log(sin(d*x + c) + 1)*sin(d*x + c) + 105*(a*cos(d*x + c)^5 - a*cos(d* x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a)*l og(-sin(d*x + c) + 1)*sin(d*x + c) + 142*a*cos(d*x + c) - 352*a)/((d*cos(d *x + c)^5 - d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 + d *cos(d*x + c) - d)*sin(d*x + c))
Timed out. \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} + 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} + 15\right )}}{\sin \left (d x + c\right )^{7}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {6 \, {\left (35 \, \tan \left (d x + c\right )^{6} + 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} + 5\right )} a}{\tan \left (d x + c\right )^{7}}}{210 \, d} \]
-1/210*(a*(2*(105*sin(d*x + c)^6 + 35*sin(d*x + c)^4 + 21*sin(d*x + c)^2 + 15)/sin(d*x + c)^7 - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1 )) + 6*(35*tan(d*x + c)^6 + 35*tan(d*x + c)^4 + 21*tan(d*x + c)^2 + 5)*a/t an(d*x + c)^7)/d
Time = 0.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6720 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6720 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3045 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1015 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{6720 \, d} \]
-1/6720*(21*a*tan(1/2*d*x + 1/2*c)^5 + 280*a*tan(1/2*d*x + 1/2*c)^3 - 6720 *a*log(abs(tan(1/2*d*x + 1/2*c) + 1)) + 6720*a*log(abs(tan(1/2*d*x + 1/2*c ) - 1)) + 3045*a*tan(1/2*d*x + 1/2*c) + (6720*a*tan(1/2*d*x + 1/2*c)^6 + 1 015*a*tan(1/2*d*x + 1/2*c)^4 + 168*a*tan(1/2*d*x + 1/2*c)^2 + 15*a)/tan(1/ 2*d*x + 1/2*c)^7)/d
Time = 14.44 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx=\frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {29\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (64\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {a}{7}\right )}{64\,d} \]