Integrand size = 19, antiderivative size = 118 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a^3}{8 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^2}{8 d (a+a \cos (c+d x))}+\frac {11 a \log (1-\cos (c+d x))}{16 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {5 a \log (1+\cos (c+d x))}{16 d} \]
-1/8*a^3/d/(a-a*cos(d*x+c))^2-1/2*a^2/d/(a-a*cos(d*x+c))-1/8*a^2/d/(a+a*co s(d*x+c))+11/16*a*ln(1-cos(d*x+c))/d-a*ln(cos(d*x+c))/d+5/16*a*ln(1+cos(d* x+c))/d
Time = 0.04 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.49 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {3 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {3 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \]
(-3*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Csc[ c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^4)/(4*d) - (3*a*Log[Cos[(c + d*x)/2]]) /(8*d) - (a*Log[Cos[c + d*x]])/d + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) + (a* Log[Sin[c + d*x]])/d + (3*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/ 2]^4)/(64*d)
Time = 0.39 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, 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.
| \(\displaystyle \int \csc ^5(c+d x) (a \sec (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\left (\csc ^5(c+d x) \sec (c+d x) (a (-\cos (c+d x))-a)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\left ((\cos (c+d x) a+a) \csc ^5(c+d x) \sec (c+d x)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^5(c+d x) \sec (c+d x) (a \cos (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {a-a \sin \left (c+d x-\frac {\pi }{2}\right )}{\sin \left (c+d x-\frac {\pi }{2}\right ) \cos \left (c+d x-\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^5 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int -\frac {\sec (c+d x)}{(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^5 \int \frac {\sec (c+d x)}{(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^6 \int \frac {\sec (c+d x)}{a (a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^6 \int \left (\frac {\sec (c+d x)}{a^6}+\frac {11}{16 a^5 (a-a \cos (c+d x))}-\frac {5}{16 a^5 (\cos (c+d x) a+a)}+\frac {1}{2 a^4 (a-a \cos (c+d x))^2}-\frac {1}{8 a^4 (\cos (c+d x) a+a)^2}+\frac {1}{4 a^3 (a-a \cos (c+d x))^3}\right )d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^6 \left (\frac {\log (a \cos (c+d x))}{a^5}-\frac {11 \log (a-a \cos (c+d x))}{16 a^5}-\frac {5 \log (a \cos (c+d x)+a)}{16 a^5}+\frac {1}{2 a^4 (a-a \cos (c+d x))}+\frac {1}{8 a^4 (a \cos (c+d x)+a)}+\frac {1}{8 a^3 (a-a \cos (c+d x))^2}\right )}{d}\) |
-((a^6*(1/(8*a^3*(a - a*Cos[c + d*x])^2) + 1/(2*a^4*(a - a*Cos[c + d*x])) + 1/(8*a^4*(a + a*Cos[c + d*x])) + Log[a*Cos[c + d*x]]/a^5 - (11*Log[a - a *Cos[c + d*x]])/(16*a^5) - (5*Log[a + a*Cos[c + d*x]])/(16*a^5)))/d)
3.1.8.3.1 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 = 0.82 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(83\) |
| default | \(\frac {a \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(83\) |
| parallelrisch | \(-\frac {a \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+10 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )\right )}{32 d}\) | \(85\) |
| norman | \(\frac {-\frac {a}{32 d}-\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {11 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(107\) |
| risch | \(\frac {a \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}+2 \,{\mathrm e}^{4 i \left (d x +c \right )}-18 \,{\mathrm e}^{3 i \left (d x +c \right )}+2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2}}+\frac {11 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(144\) |
1/d*(a*(-1/4/sin(d*x+c)^4-1/2/sin(d*x+c)^2+ln(tan(d*x+c)))+a*((-1/4*csc(d* x+c)^3-3/8*csc(d*x+c))*cot(d*x+c)+3/8*ln(-cot(d*x+c)+csc(d*x+c))))
Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.64 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {6 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - 16 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\cos \left (d x + c\right )\right ) + 5 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 11 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )}} \]
1/16*(6*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - 16*(a*cos(d*x + c)^3 - a*cos (d*x + c)^2 - a*cos(d*x + c) + a)*log(-cos(d*x + c)) + 5*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*log(1/2*cos(d*x + c) + 1/2) + 11 *(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*log(-1/2*cos(d *x + c) + 1/2) - 12*a)/(d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)
\[ \int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx=a \left (\int \csc ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.81 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {5 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - 16 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (3 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - 6 \, a\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 1}}{16 \, d} \]
1/16*(5*a*log(cos(d*x + c) + 1) + 11*a*log(cos(d*x + c) - 1) - 16*a*log(co s(d*x + c)) + 2*(3*a*cos(d*x + c)^2 + a*cos(d*x + c) - 6*a)/(cos(d*x + c)^ 3 - cos(d*x + c)^2 - cos(d*x + c) + 1))/d
Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.26 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {22 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {{\left (a - \frac {10 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {33 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{32 \, d} \]
1/32*(22*a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 32*a*log(ab s(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - (a - 10*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 33*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)*( cos(d*x + c) + 1)^2/(cos(d*x + c) - 1)^2 + 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/d
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx=\frac {11\,a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{16\,d}-\frac {a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d}+\frac {5\,a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{16\,d}+\frac {\frac {3\,a\,{\cos \left (c+d\,x\right )}^2}{8}+\frac {a\,\cos \left (c+d\,x\right )}{8}-\frac {3\,a}{4}}{d\,\left ({\cos \left (c+d\,x\right )}^3-\cos \left (c+d\,x\right )+{\sin \left (c+d\,x\right )}^2\right )} \]