Integrand size = 21, antiderivative size = 115 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^4}{4 d (a-a \cos (c+d x))^2}-\frac {5 a^3}{4 d (a-a \cos (c+d x))}+\frac {17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {a^2 \log (1+\cos (c+d x))}{8 d}+\frac {a^2 \sec (c+d x)}{d} \]
-1/4*a^4/d/(a-a*cos(d*x+c))^2-5/4*a^3/d/(a-a*cos(d*x+c))+17/8*a^2*ln(1-cos (d*x+c))/d-2*a^2*ln(cos(d*x+c))/d-1/8*a^2*ln(1+cos(d*x+c))/d+a^2*sec(d*x+c )/d
Time = 1.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (10 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right )+4 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+8 \log (\cos (c+d x))-17 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 \sec (c+d x)\right )\right )}{64 d} \]
-1/64*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(10*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 + 4*(Log[Cos[(c + d*x)/2]] + 8*Log[Cos[c + d*x]] - 17 *Log[Sin[(c + d*x)/2]] - 4*Sec[c + d*x])))/d
Time = 0.43 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96, 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.
| \(\displaystyle \int \csc ^5(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^5(c+d x) \sec ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}{\sin \left (c+d x-\frac {\pi }{2}\right )^2 \cos \left (c+d x-\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int \frac {\sec ^2(c+d x)}{(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^7 \int \frac {\sec ^2(c+d x)}{a^2 (a-a \cos (c+d x))^3 (\cos (c+d x) a+a)}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^7 \int \left (\frac {\sec ^2(c+d x)}{a^6}+\frac {2 \sec (c+d x)}{a^6}+\frac {17}{8 a^5 (a-a \cos (c+d x))}+\frac {1}{8 a^5 (\cos (c+d x) a+a)}+\frac {5}{4 a^4 (a-a \cos (c+d x))^2}+\frac {1}{2 a^3 (a-a \cos (c+d x))^3}\right )d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^7 \left (\frac {\sec (c+d x)}{a^5}-\frac {2 \log (-a \cos (c+d x))}{a^5}+\frac {17 \log (a-a \cos (c+d x))}{8 a^5}-\frac {\log (a \cos (c+d x)+a)}{8 a^5}-\frac {5}{4 a^4 (a-a \cos (c+d x))}-\frac {1}{4 a^3 (a-a \cos (c+d x))^2}\right )}{d}\) |
(a^7*(-1/4*1/(a^3*(a - a*Cos[c + d*x])^2) - 5/(4*a^4*(a - a*Cos[c + d*x])) - (2*Log[-(a*Cos[c + d*x])])/a^5 + (17*Log[a - a*Cos[c + d*x]])/(8*a^5) - Log[a + a*Cos[c + d*x]]/(8*a^5) + Sec[c + d*x]/a^5))/d
3.1.26.3.1 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 = 0.97 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03
| method | result | size |
| parallelrisch | \(\frac {17 \left (-\frac {8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )}{17}-\frac {8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )}{17}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (d x +c \right )-\frac {45 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\cos \left (d x +c \right )-\frac {3 \cos \left (2 d x +2 c \right )}{10}-\frac {59}{90}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{136}\right ) a^{2}}{4 d \cos \left (d x +c \right )}\) | \(118\) |
| norman | \(\frac {\frac {a^{2}}{16 d}+\frac {11 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 d}-\frac {11 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {17 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(134\) |
| risch | \(\frac {a^{2} \left (9 \,{\mathrm e}^{5 i \left (d x +c \right )}-28 \,{\mathrm e}^{4 i \left (d x +c \right )}+34 \,{\mathrm e}^{3 i \left (d x +c \right )}-28 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}+\frac {17 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(152\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(157\) |
| default | \(\frac {a^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(157\) |
17/4*(-8/17*ln(tan(1/2*d*x+1/2*c)-1)*cos(d*x+c)-8/17*ln(tan(1/2*d*x+1/2*c) +1)*cos(d*x+c)+ln(tan(1/2*d*x+1/2*c))*cos(d*x+c)-45/136*csc(1/2*d*x+1/2*c) ^2*(cos(d*x+c)-3/10*cos(2*d*x+2*c)-59/90)*cot(1/2*d*x+1/2*c)^2)*a^2/d/cos( d*x+c)
Time = 0.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.82 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {18 \, a^{2} \cos \left (d x + c\right )^{2} - 28 \, a^{2} \cos \left (d x + c\right ) + 8 \, a^{2} - 16 \, {\left (a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right )\right ) - {\left (a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 17 \, {\left (a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{8 \, {\left (d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}} \]
1/8*(18*a^2*cos(d*x + c)^2 - 28*a^2*cos(d*x + c) + 8*a^2 - 16*(a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(-cos(d*x + c)) - (a ^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c))*log(1/2*cos(d *x + c) + 1/2) + 17*(a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d *x + c))*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^3 - 2*d*cos(d*x + c )^2 + d*cos(d*x + c))
\[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=a^{2} \left (\int 2 \csc ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]
a**2*(Integral(2*csc(c + d*x)**5*sec(c + d*x), x) + Integral(csc(c + d*x)* *5*sec(c + d*x)**2, x) + Integral(csc(c + d*x)**5, x))
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) - 17 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) + 16 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (9 \, a^{2} \cos \left (d x + c\right )^{2} - 14 \, a^{2} \cos \left (d x + c\right ) + 4 \, a^{2}\right )}}{\cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )}}{8 \, d} \]
-1/8*(a^2*log(cos(d*x + c) + 1) - 17*a^2*log(cos(d*x + c) - 1) + 16*a^2*lo g(cos(d*x + c)) - 2*(9*a^2*cos(d*x + c)^2 - 14*a^2*cos(d*x + c) + 4*a^2)/( cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c)))/d
Time = 0.35 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.66 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {34 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {{\left (a^{2} - \frac {12 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {51 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} + \frac {32 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{16 \, d} \]
1/16*(34*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 32*a^2*lo g(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - (a^2 - 12*a^2*(cos(d* x + c) - 1)/(cos(d*x + c) + 1) + 51*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)^2/(cos(d*x + c) - 1)^2 + 32*(2*a^2 + a^2*(cos( d*x + c) - 1)/(cos(d*x + c) + 1))/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/d
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {17\,a^2\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{8\,d}-\frac {a^2\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{8\,d}+\frac {\frac {9\,a^2\,{\cos \left (c+d\,x\right )}^2}{4}-\frac {7\,a^2\,\cos \left (c+d\,x\right )}{2}+a^2}{d\,\left ({\cos \left (c+d\,x\right )}^3-2\,{\cos \left (c+d\,x\right )}^2+\cos \left (c+d\,x\right )\right )}-\frac {2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]