Integrand size = 21, antiderivative size = 160 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^5}{12 d (a-a \cos (c+d x))^3}-\frac {3 a^4}{8 d (a-a \cos (c+d x))^2}-\frac {23 a^3}{16 d (a-a \cos (c+d x))}+\frac {a^3}{16 d (a+a \cos (c+d x))}+\frac {9 a^2 \log (1-\cos (c+d x))}{4 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {a^2 \log (1+\cos (c+d x))}{4 d}+\frac {a^2 \sec (c+d x)}{d} \]
-1/12*a^5/d/(a-a*cos(d*x+c))^3-3/8*a^4/d/(a-a*cos(d*x+c))^2-23/16*a^3/d/(a -a*cos(d*x+c))+1/16*a^3/d/(a+a*cos(d*x+c))+9/4*a^2*ln(1-cos(d*x+c))/d-2*a^ 2*ln(cos(d*x+c))/d-1/4*a^2*ln(1+cos(d*x+c))/d+a^2*sec(d*x+c)/d
Time = 1.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.85 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (120 \csc ^2\left (\frac {1}{2} (c+d x)\right )+36 \csc ^4\left (\frac {1}{2} (c+d x)\right )+48 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 \log (\cos (c+d x))-9 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\csc ^6\left (\frac {1}{2} (c+d x)\right ) \left (16-3 \sec ^2\left (\frac {1}{2} (c+d x)\right ) (3+2 \sec (c+d x))\right )\right )}{384 d} \]
-1/384*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(120*Csc[(c + d*x)/2]^ 2 + 36*Csc[(c + d*x)/2]^4 + 48*(Log[Cos[(c + d*x)/2]] + 4*Log[Cos[c + d*x] ] - 9*Log[Sin[(c + d*x)/2]]) + Csc[(c + d*x)/2]^6*(16 - 3*Sec[(c + d*x)/2] ^2*(3 + 2*Sec[c + d*x]))))/d
Time = 0.45 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.93, 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.
| \(\displaystyle \int \csc ^7(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^7(c+d x) \sec ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}{\sin \left (c+d x-\frac {\pi }{2}\right )^2 \cos \left (c+d x-\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^7 \int \frac {\sec ^2(c+d x)}{(a-a \cos (c+d x))^4 (\cos (c+d x) a+a)^2}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^9 \int \frac {\sec ^2(c+d x)}{a^2 (a-a \cos (c+d x))^4 (\cos (c+d x) a+a)^2}d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^9 \int \left (\frac {\sec ^2(c+d x)}{a^8}+\frac {2 \sec (c+d x)}{a^8}+\frac {9}{4 a^7 (a-a \cos (c+d x))}+\frac {1}{4 a^7 (\cos (c+d x) a+a)}+\frac {23}{16 a^6 (a-a \cos (c+d x))^2}+\frac {1}{16 a^6 (\cos (c+d x) a+a)^2}+\frac {3}{4 a^5 (a-a \cos (c+d x))^3}+\frac {1}{4 a^4 (a-a \cos (c+d x))^4}\right )d(-a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^9 \left (\frac {\sec (c+d x)}{a^7}-\frac {2 \log (-a \cos (c+d x))}{a^7}+\frac {9 \log (a-a \cos (c+d x))}{4 a^7}-\frac {\log (a \cos (c+d x)+a)}{4 a^7}-\frac {23}{16 a^6 (a-a \cos (c+d x))}+\frac {1}{16 a^6 (a \cos (c+d x)+a)}-\frac {3}{8 a^5 (a-a \cos (c+d x))^2}-\frac {1}{12 a^4 (a-a \cos (c+d x))^3}\right )}{d}\) |
(a^9*(-1/12*1/(a^4*(a - a*Cos[c + d*x])^3) - 3/(8*a^5*(a - a*Cos[c + d*x]) ^2) - 23/(16*a^6*(a - a*Cos[c + d*x])) + 1/(16*a^6*(a + a*Cos[c + d*x])) - (2*Log[-(a*Cos[c + d*x])])/a^7 + (9*Log[a - a*Cos[c + d*x]])/(4*a^7) - Lo g[a + a*Cos[c + d*x]]/(4*a^7) + Sec[c + d*x]/a^7))/d
3.1.27.3.1 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 = 1.35 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.08
| method | result | size |
| norman | \(\frac {\frac {a^{2}}{96 d}+\frac {11 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{96 d}+\frac {13 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 d}+\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{32 d}-\frac {95 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {9 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(172\) |
| parallelrisch | \(\frac {a^{2} \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+11 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+432 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+78 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-432 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-285\right )}{96 d \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\) | \(189\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {7}{24 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {35}{48 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {35}{16 \cos \left (d x +c \right )}+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )+2 a^{2} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\) | \(195\) |
| default | \(\frac {a^{2} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {7}{24 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {35}{48 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {35}{16 \cos \left (d x +c \right )}+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )+2 a^{2} \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{2} \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\) | \(195\) |
| risch | \(\frac {a^{2} \left (15 \,{\mathrm e}^{9 i \left (d x +c \right )}-48 \,{\mathrm e}^{8 i \left (d x +c \right )}+32 \,{\mathrm e}^{7 i \left (d x +c \right )}+40 \,{\mathrm e}^{6 i \left (d x +c \right )}-62 \,{\mathrm e}^{5 i \left (d x +c \right )}+40 \,{\mathrm e}^{4 i \left (d x +c \right )}+32 \,{\mathrm e}^{3 i \left (d x +c \right )}-48 \,{\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(209\) |
(1/96/d*a^2+11/96*a^2/d*tan(1/2*d*x+1/2*c)^2+13/16*a^2/d*tan(1/2*d*x+1/2*c )^4+1/32*a^2/d*tan(1/2*d*x+1/2*c)^10-95/32*a^2/d*tan(1/2*d*x+1/2*c)^6)/tan (1/2*d*x+1/2*c)^6/(-1+tan(1/2*d*x+1/2*c)^2)+9/2/d*a^2*ln(tan(1/2*d*x+1/2*c ))-2*a^2/d*ln(tan(1/2*d*x+1/2*c)-1)-2*a^2/d*ln(tan(1/2*d*x+1/2*c)+1)
Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.81 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {30 \, a^{2} \cos \left (d x + c\right )^{4} - 48 \, a^{2} \cos \left (d x + c\right )^{3} - 14 \, a^{2} \cos \left (d x + c\right )^{2} + 46 \, a^{2} \cos \left (d x + c\right ) - 12 \, a^{2} - 24 \, {\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right )\right ) - 3 \, {\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 27 \, {\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{12 \, {\left (d \cos \left (d x + c\right )^{5} - 2 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right )\right )}} \]
1/12*(30*a^2*cos(d*x + c)^4 - 48*a^2*cos(d*x + c)^3 - 14*a^2*cos(d*x + c)^ 2 + 46*a^2*cos(d*x + c) - 12*a^2 - 24*(a^2*cos(d*x + c)^5 - 2*a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c)^2 - a^2*cos(d*x + c))*log(-cos(d*x + c)) - 3*( a^2*cos(d*x + c)^5 - 2*a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c)^2 - a^2*cos (d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 27*(a^2*cos(d*x + c)^5 - 2*a^2*co s(d*x + c)^4 + 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)^5 - 2*d*cos(d*x + c)^4 + 2*d*cos(d*x + c)^2 - d*cos(d*x + c))
Timed out. \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.89 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {3 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) - 27 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) + 24 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{4} - 24 \, a^{2} \cos \left (d x + c\right )^{3} - 7 \, a^{2} \cos \left (d x + c\right )^{2} + 23 \, a^{2} \cos \left (d x + c\right ) - 6 \, a^{2}\right )}}{\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )}}{12 \, d} \]
-1/12*(3*a^2*log(cos(d*x + c) + 1) - 27*a^2*log(cos(d*x + c) - 1) + 24*a^2 *log(cos(d*x + c)) - 2*(15*a^2*cos(d*x + c)^4 - 24*a^2*cos(d*x + c)^3 - 7* a^2*cos(d*x + c)^2 + 23*a^2*cos(d*x + c) - 6*a^2)/(cos(d*x + c)^5 - 2*cos( d*x + c)^4 + 2*cos(d*x + c)^2 - cos(d*x + c)))/d
Time = 0.37 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.49 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {216 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 192 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {3 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {{\left (a^{2} - \frac {12 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {90 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {396 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} + \frac {192 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{96 \, d} \]
1/96*(216*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 192*a^2* log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - 3*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + (a^2 - 12*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 90*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 396*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)*(cos(d*x + c) + 1)^3/(cos(d*x + c) - 1) ^3 + 192*(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 = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.92 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {9\,a^2\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{4\,d}-\frac {a^2\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{4\,d}-\frac {2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d}+\frac {-\frac {5\,a^2\,{\cos \left (c+d\,x\right )}^4}{2}+4\,a^2\,{\cos \left (c+d\,x\right )}^3+\frac {7\,a^2\,{\cos \left (c+d\,x\right )}^2}{6}-\frac {23\,a^2\,\cos \left (c+d\,x\right )}{6}+a^2}{d\,\left (-{\cos \left (c+d\,x\right )}^5+2\,{\cos \left (c+d\,x\right )}^4-2\,{\cos \left (c+d\,x\right )}^2+\cos \left (c+d\,x\right )\right )} \]