Integrand size = 21, antiderivative size = 111 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^5}{2 d (a-a \cos (c+d x))^2}-\frac {3 a^4}{d (a-a \cos (c+d x))}+\frac {6 a^3 \log (1-\cos (c+d x))}{d}-\frac {6 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
-1/2*a^5/d/(a-a*cos(d*x+c))^2-3*a^4/d/(a-a*cos(d*x+c))+6*a^3*ln(1-cos(d*x+ c))/d-6*a^3*ln(cos(d*x+c))/d+3*a^3*sec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d
Time = 0.72 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (12 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right )+48 \left (\log (\cos (c+d x))-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-24 \sec (c+d x)-4 \sec ^2(c+d x)\right )}{64 d} \]
-1/64*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(12*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 + 48*(Log[Cos[c + d*x]] - 2*Log[Sin[(c + d*x)/2]]) - 24*Sec[c + d*x] - 4*Sec[c + d*x]^2))/d
Time = 0.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 25, 27, 54, 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.
| \(\displaystyle \int \csc ^5(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^5(c+d x) \sec ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \csc ^5(c+d x) \sec ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^5(c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^5 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int -\frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^3}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^5 \int \frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^3}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^8 \int \frac {\sec ^3(c+d x)}{a^3 (a-a \cos (c+d x))^3}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {a^8 \int \left (\frac {\sec ^3(c+d x)}{a^6}+\frac {3 \sec ^2(c+d x)}{a^6}+\frac {6 \sec (c+d x)}{a^6}+\frac {6}{a^5 (a-a \cos (c+d x))}+\frac {3}{a^4 (a-a \cos (c+d x))^2}+\frac {1}{a^3 (a-a \cos (c+d x))^3}\right )d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^8 \left (-\frac {\sec ^2(c+d x)}{2 a^5}-\frac {3 \sec (c+d x)}{a^5}+\frac {6 \log (a \cos (c+d x))}{a^5}-\frac {6 \log (a-a \cos (c+d x))}{a^5}+\frac {3}{a^4 (a-a \cos (c+d x))}+\frac {1}{2 a^3 (a-a \cos (c+d x))^2}\right )}{d}\) |
-((a^8*(1/(2*a^3*(a - a*Cos[c + d*x])^2) + 3/(a^4*(a - a*Cos[c + d*x])) + (6*Log[a*Cos[c + d*x]])/a^5 - (6*Log[a - a*Cos[c + d*x]])/a^5 - (3*Sec[c + d*x])/a^5 - Sec[c + d*x]^2/(2*a^5)))/d)
3.1.45.3.1 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_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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 = 1.00 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.38
| method | result | size |
| norman | \(\frac {-\frac {a^{3}}{8 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 d}-\frac {23 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 d}+\frac {75 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {12 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {6 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(153\) |
| parallelrisch | \(\frac {12 \left (\frac {\left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {\left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+\left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {49 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\cos \left (d x +c \right )-\frac {34 \cos \left (2 d x +2 c \right )}{49}+\frac {11 \cos \left (3 d x +3 c \right )}{49}-\frac {86}{147}\right )}{128}\right ) a^{3}}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(153\) |
| risch | \(\frac {4 a^{3} \left (3 \,{\mathrm e}^{7 i \left (d x +c \right )}-9 \,{\mathrm e}^{6 i \left (d x +c \right )}+13 \,{\mathrm e}^{5 i \left (d x +c \right )}-16 \,{\mathrm e}^{4 i \left (d x +c \right )}+13 \,{\mathrm e}^{3 i \left (d x +c \right )}-9 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {12 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {6 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(154\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(218\) |
| default | \(\frac {a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {5}{8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {15}{8 \cos \left (d x +c \right )}+\frac {15 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}\) | \(218\) |
(-1/8/d*a^3-3/2*a^3/d*tan(1/2*d*x+1/2*c)^2-23/4*a^3/d*tan(1/2*d*x+1/2*c)^6 +75/8*a^3/d*tan(1/2*d*x+1/2*c)^4)/tan(1/2*d*x+1/2*c)^4/(-1+tan(1/2*d*x+1/2 *c)^2)^2+12/d*a^3*ln(tan(1/2*d*x+1/2*c))-6/d*a^3*ln(tan(1/2*d*x+1/2*c)-1)- 6/d*a^3*ln(tan(1/2*d*x+1/2*c)+1)
Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.59 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {12 \, a^{3} \cos \left (d x + c\right )^{3} - 18 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} \cos \left (d x + c\right ) + a^{3} - 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}} \]
1/2*(12*a^3*cos(d*x + c)^3 - 18*a^3*cos(d*x + c)^2 + 4*a^3*cos(d*x + c) + a^3 - 12*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)* log(-cos(d*x + c)) + 12*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^3 + a^3*c os(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^4 - 2*d*cos(d *x + c)^3 + d*cos(d*x + c)^2)
Timed out. \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {12 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 12 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {12 \, a^{3} \cos \left (d x + c\right )^{3} - 18 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}}{2 \, d} \]
1/2*(12*a^3*log(cos(d*x + c) - 1) - 12*a^3*log(cos(d*x + c)) + (12*a^3*cos (d*x + c)^3 - 18*a^3*cos(d*x + c)^2 + 4*a^3*cos(d*x + c) + a^3)/(cos(d*x + c)^4 - 2*cos(d*x + c)^3 + cos(d*x + c)^2))/d
Time = 0.39 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.68 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {48 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 48 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {a^{3} - \frac {12 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {75 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {46 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}^{2}}}{8 \, d} \]
1/8*(48*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 48*a^3*log (abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - (a^3 - 12*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 75*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 46*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + (cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)^2)/d
Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.86 \[ \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {6\,a^3\,{\cos \left (c+d\,x\right )}^3-9\,a^3\,{\cos \left (c+d\,x\right )}^2+2\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{d\,\left ({\cos \left (c+d\,x\right )}^4-2\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {12\,a^3\,\mathrm {atanh}\left (2\,\cos \left (c+d\,x\right )-1\right )}{d} \]