Integrand size = 19, antiderivative size = 84 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx=-a x-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 a \sec (c+d x))}{15 d}-\frac {\cot (c+d x) (15 a+8 a \sec (c+d x))}{15 d} \]
-a*x-1/5*cot(d*x+c)^5*(a+a*sec(d*x+c))/d+1/15*cot(d*x+c)^3*(5*a+4*a*sec(d* x+c))/d-1/15*cot(d*x+c)*(15*a+8*a*sec(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d} \]
-((a*Csc[c + d*x])/d) + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5 *d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2]) /(5*d)
Time = 0.44 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3042, 4370, 25, 3042, 4370, 25, 3042, 4370, 24}
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.
\(\displaystyle \int \cot ^6(c+d x) (a \sec (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a \csc \left (c+d x+\frac {\pi }{2}\right )+a}{\cot \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle \frac {1}{5} \int -\cot ^4(c+d x) (4 \sec (c+d x) a+5 a)dx-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{5} \int \cot ^4(c+d x) (4 \sec (c+d x) a+5 a)dx-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{5} \int \frac {4 \csc \left (c+d x+\frac {\pi }{2}\right ) a+5 a}{\cot \left (c+d x+\frac {\pi }{2}\right )^4}dx-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle \frac {1}{5} \left (\frac {\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{3 d}-\frac {1}{3} \int -\cot ^2(c+d x) (8 \sec (c+d x) a+15 a)dx\right )-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \cot ^2(c+d x) (8 \sec (c+d x) a+15 a)dx+\frac {\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{3 d}\right )-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {8 \csc \left (c+d x+\frac {\pi }{2}\right ) a+15 a}{\cot \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{3 d}\right )-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int -15 adx-\frac {\cot (c+d x) (8 a \sec (c+d x)+15 a)}{d}\right )+\frac {\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{3 d}\right )-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{5} \left (\frac {\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{3 d}+\frac {1}{3} \left (-\frac {\cot (c+d x) (8 a \sec (c+d x)+15 a)}{d}-15 a x\right )\right )-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}\) |
-1/5*(Cot[c + d*x]^5*(a + a*Sec[c + d*x]))/d + ((Cot[c + d*x]^3*(5*a + 4*a *Sec[c + d*x]))/(3*d) + (-15*a*x - (Cot[c + d*x]*(15*a + 8*a*Sec[c + d*x]) )/d)/3)/5
3.1.16.3.1 Defintions of rubi rules used
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-(e*Cot[c + d*x])^(m + 1))*((a + b*Csc[c + d*x])/( d*e*(m + 1))), x] - Simp[1/(e^2*(m + 1)) Int[(e*Cot[c + d*x])^(m + 2)*(a* (m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && L tQ[m, -1]
Result contains complex when optimal does not.
Time = 1.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.40
method | result | size |
risch | \(-a x -\frac {2 i a \left (15 \,{\mathrm e}^{7 i \left (d x +c \right )}+15 \,{\mathrm e}^{6 i \left (d x +c \right )}-65 \,{\mathrm e}^{5 i \left (d x +c \right )}+25 \,{\mathrm e}^{4 i \left (d x +c \right )}+73 \,{\mathrm e}^{3 i \left (d x +c \right )}-31 \,{\mathrm e}^{2 i \left (d x +c \right )}-31 \,{\mathrm e}^{i \left (d x +c \right )}+23\right )}{15 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(118\) |
derivativedivides | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{6}}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{6}}{15 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{6}}{5 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}\right )+a \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(129\) |
default | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{6}}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{6}}{15 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{6}}{5 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}\right )+a \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(129\) |
-a*x-2/15*I*a*(15*exp(7*I*(d*x+c))+15*exp(6*I*(d*x+c))-65*exp(5*I*(d*x+c)) +25*exp(4*I*(d*x+c))+73*exp(3*I*(d*x+c))-31*exp(2*I*(d*x+c))-31*exp(I*(d*x +c))+23)/d/(exp(I*(d*x+c))-1)^5/(exp(I*(d*x+c))+1)^3
Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.65 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {23 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{3} - 27 \, a \cos \left (d x + c\right )^{2} + 7 \, a \cos \left (d x + c\right ) + 15 \, {\left (a d x \cos \left (d x + c\right )^{3} - a d x \cos \left (d x + c\right )^{2} - a d x \cos \left (d x + c\right ) + a d x\right )} \sin \left (d x + c\right ) + 8 \, a}{15 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \]
-1/15*(23*a*cos(d*x + c)^4 - 8*a*cos(d*x + c)^3 - 27*a*cos(d*x + c)^2 + 7* a*cos(d*x + c) + 15*(a*d*x*cos(d*x + c)^3 - a*d*x*cos(d*x + c)^2 - a*d*x*c os(d*x + c) + a*d*x)*sin(d*x + c) + 8*a)/((d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)*sin(d*x + c))
\[ \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx=a \left (\int \cot ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{6}{\left (c + d x \right )}\, dx\right ) \]
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {{\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a + \frac {{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} a}{\sin \left (d x + c\right )^{5}}}{15 \, d} \]
-1/15*((15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a + (15*sin(d*x + c)^4 - 10*sin(d*x + c)^2 + 3)*a/sin(d*x + c)^5) /d
Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {5 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, {\left (d x + c\right )} a - 90 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3 \, {\left (80 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{240 \, d} \]
-1/240*(5*a*tan(1/2*d*x + 1/2*c)^3 + 240*(d*x + c)*a - 90*a*tan(1/2*d*x + 1/2*c) + 3*(80*a*tan(1/2*d*x + 1/2*c)^4 - 10*a*tan(1/2*d*x + 1/2*c)^2 + a) /tan(1/2*d*x + 1/2*c)^5)/d
Time = 14.44 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.86 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a\,\left (3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-90\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-30\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )\right )}{240\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
-(a*(3*cos(c/2 + (d*x)/2)^8 + 5*sin(c/2 + (d*x)/2)^8 - 90*cos(c/2 + (d*x)/ 2)^2*sin(c/2 + (d*x)/2)^6 + 240*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^4 - 30*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^2 + 240*cos(c/2 + (d*x)/2)^3* sin(c/2 + (d*x)/2)^5*(c + d*x)))/(240*d*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d* x)/2)^5)