Integrand size = 19, antiderivative size = 84 \[ \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx=-a x-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{15 d}-\frac {\cot (c+d x) (15 a+8 b \sec (c+d x))}{15 d} \] Output:
-a*x-1/5*cot(d*x+c)^5*(a+b*sec(d*x+c))/d+1/15*cot(d*x+c)^3*(5*a+4*b*sec(d* x+c))/d-1/15*cot(d*x+c)*(15*a+8*b*sec(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {b \csc (c+d x)}{d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d}-\frac {a \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d} \] Input:
Integrate[Cot[c + d*x]^6*(a + b*Sec[c + d*x]),x]
Output:
-((b*Csc[c + d*x])/d) + (2*b*Csc[c + d*x]^3)/(3*d) - (b*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 = 0.42 (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.
\(\displaystyle \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle \frac {1}{5} \int -\cot ^4(c+d x) (5 a+4 b \sec (c+d x))dx-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{5} \int \cot ^4(c+d x) (5 a+4 b \sec (c+d x))dx-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{5} \int \frac {5 a+4 b \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^4}dx-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle \frac {1}{5} \left (\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{3 d}-\frac {1}{3} \int -\cot ^2(c+d x) (15 a+8 b \sec (c+d x))dx\right )-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \cot ^2(c+d x) (15 a+8 b \sec (c+d x))dx+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{3 d}\right )-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {15 a+8 b \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{3 d}\right )-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int -15 adx-\frac {\cot (c+d x) (15 a+8 b \sec (c+d x))}{d}\right )+\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{3 d}\right )-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{5} \left (\frac {\cot ^3(c+d x) (5 a+4 b \sec (c+d x))}{3 d}+\frac {1}{3} \left (-\frac {\cot (c+d x) (15 a+8 b \sec (c+d x))}{d}-15 a x\right )\right )-\frac {\cot ^5(c+d x) (a+b \sec (c+d x))}{5 d}\) |
Input:
Int[Cot[c + d*x]^6*(a + b*Sec[c + d*x]),x]
Output:
-1/5*(Cot[c + d*x]^5*(a + b*Sec[c + d*x]))/d + ((Cot[c + d*x]^3*(5*a + 4*b *Sec[c + d*x]))/(3*d) + (-15*a*x - (Cot[c + d*x]*(15*a + 8*b*Sec[c + d*x]) )/d)/3)/5
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]
Time = 0.53 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.54
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+b \left (-\frac {\cos \left (d x +c \right )^{6}}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{6}}{15 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{6}}{5 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}\right )}{d}\) | \(129\) |
default | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+b \left (-\frac {\cos \left (d x +c \right )^{6}}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{6}}{15 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{6}}{5 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}\right )}{d}\) | \(129\) |
risch | \(-a x -\frac {2 i \left (15 b \,{\mathrm e}^{9 i \left (d x +c \right )}+45 a \,{\mathrm e}^{8 i \left (d x +c \right )}-20 b \,{\mathrm e}^{7 i \left (d x +c \right )}-90 a \,{\mathrm e}^{6 i \left (d x +c \right )}+58 b \,{\mathrm e}^{5 i \left (d x +c \right )}+140 a \,{\mathrm e}^{4 i \left (d x +c \right )}-20 b \,{\mathrm e}^{3 i \left (d x +c \right )}-70 a \,{\mathrm e}^{2 i \left (d x +c \right )}+15 b \,{\mathrm e}^{i \left (d x +c \right )}+23 a \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}\) | \(137\) |
Input:
int(cot(d*x+c)^6*(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+b*(-1/5/sin(d *x+c)^5*cos(d*x+c)^6+1/15/sin(d*x+c)^3*cos(d*x+c)^6-1/5/sin(d*x+c)*cos(d*x +c)^6-1/5*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))
Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.55 \[ \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {23 \, a \cos \left (d x + c\right )^{5} + 15 \, b \cos \left (d x + c\right )^{4} - 35 \, a \cos \left (d x + c\right )^{3} - 20 \, b \cos \left (d x + c\right )^{2} + 15 \, a \cos \left (d x + c\right ) + 15 \, {\left (a d x \cos \left (d x + c\right )^{4} - 2 \, a d x \cos \left (d x + c\right )^{2} + a d x\right )} \sin \left (d x + c\right ) + 8 \, b}{15 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm="fricas")
Output:
-1/15*(23*a*cos(d*x + c)^5 + 15*b*cos(d*x + c)^4 - 35*a*cos(d*x + c)^3 - 2 0*b*cos(d*x + c)^2 + 15*a*cos(d*x + c) + 15*(a*d*x*cos(d*x + c)^4 - 2*a*d* x*cos(d*x + c)^2 + a*d*x)*sin(d*x + c) + 8*b)/((d*cos(d*x + c)^4 - 2*d*cos (d*x + c)^2 + d)*sin(d*x + c))
\[ \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{6}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**6*(a+b*sec(d*x+c)),x)
Output:
Integral((a + b*sec(c + d*x))*cot(c + d*x)**6, x)
Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {{\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a + \frac {{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} b}{\sin \left (d x + c\right )^{5}}}{15 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm="maxima")
Output:
-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)*b/sin(d*x + c)^5) /d
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (78) = 156\).
Time = 0.16 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.02 \[ \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 25 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, {\left (d x + c\right )} a + 330 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 150 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {330 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 150 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 25 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a + 3 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm="giac")
Output:
1/480*(3*a*tan(1/2*d*x + 1/2*c)^5 - 3*b*tan(1/2*d*x + 1/2*c)^5 - 35*a*tan( 1/2*d*x + 1/2*c)^3 + 25*b*tan(1/2*d*x + 1/2*c)^3 - 480*(d*x + c)*a + 330*a *tan(1/2*d*x + 1/2*c) - 150*b*tan(1/2*d*x + 1/2*c) - (330*a*tan(1/2*d*x + 1/2*c)^4 + 150*b*tan(1/2*d*x + 1/2*c)^4 - 35*a*tan(1/2*d*x + 1/2*c)^2 - 25 *b*tan(1/2*d*x + 1/2*c)^2 + 3*a + 3*b)/tan(1/2*d*x + 1/2*c)^5)/d
Time = 10.42 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.57 \[ \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a}{160}-\frac {b}{160}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\left (22\,a+10\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {7\,a}{3}-\frac {5\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{5}+\frac {b}{5}\right )}{32\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {7\,a}{96}-\frac {5\,b}{96}\right )}{d}-a\,x+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {11\,a}{16}-\frac {5\,b}{16}\right )}{d} \] Input:
int(cot(c + d*x)^6*(a + b/cos(c + d*x)),x)
Output:
(tan(c/2 + (d*x)/2)^5*(a/160 - b/160))/d - (cot(c/2 + (d*x)/2)^5*(a/5 + b/ 5 - tan(c/2 + (d*x)/2)^2*((7*a)/3 + (5*b)/3) + tan(c/2 + (d*x)/2)^4*(22*a + 10*b)))/(32*d) - (tan(c/2 + (d*x)/2)^3*((7*a)/96 - (5*b)/96))/d - a*x + (tan(c/2 + (d*x)/2)*((11*a)/16 - (5*b)/16))/d
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.13 \[ \int \cot ^6(c+d x) (a+b \sec (c+d x)) \, dx=\frac {-23 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a +11 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a -3 \cos \left (d x +c \right ) a -15 \sin \left (d x +c \right )^{5} a d x -15 \sin \left (d x +c \right )^{4} b +10 \sin \left (d x +c \right )^{2} b -3 b}{15 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^6*(a+b*sec(d*x+c)),x)
Output:
( - 23*cos(c + d*x)*sin(c + d*x)**4*a + 11*cos(c + d*x)*sin(c + d*x)**2*a - 3*cos(c + d*x)*a - 15*sin(c + d*x)**5*a*d*x - 15*sin(c + d*x)**4*b + 10* sin(c + d*x)**2*b - 3*b)/(15*sin(c + d*x)**5*d)