Integrand size = 21, antiderivative size = 137 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\log (\cos (c+d x))}{a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {\sec ^2(c+d x)}{2 a^3 d}-\frac {5 \sec ^3(c+d x)}{3 a^3 d}+\frac {5 \sec ^4(c+d x)}{4 a^3 d}+\frac {\sec ^5(c+d x)}{5 a^3 d}-\frac {\sec ^6(c+d x)}{2 a^3 d}+\frac {\sec ^7(c+d x)}{7 a^3 d} \] Output:
ln(cos(d*x+c))/a^3/d+3*sec(d*x+c)/a^3/d-1/2*sec(d*x+c)^2/a^3/d-5/3*sec(d*x +c)^3/a^3/d+5/4*sec(d*x+c)^4/a^3/d+1/5*sec(d*x+c)^5/a^3/d-1/2*sec(d*x+c)^6 /a^3/d+1/7*sec(d*x+c)^7/a^3/d
Time = 0.42 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.02 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {(3732+4522 \cos (2 (c+d x))+1050 \cos (3 (c+d x))+2380 \cos (4 (c+d x))-210 \cos (5 (c+d x))+630 \cos (6 (c+d x))+2205 \cos (3 (c+d x)) \log (\cos (c+d x))+735 \cos (5 (c+d x)) \log (\cos (c+d x))+105 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (c+d x) (8+35 \log (\cos (c+d x)))) \sec ^7(c+d x)}{6720 a^3 d} \] Input:
Integrate[Tan[c + d*x]^11/(a + a*Sec[c + d*x])^3,x]
Output:
((3732 + 4522*Cos[2*(c + d*x)] + 1050*Cos[3*(c + d*x)] + 2380*Cos[4*(c + d *x)] - 210*Cos[5*(c + d*x)] + 630*Cos[6*(c + d*x)] + 2205*Cos[3*(c + d*x)] *Log[Cos[c + d*x]] + 735*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[c + d*x]*(8 + 35*Log[Cos[c + d*x]]))* Sec[c + d*x]^7)/(6720*a^3*d)
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.72, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4367, 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.
| \(\displaystyle \int \frac {\tan ^{11}(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cot \left (c+d x+\frac {\pi }{2}\right )^{11}}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^{11}}{\left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {\int a^7 (1-\cos (c+d x))^5 (\cos (c+d x)+1)^2 \sec ^8(c+d x)d\cos (c+d x)}{a^{10} d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int (1-\cos (c+d x))^5 (\cos (c+d x)+1)^2 \sec ^8(c+d x)d\cos (c+d x)}{a^3 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (\sec ^8(c+d x)-3 \sec ^7(c+d x)+\sec ^6(c+d x)+5 \sec ^5(c+d x)-5 \sec ^4(c+d x)-\sec ^3(c+d x)+3 \sec ^2(c+d x)-\sec (c+d x)\right )d\cos (c+d x)}{a^3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{7} \sec ^7(c+d x)+\frac {1}{2} \sec ^6(c+d x)-\frac {1}{5} \sec ^5(c+d x)-\frac {5}{4} \sec ^4(c+d x)+\frac {5}{3} \sec ^3(c+d x)+\frac {1}{2} \sec ^2(c+d x)-3 \sec (c+d x)-\log (\cos (c+d x))}{a^3 d}\) |
Input:
Int[Tan[c + d*x]^11/(a + a*Sec[c + d*x])^3,x]
Output:
-((-Log[Cos[c + d*x]] - 3*Sec[c + d*x] + Sec[c + d*x]^2/2 + (5*Sec[c + d*x ]^3)/3 - (5*Sec[c + d*x]^4)/4 - Sec[c + d*x]^5/5 + Sec[c + d*x]^6/2 - Sec[ c + d*x]^7/7)/(a^3*d))
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[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[1/(a^(m - n - 1)*b^n*d) Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Time = 3.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.63
| method | result | size |
| derivativedivides | \(\frac {\frac {3}{\cos \left (d x +c \right )}-\frac {1}{2 \cos \left (d x +c \right )^{6}}+\ln \left (\cos \left (d x +c \right )\right )+\frac {1}{7 \cos \left (d x +c \right )^{7}}+\frac {1}{5 \cos \left (d x +c \right )^{5}}-\frac {5}{3 \cos \left (d x +c \right )^{3}}+\frac {5}{4 \cos \left (d x +c \right )^{4}}-\frac {1}{2 \cos \left (d x +c \right )^{2}}}{d \,a^{3}}\) | \(86\) |
| default | \(\frac {\frac {3}{\cos \left (d x +c \right )}-\frac {1}{2 \cos \left (d x +c \right )^{6}}+\ln \left (\cos \left (d x +c \right )\right )+\frac {1}{7 \cos \left (d x +c \right )^{7}}+\frac {1}{5 \cos \left (d x +c \right )^{5}}-\frac {5}{3 \cos \left (d x +c \right )^{3}}+\frac {5}{4 \cos \left (d x +c \right )^{4}}-\frac {1}{2 \cos \left (d x +c \right )^{2}}}{d \,a^{3}}\) | \(86\) |
| risch | \(-\frac {i x}{a^{3}}-\frac {2 i c}{d \,a^{3}}+\frac {6 \,{\mathrm e}^{13 i \left (d x +c \right )}-2 \,{\mathrm e}^{12 i \left (d x +c \right )}+\frac {68 \,{\mathrm e}^{11 i \left (d x +c \right )}}{3}+10 \,{\mathrm e}^{10 i \left (d x +c \right )}+\frac {646 \,{\mathrm e}^{9 i \left (d x +c \right )}}{15}+8 \,{\mathrm e}^{8 i \left (d x +c \right )}+\frac {2488 \,{\mathrm e}^{7 i \left (d x +c \right )}}{35}+8 \,{\mathrm e}^{6 i \left (d x +c \right )}+\frac {646 \,{\mathrm e}^{5 i \left (d x +c \right )}}{15}+10 \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {68 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}-2 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \,a^{3}}\) | \(203\) |
Input:
int(tan(d*x+c)^11/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(3/cos(d*x+c)-1/2/cos(d*x+c)^6+ln(cos(d*x+c))+1/7/cos(d*x+c)^7+1/5 /cos(d*x+c)^5-5/3/cos(d*x+c)^3+5/4/cos(d*x+c)^4-1/2/cos(d*x+c)^2)
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.69 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {420 \, \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) + 1260 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{5} - 700 \, \cos \left (d x + c\right )^{4} + 525 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) + 60}{420 \, a^{3} d \cos \left (d x + c\right )^{7}} \] Input:
integrate(tan(d*x+c)^11/(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/420*(420*cos(d*x + c)^7*log(-cos(d*x + c)) + 1260*cos(d*x + c)^6 - 210*c os(d*x + c)^5 - 700*cos(d*x + c)^4 + 525*cos(d*x + c)^3 + 84*cos(d*x + c)^ 2 - 210*cos(d*x + c) + 60)/(a^3*d*cos(d*x + c)^7)
\[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\tan ^{11}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(tan(d*x+c)**11/(a+a*sec(d*x+c))**3,x)
Output:
Integral(tan(c + d*x)**11/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.66 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {420 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} + \frac {1260 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{5} - 700 \, \cos \left (d x + c\right )^{4} + 525 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) + 60}{a^{3} \cos \left (d x + c\right )^{7}}}{420 \, d} \] Input:
integrate(tan(d*x+c)^11/(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/420*(420*log(cos(d*x + c))/a^3 + (1260*cos(d*x + c)^6 - 210*cos(d*x + c) ^5 - 700*cos(d*x + c)^4 + 525*cos(d*x + c)^3 + 84*cos(d*x + c)^2 - 210*cos (d*x + c) + 60)/(a^3*cos(d*x + c)^7))/d
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.64 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {1260 \, \cos \left (d x + c\right )^{6} - 210 \, \cos \left (d x + c\right )^{5} - 700 \, \cos \left (d x + c\right )^{4} + 525 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} - 210 \, \cos \left (d x + c\right ) + 60}{\cos \left (d x + c\right )^{7}} + 420 \, \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{420 \, a^{3} d} \] Input:
integrate(tan(d*x+c)^11/(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
1/420*((1260*cos(d*x + c)^6 - 210*cos(d*x + c)^5 - 700*cos(d*x + c)^4 + 52 5*cos(d*x + c)^3 + 84*cos(d*x + c)^2 - 210*cos(d*x + c) + 60)/cos(d*x + c) ^7 + 420*log(abs(cos(d*x + c))))/(a^3*d)
Time = 15.56 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.64 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^3\,d}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {224\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {282\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {322\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {352}{105}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^3\right )} \] Input:
int(tan(c + d*x)^11/(a + a/cos(c + d*x))^3,x)
Output:
- (2*atanh(tan(c/2 + (d*x)/2)^2))/(a^3*d) - ((282*tan(c/2 + (d*x)/2)^4)/5 - (322*tan(c/2 + (d*x)/2)^2)/15 - (224*tan(c/2 + (d*x)/2)^6)/3 + (128*tan( c/2 + (d*x)/2)^8)/3 + 14*tan(c/2 + (d*x)/2)^10 - 2*tan(c/2 + (d*x)/2)^12 + 352/105)/(d*(7*a^3*tan(c/2 + (d*x)/2)^2 - 21*a^3*tan(c/2 + (d*x)/2)^4 + 3 5*a^3*tan(c/2 + (d*x)/2)^6 - 35*a^3*tan(c/2 + (d*x)/2)^8 + 21*a^3*tan(c/2 + (d*x)/2)^10 - 7*a^3*tan(c/2 + (d*x)/2)^12 + a^3*tan(c/2 + (d*x)/2)^14 - a^3))
Time = 0.16 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.33 \[ \int \frac {\tan ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {-420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{6}+1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4}-1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{2}+420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{6}-1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4}+1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}-420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{6}-1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}+1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}-420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-809 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+2637 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-2322 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+704 \cos \left (d x +c \right )+1260 \sin \left (d x +c \right )^{6}-3080 \sin \left (d x +c \right )^{4}+2464 \sin \left (d x +c \right )^{2}-704}{420 \cos \left (d x +c \right ) a^{3} d \left (\sin \left (d x +c \right )^{6}-3 \sin \left (d x +c \right )^{4}+3 \sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(tan(d*x+c)^11/(a+a*sec(d*x+c))^3,x)
Output:
( - 420*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**6 + 1260*c os(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4 - 1260*cos(c + d* x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2 + 420*cos(c + d*x)*log(tan ((c + d*x)/2)**2 + 1) + 420*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6 - 1260*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 + 1 260*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2 - 420*cos(c + d *x)*log(tan((c + d*x)/2) - 1) + 420*cos(c + d*x)*log(tan((c + d*x)/2) + 1) *sin(c + d*x)**6 - 1260*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x )**4 + 1260*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 - 420*c os(c + d*x)*log(tan((c + d*x)/2) + 1) - 809*cos(c + d*x)*sin(c + d*x)**6 + 2637*cos(c + d*x)*sin(c + d*x)**4 - 2322*cos(c + d*x)*sin(c + d*x)**2 + 7 04*cos(c + d*x) + 1260*sin(c + d*x)**6 - 3080*sin(c + d*x)**4 + 2464*sin(c + d*x)**2 - 704)/(420*cos(c + d*x)*a**3*d*(sin(c + d*x)**6 - 3*sin(c + d* x)**4 + 3*sin(c + d*x)**2 - 1))