Integrand size = 21, antiderivative size = 135 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{a d}-\frac {\sec (c+d x)}{a d}-\frac {3 \sec ^2(c+d x)}{2 a d}+\frac {\sec ^3(c+d x)}{a d}+\frac {3 \sec ^4(c+d x)}{4 a d}-\frac {3 \sec ^5(c+d x)}{5 a d}-\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^7(c+d x)}{7 a d} \]
-ln(cos(d*x+c))/a/d-sec(d*x+c)/a/d-3/2*sec(d*x+c)^2/a/d+sec(d*x+c)^3/a/d+3 /4*sec(d*x+c)^4/a/d-3/5*sec(d*x+c)^5/a/d-1/6*sec(d*x+c)^6/a/d+1/7*sec(d*x+ c)^7/a/d
Time = 0.41 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {(35 \cos (c+d x) (104+105 \log (\cos (c+d x)))+3 (212+602 \cos (2 (c+d x))+140 \cos (4 (c+d x))+210 \cos (5 (c+d x))+70 \cos (6 (c+d x))+245 \cos (5 (c+d x)) \log (\cos (c+d x))+35 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (3 (c+d x)) (6+7 \log (\cos (c+d x))))) \sec ^7(c+d x)}{6720 a d} \]
-1/6720*((35*Cos[c + d*x]*(104 + 105*Log[Cos[c + d*x]]) + 3*(212 + 602*Cos [2*(c + d*x)] + 140*Cos[4*(c + d*x)] + 210*Cos[5*(c + d*x)] + 70*Cos[6*(c + d*x)] + 245*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 35*Cos[7*(c + d*x)]*Log [Cos[c + d*x]] + 105*Cos[3*(c + d*x)]*(6 + 7*Log[Cos[c + d*x]])))*Sec[c + d*x]^7)/(a*d)
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68, 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.
\(\displaystyle \int \frac {\tan ^9(c+d x)}{a \sec (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cot \left (c+d x+\frac {\pi }{2}\right )^9}{a \csc \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^9}{\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a}dx\) |
\(\Big \downarrow \) 4367 |
\(\displaystyle -\frac {\int a^7 (1-\cos (c+d x))^4 (\cos (c+d x)+1)^3 \sec ^8(c+d x)d\cos (c+d x)}{a^8 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int (1-\cos (c+d x))^4 (\cos (c+d x)+1)^3 \sec ^8(c+d x)d\cos (c+d x)}{a d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {\int \left (\sec ^8(c+d x)-\sec ^7(c+d x)-3 \sec ^6(c+d x)+3 \sec ^5(c+d x)+3 \sec ^4(c+d x)-3 \sec ^3(c+d x)-\sec ^2(c+d x)+\sec (c+d x)\right )d\cos (c+d x)}{a d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{7} \sec ^7(c+d x)+\frac {1}{6} \sec ^6(c+d x)+\frac {3}{5} \sec ^5(c+d x)-\frac {3}{4} \sec ^4(c+d x)-\sec ^3(c+d x)+\frac {3}{2} \sec ^2(c+d x)+\sec (c+d x)+\log (\cos (c+d x))}{a d}\) |
-((Log[Cos[c + d*x]] + Sec[c + d*x] + (3*Sec[c + d*x]^2)/2 - Sec[c + d*x]^ 3 - (3*Sec[c + d*x]^4)/4 + (3*Sec[c + d*x]^5)/5 + Sec[c + d*x]^6/6 - Sec[c + d*x]^7/7)/(a*d))
3.1.56.3.1 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[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 = 1.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(\frac {-\frac {3}{2 \cos \left (d x +c \right )^{2}}-\ln \left (\cos \left (d x +c \right )\right )+\frac {1}{\cos \left (d x +c \right )^{3}}+\frac {3}{4 \cos \left (d x +c \right )^{4}}-\frac {3}{5 \cos \left (d x +c \right )^{5}}-\frac {1}{6 \cos \left (d x +c \right )^{6}}+\frac {1}{7 \cos \left (d x +c \right )^{7}}-\frac {1}{\cos \left (d x +c \right )}}{d a}\) | \(86\) |
default | \(\frac {-\frac {3}{2 \cos \left (d x +c \right )^{2}}-\ln \left (\cos \left (d x +c \right )\right )+\frac {1}{\cos \left (d x +c \right )^{3}}+\frac {3}{4 \cos \left (d x +c \right )^{4}}-\frac {3}{5 \cos \left (d x +c \right )^{5}}-\frac {1}{6 \cos \left (d x +c \right )^{6}}+\frac {1}{7 \cos \left (d x +c \right )^{7}}-\frac {1}{\cos \left (d x +c \right )}}{d a}\) | \(86\) |
risch | \(\frac {i x}{a}+\frac {2 i c}{d a}-\frac {2 \left (105 \,{\mathrm e}^{13 i \left (d x +c \right )}+315 \,{\mathrm e}^{12 i \left (d x +c \right )}+210 \,{\mathrm e}^{11 i \left (d x +c \right )}+945 \,{\mathrm e}^{10 i \left (d x +c \right )}+903 \,{\mathrm e}^{9 i \left (d x +c \right )}+1820 \,{\mathrm e}^{8 i \left (d x +c \right )}+636 \,{\mathrm e}^{7 i \left (d x +c \right )}+1820 \,{\mathrm e}^{6 i \left (d x +c \right )}+903 \,{\mathrm e}^{5 i \left (d x +c \right )}+945 \,{\mathrm e}^{4 i \left (d x +c \right )}+210 \,{\mathrm e}^{3 i \left (d x +c \right )}+315 \,{\mathrm e}^{2 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{105 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d a}\) | \(204\) |
1/d/a*(-3/2/cos(d*x+c)^2-ln(cos(d*x+c))+1/cos(d*x+c)^3+3/4/cos(d*x+c)^4-3/ 5/cos(d*x+c)^5-1/6/cos(d*x+c)^6+1/7/cos(d*x+c)^7-1/cos(d*x+c))
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.70 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {420 \, \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) + 420 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{5} - 420 \, \cos \left (d x + c\right )^{4} - 315 \, \cos \left (d x + c\right )^{3} + 252 \, \cos \left (d x + c\right )^{2} + 70 \, \cos \left (d x + c\right ) - 60}{420 \, a d \cos \left (d x + c\right )^{7}} \]
-1/420*(420*cos(d*x + c)^7*log(-cos(d*x + c)) + 420*cos(d*x + c)^6 + 630*c os(d*x + c)^5 - 420*cos(d*x + c)^4 - 315*cos(d*x + c)^3 + 252*cos(d*x + c) ^2 + 70*cos(d*x + c) - 60)/(a*d*cos(d*x + c)^7)
\[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\tan ^{9}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.67 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {420 \, \log \left (\cos \left (d x + c\right )\right )}{a} + \frac {420 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{5} - 420 \, \cos \left (d x + c\right )^{4} - 315 \, \cos \left (d x + c\right )^{3} + 252 \, \cos \left (d x + c\right )^{2} + 70 \, \cos \left (d x + c\right ) - 60}{a \cos \left (d x + c\right )^{7}}}{420 \, d} \]
-1/420*(420*log(cos(d*x + c))/a + (420*cos(d*x + c)^6 + 630*cos(d*x + c)^5 - 420*cos(d*x + c)^4 - 315*cos(d*x + c)^3 + 252*cos(d*x + c)^2 + 70*cos(d *x + c) - 60)/(a*cos(d*x + c)^7))/d
Time = 5.98 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.81 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {420 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a} + \frac {\frac {5775 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {20685 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {42595 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {28749 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1089 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 705}{a {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{7}}}{420 \, d} \]
1/420*(420*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a - 420*lo g(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/a + (5775*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 20685*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 42595*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 56035*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 28749*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1) ^5 + 8463*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1089*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 705)/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1 ) + 1)^7))/d
Time = 18.11 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.54 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {32}{35}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-21\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-35\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \]
(2*atanh(tan(c/2 + (d*x)/2)^2))/(a*d) - ((26*tan(c/2 + (d*x)/2)^4)/5 - (22 *tan(c/2 + (d*x)/2)^2)/5 + (32*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 + 32/35)/ (d*(a - 7*a*tan(c/2 + (d*x)/2)^2 + 21*a*tan(c/2 + (d*x)/2)^4 - 35*a*tan(c/ 2 + (d*x)/2)^6 + 35*a*tan(c/2 + (d*x)/2)^8 - 21*a*tan(c/2 + (d*x)/2)^10 + 7*a*tan(c/2 + (d*x)/2)^12 - a*tan(c/2 + (d*x)/2)^14))