Integrand size = 21, antiderivative size = 210 \[ \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx=-\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \sec ^2(c+d x)}{2 d}-\frac {11 a^3 \sec ^3(c+d x)}{3 d}-\frac {3 a^3 \sec ^4(c+d x)}{2 d}+\frac {14 a^3 \sec ^5(c+d x)}{5 d}+\frac {7 a^3 \sec ^6(c+d x)}{3 d}-\frac {6 a^3 \sec ^7(c+d x)}{7 d}-\frac {11 a^3 \sec ^8(c+d x)}{8 d}-\frac {a^3 \sec ^9(c+d x)}{9 d}+\frac {3 a^3 \sec ^{10}(c+d x)}{10 d}+\frac {a^3 \sec ^{11}(c+d x)}{11 d} \] Output:
-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-11/3*a^3*s ec(d*x+c)^3/d-3/2*a^3*sec(d*x+c)^4/d+14/5*a^3*sec(d*x+c)^5/d+7/3*a^3*sec(d *x+c)^6/d-6/7*a^3*sec(d*x+c)^7/d-11/8*a^3*sec(d*x+c)^8/d-1/9*a^3*sec(d*x+c )^9/d+3/10*a^3*sec(d*x+c)^10/d+1/11*a^3*sec(d*x+c)^11/d
Time = 0.69 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.02 \[ \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx=-\frac {a^3 (-1151740-1613260 \cos (2 (c+d x))+960960 \cos (3 (c+d x))-1131504 \cos (4 (c+d x))+314160 \cos (5 (c+d x))-432894 \cos (6 (c+d x))+145530 \cos (7 (c+d x))-106260 \cos (8 (c+d x))+6930 \cos (9 (c+d x))-20790 \cos (10 (c+d x))+1143450 \cos (3 (c+d x)) \log (\cos (c+d x))+571725 \cos (5 (c+d x)) \log (\cos (c+d x))+190575 \cos (7 (c+d x)) \log (\cos (c+d x))+38115 \cos (9 (c+d x)) \log (\cos (c+d x))+3465 \cos (11 (c+d x)) \log (\cos (c+d x))+462 \cos (c+d x) (2606+3465 \log (\cos (c+d x)))) \sec ^{11}(c+d x)}{3548160 d} \] Input:
Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^9,x]
Output:
-1/3548160*(a^3*(-1151740 - 1613260*Cos[2*(c + d*x)] + 960960*Cos[3*(c + d *x)] - 1131504*Cos[4*(c + d*x)] + 314160*Cos[5*(c + d*x)] - 432894*Cos[6*( c + d*x)] + 145530*Cos[7*(c + d*x)] - 106260*Cos[8*(c + d*x)] + 6930*Cos[9 *(c + d*x)] - 20790*Cos[10*(c + d*x)] + 1143450*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + 571725*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 190575*Cos[7*(c + d *x)]*Log[Cos[c + d*x]] + 38115*Cos[9*(c + d*x)]*Log[Cos[c + d*x]] + 3465*C os[11*(c + d*x)]*Log[Cos[c + d*x]] + 462*Cos[c + d*x]*(2606 + 3465*Log[Cos [c + d*x]]))*Sec[c + d*x]^11)/d
Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.69, 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.
| \(\displaystyle \int \tan ^9(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\cot \left (c+d x+\frac {\pi }{2}\right )^9 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^9 \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {\int a^{11} (1-\cos (c+d x))^4 (\cos (c+d x)+1)^7 \sec ^{12}(c+d x)d\cos (c+d x)}{a^8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \int (1-\cos (c+d x))^4 (\cos (c+d x)+1)^7 \sec ^{12}(c+d x)d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^3 \int \left (\sec ^{12}(c+d x)+3 \sec ^{11}(c+d x)-\sec ^{10}(c+d x)-11 \sec ^9(c+d x)-6 \sec ^8(c+d x)+14 \sec ^7(c+d x)+14 \sec ^6(c+d x)-6 \sec ^5(c+d x)-11 \sec ^4(c+d x)-\sec ^3(c+d x)+3 \sec ^2(c+d x)+\sec (c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \left (-\frac {1}{11} \sec ^{11}(c+d x)-\frac {3}{10} \sec ^{10}(c+d x)+\frac {1}{9} \sec ^9(c+d x)+\frac {11}{8} \sec ^8(c+d x)+\frac {6}{7} \sec ^7(c+d x)-\frac {7}{3} \sec ^6(c+d x)-\frac {14}{5} \sec ^5(c+d x)+\frac {3}{2} \sec ^4(c+d x)+\frac {11}{3} \sec ^3(c+d x)+\frac {1}{2} \sec ^2(c+d x)-3 \sec (c+d x)+\log (\cos (c+d x))\right )}{d}\) |
Input:
Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^9,x]
Output:
-((a^3*(Log[Cos[c + d*x]] - 3*Sec[c + d*x] + Sec[c + d*x]^2/2 + (11*Sec[c + d*x]^3)/3 + (3*Sec[c + d*x]^4)/2 - (14*Sec[c + d*x]^5)/5 - (7*Sec[c + d* x]^6)/3 + (6*Sec[c + d*x]^7)/7 + (11*Sec[c + d*x]^8)/8 + Sec[c + d*x]^9/9 - (3*Sec[c + d*x]^10)/10 - Sec[c + d*x]^11/11))/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 = 1.96 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.59
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right )^{11}}{11}+\frac {3 \sec \left (d x +c \right )^{10}}{10}-\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {11 \sec \left (d x +c \right )^{8}}{8}-\frac {6 \sec \left (d x +c \right )^{7}}{7}+\frac {7 \sec \left (d x +c \right )^{6}}{3}+\frac {14 \sec \left (d x +c \right )^{5}}{5}-\frac {3 \sec \left (d x +c \right )^{4}}{2}-\frac {11 \sec \left (d x +c \right )^{3}}{3}-\frac {\sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )\right )}{d}\) | \(124\) |
| default | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right )^{11}}{11}+\frac {3 \sec \left (d x +c \right )^{10}}{10}-\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {11 \sec \left (d x +c \right )^{8}}{8}-\frac {6 \sec \left (d x +c \right )^{7}}{7}+\frac {7 \sec \left (d x +c \right )^{6}}{3}+\frac {14 \sec \left (d x +c \right )^{5}}{5}-\frac {3 \sec \left (d x +c \right )^{4}}{2}-\frac {11 \sec \left (d x +c \right )^{3}}{3}-\frac {\sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )\right )}{d}\) | \(124\) |
| parts | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{8}}{8}-\frac {\tan \left (d x +c \right )^{6}}{6}+\frac {\tan \left (d x +c \right )^{4}}{4}-\frac {\tan \left (d x +c \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {a^{3} \left (\frac {\sec \left (d x +c \right )^{11}}{11}-\frac {4 \sec \left (d x +c \right )^{9}}{9}+\frac {6 \sec \left (d x +c \right )^{7}}{7}-\frac {4 \sec \left (d x +c \right )^{5}}{5}+\frac {\sec \left (d x +c \right )^{3}}{3}\right )}{d}+\frac {3 a^{3} \left (\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {4 \sec \left (d x +c \right )^{7}}{7}+\frac {6 \sec \left (d x +c \right )^{5}}{5}-\frac {4 \sec \left (d x +c \right )^{3}}{3}+\sec \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \tan \left (d x +c \right )^{10}}{10 d}\) | \(192\) |
| risch | \(i a^{3} x +\frac {2 i a^{3} c}{d}+\frac {2 a^{3} \left (10395 \,{\mathrm e}^{21 i \left (d x +c \right )}-3465 \,{\mathrm e}^{20 i \left (d x +c \right )}+53130 \,{\mathrm e}^{19 i \left (d x +c \right )}-72765 \,{\mathrm e}^{18 i \left (d x +c \right )}+216447 \,{\mathrm e}^{17 i \left (d x +c \right )}-157080 \,{\mathrm e}^{16 i \left (d x +c \right )}+565752 \,{\mathrm e}^{15 i \left (d x +c \right )}-480480 \,{\mathrm e}^{14 i \left (d x +c \right )}+806630 \,{\mathrm e}^{13 i \left (d x +c \right )}-601986 \,{\mathrm e}^{12 i \left (d x +c \right )}+1151740 \,{\mathrm e}^{11 i \left (d x +c \right )}-601986 \,{\mathrm e}^{10 i \left (d x +c \right )}+806630 \,{\mathrm e}^{9 i \left (d x +c \right )}-480480 \,{\mathrm e}^{8 i \left (d x +c \right )}+565752 \,{\mathrm e}^{7 i \left (d x +c \right )}-157080 \,{\mathrm e}^{6 i \left (d x +c \right )}+216447 \,{\mathrm e}^{5 i \left (d x +c \right )}-72765 \,{\mathrm e}^{4 i \left (d x +c \right )}+53130 \,{\mathrm e}^{3 i \left (d x +c \right )}-3465 \,{\mathrm e}^{2 i \left (d x +c \right )}+10395 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3465 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{11}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(292\) |
Input:
int((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x,method=_RETURNVERBOSE)
Output:
a^3/d*(1/11*sec(d*x+c)^11+3/10*sec(d*x+c)^10-1/9*sec(d*x+c)^9-11/8*sec(d*x +c)^8-6/7*sec(d*x+c)^7+7/3*sec(d*x+c)^6+14/5*sec(d*x+c)^5-3/2*sec(d*x+c)^4 -11/3*sec(d*x+c)^3-1/2*sec(d*x+c)^2+3*sec(d*x+c)+ln(sec(d*x+c)))
Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.80 \[ \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx=-\frac {27720 \, a^{3} \cos \left (d x + c\right )^{11} \log \left (-\cos \left (d x + c\right )\right ) - 83160 \, a^{3} \cos \left (d x + c\right )^{10} + 13860 \, a^{3} \cos \left (d x + c\right )^{9} + 101640 \, a^{3} \cos \left (d x + c\right )^{8} + 41580 \, a^{3} \cos \left (d x + c\right )^{7} - 77616 \, a^{3} \cos \left (d x + c\right )^{6} - 64680 \, a^{3} \cos \left (d x + c\right )^{5} + 23760 \, a^{3} \cos \left (d x + c\right )^{4} + 38115 \, a^{3} \cos \left (d x + c\right )^{3} + 3080 \, a^{3} \cos \left (d x + c\right )^{2} - 8316 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3}}{27720 \, d \cos \left (d x + c\right )^{11}} \] Input:
integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="fricas")
Output:
-1/27720*(27720*a^3*cos(d*x + c)^11*log(-cos(d*x + c)) - 83160*a^3*cos(d*x + c)^10 + 13860*a^3*cos(d*x + c)^9 + 101640*a^3*cos(d*x + c)^8 + 41580*a^ 3*cos(d*x + c)^7 - 77616*a^3*cos(d*x + c)^6 - 64680*a^3*cos(d*x + c)^5 + 2 3760*a^3*cos(d*x + c)^4 + 38115*a^3*cos(d*x + c)^3 + 3080*a^3*cos(d*x + c) ^2 - 8316*a^3*cos(d*x + c) - 2520*a^3)/(d*cos(d*x + c)^11)
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (190) = 380\).
Time = 3.75 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.09 \[ \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx=\begin {cases} \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{11 d} + \frac {3 a^{3} \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{3 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )}}{8 d} - \frac {8 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{99 d} - \frac {3 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac {8 a^{3} \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{21 d} - \frac {a^{3} \tan ^{6}{\left (c + d x \right )}}{6 d} + \frac {16 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{231 d} + \frac {3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {16 a^{3} \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {64 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{1155 d} - \frac {3 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac {64 a^{3} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{105 d} - \frac {a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {128 a^{3} \sec ^{3}{\left (c + d x \right )}}{3465 d} + \frac {3 a^{3} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {128 a^{3} \sec {\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\left (c \right )} + a\right )^{3} \tan ^{9}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate((a+a*sec(d*x+c))**3*tan(d*x+c)**9,x)
Output:
Piecewise((a**3*log(tan(c + d*x)**2 + 1)/(2*d) + a**3*tan(c + d*x)**8*sec( c + d*x)**3/(11*d) + 3*a**3*tan(c + d*x)**8*sec(c + d*x)**2/(10*d) + a**3* tan(c + d*x)**8*sec(c + d*x)/(3*d) + a**3*tan(c + d*x)**8/(8*d) - 8*a**3*t an(c + d*x)**6*sec(c + d*x)**3/(99*d) - 3*a**3*tan(c + d*x)**6*sec(c + d*x )**2/(10*d) - 8*a**3*tan(c + d*x)**6*sec(c + d*x)/(21*d) - a**3*tan(c + d* x)**6/(6*d) + 16*a**3*tan(c + d*x)**4*sec(c + d*x)**3/(231*d) + 3*a**3*tan (c + d*x)**4*sec(c + d*x)**2/(10*d) + 16*a**3*tan(c + d*x)**4*sec(c + d*x) /(35*d) + a**3*tan(c + d*x)**4/(4*d) - 64*a**3*tan(c + d*x)**2*sec(c + d*x )**3/(1155*d) - 3*a**3*tan(c + d*x)**2*sec(c + d*x)**2/(10*d) - 64*a**3*ta n(c + d*x)**2*sec(c + d*x)/(105*d) - a**3*tan(c + d*x)**2/(2*d) + 128*a**3 *sec(c + d*x)**3/(3465*d) + 3*a**3*sec(c + d*x)**2/(10*d) + 128*a**3*sec(c + d*x)/(105*d), Ne(d, 0)), (x*(a*sec(c) + a)**3*tan(c)**9, True))
Time = 0.04 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.77 \[ \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx=-\frac {27720 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {83160 \, a^{3} \cos \left (d x + c\right )^{10} - 13860 \, a^{3} \cos \left (d x + c\right )^{9} - 101640 \, a^{3} \cos \left (d x + c\right )^{8} - 41580 \, a^{3} \cos \left (d x + c\right )^{7} + 77616 \, a^{3} \cos \left (d x + c\right )^{6} + 64680 \, a^{3} \cos \left (d x + c\right )^{5} - 23760 \, a^{3} \cos \left (d x + c\right )^{4} - 38115 \, a^{3} \cos \left (d x + c\right )^{3} - 3080 \, a^{3} \cos \left (d x + c\right )^{2} + 8316 \, a^{3} \cos \left (d x + c\right ) + 2520 \, a^{3}}{\cos \left (d x + c\right )^{11}}}{27720 \, d} \] Input:
integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="maxima")
Output:
-1/27720*(27720*a^3*log(cos(d*x + c)) - (83160*a^3*cos(d*x + c)^10 - 13860 *a^3*cos(d*x + c)^9 - 101640*a^3*cos(d*x + c)^8 - 41580*a^3*cos(d*x + c)^7 + 77616*a^3*cos(d*x + c)^6 + 64680*a^3*cos(d*x + c)^5 - 23760*a^3*cos(d*x + c)^4 - 38115*a^3*cos(d*x + c)^3 - 3080*a^3*cos(d*x + c)^2 + 8316*a^3*co s(d*x + c) + 2520*a^3)/cos(d*x + c)^11)/d
Time = 0.51 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.78 \[ \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx=-\frac {27720 \, a^{3} \log \left ({\left | \cos \left (d x + c\right ) \right |}\right ) - \frac {83160 \, a^{3} \cos \left (d x + c\right )^{10} - 13860 \, a^{3} \cos \left (d x + c\right )^{9} - 101640 \, a^{3} \cos \left (d x + c\right )^{8} - 41580 \, a^{3} \cos \left (d x + c\right )^{7} + 77616 \, a^{3} \cos \left (d x + c\right )^{6} + 64680 \, a^{3} \cos \left (d x + c\right )^{5} - 23760 \, a^{3} \cos \left (d x + c\right )^{4} - 38115 \, a^{3} \cos \left (d x + c\right )^{3} - 3080 \, a^{3} \cos \left (d x + c\right )^{2} + 8316 \, a^{3} \cos \left (d x + c\right ) + 2520 \, a^{3}}{\cos \left (d x + c\right )^{11}}}{27720 \, d} \] Input:
integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="giac")
Output:
-1/27720*(27720*a^3*log(abs(cos(d*x + c))) - (83160*a^3*cos(d*x + c)^10 - 13860*a^3*cos(d*x + c)^9 - 101640*a^3*cos(d*x + c)^8 - 41580*a^3*cos(d*x + c)^7 + 77616*a^3*cos(d*x + c)^6 + 64680*a^3*cos(d*x + c)^5 - 23760*a^3*co s(d*x + c)^4 - 38115*a^3*cos(d*x + c)^3 - 3080*a^3*cos(d*x + c)^2 + 8316*a ^3*cos(d*x + c) + 2520*a^3)/cos(d*x + c)^11)/d
Time = 17.83 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.60 \[ \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx=\frac {2\,a^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}-22\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+\frac {332\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{3}-\frac {1012\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {10456\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{15}-\frac {5192\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}+\frac {8164\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}-\frac {3676\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {10090\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{63}-\frac {9334\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{315}+\frac {8704\,a^3}{3465}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}-11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+55\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-165\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+330\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-462\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+462\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-330\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+165\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-55\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \] Input:
int(tan(c + d*x)^9*(a + a/cos(c + d*x))^3,x)
Output:
(2*a^3*atanh(tan(c/2 + (d*x)/2)^2))/d - ((10090*a^3*tan(c/2 + (d*x)/2)^4)/ 63 - (9334*a^3*tan(c/2 + (d*x)/2)^2)/315 - (3676*a^3*tan(c/2 + (d*x)/2)^6) /7 + (8164*a^3*tan(c/2 + (d*x)/2)^8)/7 - (5192*a^3*tan(c/2 + (d*x)/2)^10)/ 5 + (10456*a^3*tan(c/2 + (d*x)/2)^12)/15 - (1012*a^3*tan(c/2 + (d*x)/2)^14 )/3 + (332*a^3*tan(c/2 + (d*x)/2)^16)/3 - 22*a^3*tan(c/2 + (d*x)/2)^18 + 2 *a^3*tan(c/2 + (d*x)/2)^20 + (8704*a^3)/3465)/(d*(11*tan(c/2 + (d*x)/2)^2 - 55*tan(c/2 + (d*x)/2)^4 + 165*tan(c/2 + (d*x)/2)^6 - 330*tan(c/2 + (d*x) /2)^8 + 462*tan(c/2 + (d*x)/2)^10 - 462*tan(c/2 + (d*x)/2)^12 + 330*tan(c/ 2 + (d*x)/2)^14 - 165*tan(c/2 + (d*x)/2)^16 + 55*tan(c/2 + (d*x)/2)^18 - 1 1*tan(c/2 + (d*x)/2)^20 + tan(c/2 + (d*x)/2)^22 - 1))
Time = 0.16 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.42 \[ \int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx=\frac {a^{3} \left (13860 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right )+2520 \sec \left (d x +c \right )^{3} \tan \left (d x +c \right )^{8}-2240 \sec \left (d x +c \right )^{3} \tan \left (d x +c \right )^{6}+1920 \sec \left (d x +c \right )^{3} \tan \left (d x +c \right )^{4}-1536 \sec \left (d x +c \right )^{3} \tan \left (d x +c \right )^{2}+1024 \sec \left (d x +c \right )^{3}+8316 \sec \left (d x +c \right )^{2} \tan \left (d x +c \right )^{8}-8316 \sec \left (d x +c \right )^{2} \tan \left (d x +c \right )^{6}+8316 \sec \left (d x +c \right )^{2} \tan \left (d x +c \right )^{4}-8316 \sec \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2}+8316 \sec \left (d x +c \right )^{2}+9240 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{8}-10560 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{6}+12672 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{4}-16896 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2}+33792 \sec \left (d x +c \right )+3465 \tan \left (d x +c \right )^{8}-4620 \tan \left (d x +c \right )^{6}+6930 \tan \left (d x +c \right )^{4}-13860 \tan \left (d x +c \right )^{2}\right )}{27720 d} \] Input:
int((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x)
Output:
(a**3*(13860*log(tan(c + d*x)**2 + 1) + 2520*sec(c + d*x)**3*tan(c + d*x)* *8 - 2240*sec(c + d*x)**3*tan(c + d*x)**6 + 1920*sec(c + d*x)**3*tan(c + d *x)**4 - 1536*sec(c + d*x)**3*tan(c + d*x)**2 + 1024*sec(c + d*x)**3 + 831 6*sec(c + d*x)**2*tan(c + d*x)**8 - 8316*sec(c + d*x)**2*tan(c + d*x)**6 + 8316*sec(c + d*x)**2*tan(c + d*x)**4 - 8316*sec(c + d*x)**2*tan(c + d*x)* *2 + 8316*sec(c + d*x)**2 + 9240*sec(c + d*x)*tan(c + d*x)**8 - 10560*sec( c + d*x)*tan(c + d*x)**6 + 12672*sec(c + d*x)*tan(c + d*x)**4 - 16896*sec( c + d*x)*tan(c + d*x)**2 + 33792*sec(c + d*x) + 3465*tan(c + d*x)**8 - 462 0*tan(c + d*x)**6 + 6930*tan(c + d*x)**4 - 13860*tan(c + d*x)**2))/(27720* d)