Integrand size = 24, antiderivative size = 124 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {7 \text {arctanh}(\sin (c+d x))}{16 a^2 d}+\frac {7 \sec (c+d x) \tan (c+d x)}{16 a^2 d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{24 a^2 d}+\frac {7 \sec ^5(c+d x) \tan (c+d x)}{30 a^2 d}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )} \] Output:
7/16*arctanh(sin(d*x+c))/a^2/d+7/16*sec(d*x+c)*tan(d*x+c)/a^2/d+7/24*sec(d *x+c)^3*tan(d*x+c)/a^2/d+7/30*sec(d*x+c)^5*tan(d*x+c)/a^2/d-2/5*I*sec(d*x+ c)^7/d/(a^2+I*a^2*tan(d*x+c))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(294\) vs. \(2(124)=248\).
Time = 2.23 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.37 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\sec ^6(c+d x) \left (3072 i \cos (c+d x)+5 \left (210 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+21 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+315 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+126 \cos (4 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-210 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-21 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \sin (c+d x)-238 \sin (3 (c+d x))-42 \sin (5 (c+d x))\right )\right )}{7680 a^2 d} \] Input:
Integrate[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^2,x]
Output:
-1/7680*(Sec[c + d*x]^6*((3072*I)*Cos[c + d*x] + 5*(210*Log[Cos[(c + d*x)/ 2] - Sin[(c + d*x)/2]] + 21*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 315*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2 ]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 126*Cos[4*(c + d*x)]*(Log [Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d* x)/2]]) - 210*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 21*Cos[6*(c + d*x )]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*Sin[c + d*x] - 238*Sin[3* (c + d*x)] - 42*Sin[5*(c + d*x)])))/(a^2*d)
Time = 0.61 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 3981, 3042, 4255, 3042, 4255, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)^9}{(a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3981 |
| \(\displaystyle \frac {7 \int \sec ^7(c+d x)dx}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \int \csc \left (c+d x+\frac {\pi }{2}\right )^7dx}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {7 \left (\frac {5}{6} \int \sec ^5(c+d x)dx+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5}{6} \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {7 \left (\frac {5}{6} \left (\frac {3}{4} \int \sec ^3(c+d x)dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5}{6} \left (\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {7 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {7 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )}{5 a^2}-\frac {2 i \sec ^7(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\) |
Input:
Int[Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^2,x]
Output:
(((-2*I)/5)*Sec[c + d*x]^7)/(d*(a^2 + I*a^2*Tan[c + d*x])) + (7*((Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (5*((Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3* (ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/4))/6)) /(5*a^2)
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_), x_Symbol] :> Simp[2*d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + 2*n))), x] - Simp[d^2*((m - 2)/(b^2*(m + 2*n))) Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[ {a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (IntegersQ[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.83 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(-\frac {i \left (105 \,{\mathrm e}^{11 i \left (d x +c \right )}+595 \,{\mathrm e}^{9 i \left (d x +c \right )}+1386 \,{\mathrm e}^{7 i \left (d x +c \right )}+1686 \,{\mathrm e}^{5 i \left (d x +c \right )}-595 \,{\mathrm e}^{3 i \left (d x +c \right )}-105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{120 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 a^{2} d}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 a^{2} d}\) | \(133\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {1}{4}+\frac {i}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2 \left (\frac {9}{32}-\frac {3 i}{8}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 \left (-\frac {1}{12}-\frac {3 i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (-\frac {1}{4}-\frac {i}{5}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2 \left (-\frac {9}{32}+\frac {5 i}{8}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16}+\frac {2 \left (-\frac {1}{4}+\frac {i}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {2 \left (\frac {9}{32}+\frac {5 i}{8}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (\frac {9}{32}+\frac {3 i}{8}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {2 \left (-\frac {1}{12}+\frac {3 i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (-\frac {1}{4}+\frac {i}{5}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16}}{a^{2} d}\) | \(238\) |
| default | \(\frac {\frac {2 \left (\frac {1}{4}+\frac {i}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2 \left (\frac {9}{32}-\frac {3 i}{8}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 \left (-\frac {1}{12}-\frac {3 i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (-\frac {1}{4}-\frac {i}{5}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2 \left (-\frac {9}{32}+\frac {5 i}{8}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16}+\frac {2 \left (-\frac {1}{4}+\frac {i}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {2 \left (\frac {9}{32}+\frac {5 i}{8}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (\frac {9}{32}+\frac {3 i}{8}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {2 \left (-\frac {1}{12}+\frac {3 i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (-\frac {1}{4}+\frac {i}{5}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16}}{a^{2} d}\) | \(238\) |
Input:
int(sec(d*x+c)^9/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/120*I/d/a^2/(exp(2*I*(d*x+c))+1)^6*(105*exp(11*I*(d*x+c))+595*exp(9*I*( d*x+c))+1386*exp(7*I*(d*x+c))+1686*exp(5*I*(d*x+c))-595*exp(3*I*(d*x+c))-1 05*exp(I*(d*x+c)))-7/16/a^2/d*ln(exp(I*(d*x+c))-I)+7/16/a^2/d*ln(exp(I*(d* x+c))+I)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (110) = 220\).
Time = 0.11 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.63 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {105 \, {\left (e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, {\left (e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 210 i \, e^{\left (11 i \, d x + 11 i \, c\right )} - 1190 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 2772 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 3372 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 1190 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, e^{\left (i \, d x + i \, c\right )}}{240 \, {\left (a^{2} d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{2} d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \] Input:
integrate(sec(d*x+c)^9/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
1/240*(105*(e^(12*I*d*x + 12*I*c) + 6*e^(10*I*d*x + 10*I*c) + 15*e^(8*I*d* x + 8*I*c) + 20*e^(6*I*d*x + 6*I*c) + 15*e^(4*I*d*x + 4*I*c) + 6*e^(2*I*d* x + 2*I*c) + 1)*log(e^(I*d*x + I*c) + I) - 105*(e^(12*I*d*x + 12*I*c) + 6* e^(10*I*d*x + 10*I*c) + 15*e^(8*I*d*x + 8*I*c) + 20*e^(6*I*d*x + 6*I*c) + 15*e^(4*I*d*x + 4*I*c) + 6*e^(2*I*d*x + 2*I*c) + 1)*log(e^(I*d*x + I*c) - I) - 210*I*e^(11*I*d*x + 11*I*c) - 1190*I*e^(9*I*d*x + 9*I*c) - 2772*I*e^( 7*I*d*x + 7*I*c) - 3372*I*e^(5*I*d*x + 5*I*c) + 1190*I*e^(3*I*d*x + 3*I*c) + 210*I*e^(I*d*x + I*c))/(a^2*d*e^(12*I*d*x + 12*I*c) + 6*a^2*d*e^(10*I*d *x + 10*I*c) + 15*a^2*d*e^(8*I*d*x + 8*I*c) + 20*a^2*d*e^(6*I*d*x + 6*I*c) + 15*a^2*d*e^(4*I*d*x + 4*I*c) + 6*a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)
\[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {\sec ^{9}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \] Input:
integrate(sec(d*x+c)**9/(a+I*a*tan(d*x+c))**2,x)
Output:
-Integral(sec(c + d*x)**9/(tan(c + d*x)**2 - 2*I*tan(c + d*x) - 1), x)/a** 2
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (110) = 220\).
Time = 0.05 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.40 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {\frac {2 \, {\left (\frac {135 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {96 i \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {445 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {960 i \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {330 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {960 i \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {330 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {480 i \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {445 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {480 i \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {135 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 96 i\right )}}{a^{2} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {105 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} - \frac {105 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}}{240 \, d} \] Input:
integrate(sec(d*x+c)^9/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
1/240*(2*(135*sin(d*x + c)/(cos(d*x + c) + 1) + 96*I*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 - 445*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 960*I*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 330*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 960 *I*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 330*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 480*I*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 445*sin(d*x + c)^9/(c os(d*x + c) + 1)^9 + 480*I*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 135*sin (d*x + c)^11/(cos(d*x + c) + 1)^11 - 96*I)/(a^2 - 6*a^2*sin(d*x + c)^2/(co s(d*x + c) + 1)^2 + 15*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 20*a^2*si n(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 6*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^2*sin(d*x + c)^12/( cos(d*x + c) + 1)^12) + 105*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^2 - 105*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^2)/d
Time = 0.25 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.64 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {\frac {105 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2}} - \frac {105 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{2}} + \frac {2 \, {\left (135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 480 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 445 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 480 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 960 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 960 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 445 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6} a^{2}}}{240 \, d} \] Input:
integrate(sec(d*x+c)^9/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
1/240*(105*log(tan(1/2*d*x + 1/2*c) + 1)/a^2 - 105*log(tan(1/2*d*x + 1/2*c ) - 1)/a^2 + 2*(135*tan(1/2*d*x + 1/2*c)^11 + 480*I*tan(1/2*d*x + 1/2*c)^1 0 - 445*tan(1/2*d*x + 1/2*c)^9 - 480*I*tan(1/2*d*x + 1/2*c)^8 - 330*tan(1/ 2*d*x + 1/2*c)^7 + 960*I*tan(1/2*d*x + 1/2*c)^6 - 330*tan(1/2*d*x + 1/2*c) ^5 - 960*I*tan(1/2*d*x + 1/2*c)^4 - 445*tan(1/2*d*x + 1/2*c)^3 + 96*I*tan( 1/2*d*x + 1/2*c)^2 + 135*tan(1/2*d*x + 1/2*c) - 96*I)/((tan(1/2*d*x + 1/2* c)^2 - 1)^6*a^2))/d
Time = 3.38 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.54 \[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a^2\,d}-\frac {-\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,4{}\mathrm {i}+\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,4{}\mathrm {i}+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,8{}\mathrm {i}+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,8{}\mathrm {i}+\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,4{}\mathrm {i}}{5}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {4}{5}{}\mathrm {i}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^6} \] Input:
int(1/(cos(c + d*x)^9*(a + a*tan(c + d*x)*1i)^2),x)
Output:
(7*atanh(tan(c/2 + (d*x)/2)))/(8*a^2*d) - ((89*tan(c/2 + (d*x)/2)^3)/24 - (tan(c/2 + (d*x)/2)^2*4i)/5 - (9*tan(c/2 + (d*x)/2))/8 + tan(c/2 + (d*x)/2 )^4*8i + (11*tan(c/2 + (d*x)/2)^5)/4 - tan(c/2 + (d*x)/2)^6*8i + (11*tan(c /2 + (d*x)/2)^7)/4 + tan(c/2 + (d*x)/2)^8*4i + (89*tan(c/2 + (d*x)/2)^9)/2 4 - tan(c/2 + (d*x)/2)^10*4i - (9*tan(c/2 + (d*x)/2)^11)/8 + 4i/5)/(a^2*d* (tan(c/2 + (d*x)/2)^2 - 1)^6)
\[ \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\int \frac {\sec \left (d x +c \right )^{9}}{\tan \left (d x +c \right )^{2}-2 \tan \left (d x +c \right ) i -1}d x}{a^{2}} \] Input:
int(sec(d*x+c)^9/(a+I*a*tan(d*x+c))^2,x)
Output:
( - int(sec(c + d*x)**9/(tan(c + d*x)**2 - 2*tan(c + d*x)*i - 1),x))/a**2