Integrand size = 24, antiderivative size = 152 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \text {arctanh}(\sin (c+d x))}{d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}-\frac {2 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d} \]
a^8*arctanh(sin(d*x+c))/d+2/5*I*a^3*cos(d*x+c)^5*(a+I*a*tan(d*x+c))^5/d-2/ 7*I*a*cos(d*x+c)^7*(a+I*a*tan(d*x+c))^7/d-2/3*I*a^2*cos(d*x+c)^3*(a^2+I*a^ 2*tan(d*x+c))^3/d+2*I*cos(d*x+c)*(a^8+I*a^8*tan(d*x+c))/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(305\) vs. \(2(152)=304\).
Time = 3.17 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.01 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \left (-70 i \cos \left (\frac {1}{2} (c+d x)\right )+42 i \cos \left (\frac {3}{2} (c+d x)\right )+210 i \cos \left (\frac {5}{2} (c+d x)\right )-30 i \cos \left (\frac {7}{2} (c+d x)\right )-105 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-70 \sin \left (\frac {1}{2} (c+d x)\right )-42 \sin \left (\frac {3}{2} (c+d x)\right )+210 \sin \left (\frac {5}{2} (c+d x)\right )+30 \sin \left (\frac {7}{2} (c+d x)\right )+105 i \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {7}{2} (c+d x)\right )-105 i \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {7}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (7 c+23 d x)\right )+i \sin \left (\frac {1}{2} (7 c+23 d x)\right )\right )}{105 d (\cos (d x)+i \sin (d x))^8} \]
(a^8*((-70*I)*Cos[(c + d*x)/2] + (42*I)*Cos[(3*(c + d*x))/2] + (210*I)*Cos [(5*(c + d*x))/2] - (30*I)*Cos[(7*(c + d*x))/2] - 105*Cos[(7*(c + d*x))/2] *Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[(7*(c + d*x))/2]*Log[C os[(c + d*x)/2] + Sin[(c + d*x)/2]] - 70*Sin[(c + d*x)/2] - 42*Sin[(3*(c + d*x))/2] + 210*Sin[(5*(c + d*x))/2] + 30*Sin[(7*(c + d*x))/2] + (105*I)*L og[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2] - (105*I)*Log [Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2])*(Cos[(7*c + 23 *d*x)/2] + I*Sin[(7*c + 23*d*x)/2]))/(105*d*(Cos[d*x] + I*Sin[d*x])^8)
Time = 0.73 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 3977, 3042, 3977, 3042, 3977, 3042, 3977, 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.
| \(\displaystyle \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^8}{\sec (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle -a^2 \int \cos ^5(c+d x) (i \tan (c+d x) a+a)^6dx-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^2 \int \frac {(i \tan (c+d x) a+a)^6}{\sec (c+d x)^5}dx-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle -a^2 \left (-a^2 \int \cos ^3(c+d x) (i \tan (c+d x) a+a)^4dx-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}\right )-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^2 \left (-a^2 \int \frac {(i \tan (c+d x) a+a)^4}{\sec (c+d x)^3}dx-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}\right )-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle -a^2 \left (-a^2 \left (-a^2 \int \cos (c+d x) (i \tan (c+d x) a+a)^2dx-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\right )-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}\right )-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^2 \left (-a^2 \left (-a^2 \int \frac {(i \tan (c+d x) a+a)^2}{\sec (c+d x)}dx-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\right )-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}\right )-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle -a^2 \left (-a^2 \left (-a^2 \left (a^2 (-\int \sec (c+d x)dx)-\frac {2 i \cos (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{d}\right )-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\right )-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}\right )-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^2 \left (-a^2 \left (-a^2 \left (a^2 \left (-\int \csc \left (c+d x+\frac {\pi }{2}\right )dx\right )-\frac {2 i \cos (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{d}\right )-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\right )-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}\right )-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -a^2 \left (-a^2 \left (-a^2 \left (-\frac {a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 i \cos (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{d}\right )-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\right )-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}\right )-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}\) |
(((-2*I)/7)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^7)/d - a^2*((((-2*I)/5 )*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/d - a^2*((((-2*I)/3)*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/d - a^2*(-((a^2*ArcTanh[Sin[c + d*x]])/ d) - ((2*I)*Cos[c + d*x]*(a^2 + I*a^2*Tan[c + d*x]))/d)))
3.1.94.3.1 Defintions of rubi rules used
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_), x_Symbol] :> Simp[2*b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^( n - 1)/(f*m)), x] - Simp[b^2*((m + 2*n - 2)/(d^2*m)) Int[(d*Sec[e + f*x]) ^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILtQ[m, 0] & & LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (138 ) = 276\).
Time = 1.56 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.53
\[-\frac {29 a^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{7 d}-\frac {a^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{5 d}-\frac {a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}+\frac {139 a^{8} \sin \left (d x +c \right )}{105 d}+\frac {a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {128 i a^{8} \cos \left (d x +c \right )}{35 d}-\frac {8 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{d}-\frac {10 a^{8} \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{d}-\frac {232 a^{8} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35 d}+\frac {122 a^{8} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{105 d}-\frac {64 i a^{8} \left (\cos ^{3}\left (d x +c \right )\right )}{15 d}+\frac {48 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}+\frac {29 a^{8} \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7 d}-\frac {32 i a^{8} \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5 d}+\frac {8 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{7 d}+\frac {64 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35 d}-\frac {8 i a^{8} \left (\cos ^{7}\left (d x +c \right )\right )}{7 d}+\frac {8 i a^{8} \left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {16 i a^{8} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d}\]
-29/7*a^8*sin(d*x+c)^7/d-1/5*a^8*sin(d*x+c)^5/d-1/3*a^8*sin(d*x+c)^3/d+139 /105*a^8*sin(d*x+c)/d+1/d*a^8*ln(sec(d*x+c)+tan(d*x+c))+128/35*I/d*a^8*cos (d*x+c)-8*I/d*a^8*sin(d*x+c)^4*cos(d*x+c)^3-10/d*a^8*sin(d*x+c)^3*cos(d*x+ c)^4-232/35/d*a^8*cos(d*x+c)^4*sin(d*x+c)+122/105/d*a^8*cos(d*x+c)^2*sin(d *x+c)-64/15*I/d*a^8*cos(d*x+c)^3+48/35*I/d*a^8*cos(d*x+c)*sin(d*x+c)^4+29/ 7/d*a^8*cos(d*x+c)^6*sin(d*x+c)-32/5*I/d*a^8*cos(d*x+c)^3*sin(d*x+c)^2+8/7 *I/d*a^8*cos(d*x+c)*sin(d*x+c)^6+64/35*I/d*a^8*cos(d*x+c)*sin(d*x+c)^2-8/7 *I/d*a^8*cos(d*x+c)^7+8*I/d*a^8*cos(d*x+c)^5*sin(d*x+c)^2+16/5*I/d*a^8*cos (d*x+c)^5
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.63 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-30 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} + 42 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 70 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, a^{8} e^{\left (i \, d x + i \, c\right )} + 105 \, a^{8} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, a^{8} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{105 \, d} \]
1/105*(-30*I*a^8*e^(7*I*d*x + 7*I*c) + 42*I*a^8*e^(5*I*d*x + 5*I*c) - 70*I *a^8*e^(3*I*d*x + 3*I*c) + 210*I*a^8*e^(I*d*x + I*c) + 105*a^8*log(e^(I*d* x + I*c) + I) - 105*a^8*log(e^(I*d*x + I*c) - I))/d
Time = 0.48 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.23 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^{8} \left (- \log {\left (e^{i d x} - i e^{- i c} \right )} + \log {\left (e^{i d x} + i e^{- i c} \right )}\right )}{d} + \begin {cases} \frac {- 30 i a^{8} d^{3} e^{7 i c} e^{7 i d x} + 42 i a^{8} d^{3} e^{5 i c} e^{5 i d x} - 70 i a^{8} d^{3} e^{3 i c} e^{3 i d x} + 210 i a^{8} d^{3} e^{i c} e^{i d x}}{105 d^{4}} & \text {for}\: d^{4} \neq 0 \\x \left (2 a^{8} e^{7 i c} - 2 a^{8} e^{5 i c} + 2 a^{8} e^{3 i c} - 2 a^{8} e^{i c}\right ) & \text {otherwise} \end {cases} \]
a**8*(-log(exp(I*d*x) - I*exp(-I*c)) + log(exp(I*d*x) + I*exp(-I*c)))/d + Piecewise(((-30*I*a**8*d**3*exp(7*I*c)*exp(7*I*d*x) + 42*I*a**8*d**3*exp(5 *I*c)*exp(5*I*d*x) - 70*I*a**8*d**3*exp(3*I*c)*exp(3*I*d*x) + 210*I*a**8*d **3*exp(I*c)*exp(I*d*x))/(105*d**4), Ne(d**4, 0)), (x*(2*a**8*exp(7*I*c) - 2*a**8*exp(5*I*c) + 2*a**8*exp(3*I*c) - 2*a**8*exp(I*c)), True))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (130) = 260\).
Time = 0.27 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.03 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {240 i \, a^{8} \cos \left (d x + c\right )^{7} + 840 \, a^{8} \sin \left (d x + c\right )^{7} + 112 i \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 336 i \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 48 i \, {\left (5 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )\right )} a^{8} + {\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{8} + 56 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + 420 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 6 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{8}}{210 \, d} \]
-1/210*(240*I*a^8*cos(d*x + c)^7 + 840*a^8*sin(d*x + c)^7 + 112*I*(15*cos( d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^8 + 336*I*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^8 + 48*I*(5*cos(d*x + c)^7 - 21*cos(d*x + c) ^5 + 35*cos(d*x + c)^3 - 35*cos(d*x + c))*a^8 + (30*sin(d*x + c)^7 + 42*si n(d*x + c)^5 + 70*sin(d*x + c)^3 - 105*log(sin(d*x + c) + 1) + 105*log(sin (d*x + c) - 1) + 210*sin(d*x + c))*a^8 + 56*(15*sin(d*x + c)^7 - 42*sin(d* x + c)^5 + 35*sin(d*x + c)^3)*a^8 + 420*(5*sin(d*x + c)^7 - 7*sin(d*x + c) ^5)*a^8 + 6*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35 *sin(d*x + c))*a^8)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2863 vs. \(2 (130) = 260\).
Time = 1.54 (sec) , antiderivative size = 2863, normalized size of antiderivative = 18.84 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]
1/55050240*(1635552135*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1 ) + 22897729890*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 148 835244285*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 595340977 140*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1637187687135*a^ 8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3274375374270*a^8*e^(1 8*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 4911563061405*a^8*e^(16*I*d* x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 4911563061405*a^8*e^(12*I*d*x - 2* I*c)*log(I*e^(I*d*x + I*c) + 1) + 3274375374270*a^8*e^(10*I*d*x - 4*I*c)*l og(I*e^(I*d*x + I*c) + 1) + 1637187687135*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^ (I*d*x + I*c) + 1) + 595340977140*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 148835244285*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 22897729890*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 56 13214927320*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 1635552135*a^8*e ^(-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1690450650*a^8*e^(28*I*d*x + 14*I* c)*log(I*e^(I*d*x + I*c) - 1) + 23666309100*a^8*e^(26*I*d*x + 12*I*c)*log( I*e^(I*d*x + I*c) - 1) + 153831009150*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I *d*x + I*c) - 1) + 615324036600*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1692141100650*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 3384282201300*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5076423301950*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 507...
Time = 8.25 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.36 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {2\,a^8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,16{}\mathrm {i}-\frac {80\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,224{}\mathrm {i}}{3}+\frac {224\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,304{}\mathrm {i}}{15}-\frac {304\,a^8}{105}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,7{}\mathrm {i}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,35{}\mathrm {i}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,21{}\mathrm {i}+7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
(2*a^8*atanh(tan(c/2 + (d*x)/2)))/d + ((224*a^8*tan(c/2 + (d*x)/2)^2)/5 - (a^8*tan(c/2 + (d*x)/2)^3*224i)/3 - (80*a^8*tan(c/2 + (d*x)/2)^4)/3 + a^8* tan(c/2 + (d*x)/2)^5*16i - (304*a^8)/105 + (a^8*tan(c/2 + (d*x)/2)*304i)/1 5)/(d*(7*tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2*21i - 35*tan(c/2 + (d*x )/2)^3 + tan(c/2 + (d*x)/2)^4*35i + 21*tan(c/2 + (d*x)/2)^5 - tan(c/2 + (d *x)/2)^6*7i - tan(c/2 + (d*x)/2)^7 + 1i))