Integrand size = 24, antiderivative size = 55 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i (a+i a \tan (c+d x))^{10}}{5 a^2 d}+\frac {i (a+i a \tan (c+d x))^{11}}{11 a^3 d} \]
Time = 0.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.62 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 (-i+\tan (c+d x))^{10} (6 i+5 \tan (c+d x))}{55 d} \]
Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3968, 49, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^4 (a+i a \tan (c+d x))^8dx\) |
| \(\Big \downarrow \) 3968 |
| \(\displaystyle -\frac {i \int (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^9d(i a \tan (c+d x))}{a^3 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {i \int \left (2 a (i \tan (c+d x) a+a)^9-(i \tan (c+d x) a+a)^{10}\right )d(i a \tan (c+d x))}{a^3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i \left (\frac {1}{5} a (a+i a \tan (c+d x))^{10}-\frac {1}{11} (a+i a \tan (c+d x))^{11}\right )}{a^3 d}\) |
3.1.79.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(a^(m - 2)*b*f) Subst[Int[(a - x)^(m/2 - 1)*(a + x )^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (47 ) = 94\).
Time = 286.48 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.25
| method | result | size |
| risch | \(\frac {1024 i a^{8} \left (55 \,{\mathrm e}^{18 i \left (d x +c \right )}+165 \,{\mathrm e}^{16 i \left (d x +c \right )}+330 \,{\mathrm e}^{14 i \left (d x +c \right )}+462 \,{\mathrm e}^{12 i \left (d x +c \right )}+462 \,{\mathrm e}^{10 i \left (d x +c \right )}+330 \,{\mathrm e}^{8 i \left (d x +c \right )}+165 \,{\mathrm e}^{6 i \left (d x +c \right )}+55 \,{\mathrm e}^{4 i \left (d x +c \right )}+11 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{55 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{11}}\) | \(124\) |
| derivativedivides | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}\right )+\frac {2 i a^{8}}{\cos \left (d x +c \right )^{4}}+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )-28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{8}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{8}}\right )-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(339\) |
| default | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}\right )+\frac {2 i a^{8}}{\cos \left (d x +c \right )^{4}}+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )-28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{8}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{8}}\right )-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(339\) |
1024/55*I*a^8*(55*exp(18*I*(d*x+c))+165*exp(16*I*(d*x+c))+330*exp(14*I*(d* x+c))+462*exp(12*I*(d*x+c))+462*exp(10*I*(d*x+c))+330*exp(8*I*(d*x+c))+165 *exp(6*I*(d*x+c))+55*exp(4*I*(d*x+c))+11*exp(2*I*(d*x+c))+1)/d/(exp(2*I*(d *x+c))+1)^11
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (43) = 86\).
Time = 0.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.89 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {1024 \, {\left (-55 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 165 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 330 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 462 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 462 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 330 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 165 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 55 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 11 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )}}{55 \, {\left (d e^{\left (22 i \, d x + 22 i \, c\right )} + 11 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 55 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 165 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 330 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 462 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 462 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 330 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 165 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 55 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 11 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-1024/55*(-55*I*a^8*e^(18*I*d*x + 18*I*c) - 165*I*a^8*e^(16*I*d*x + 16*I*c ) - 330*I*a^8*e^(14*I*d*x + 14*I*c) - 462*I*a^8*e^(12*I*d*x + 12*I*c) - 46 2*I*a^8*e^(10*I*d*x + 10*I*c) - 330*I*a^8*e^(8*I*d*x + 8*I*c) - 165*I*a^8* e^(6*I*d*x + 6*I*c) - 55*I*a^8*e^(4*I*d*x + 4*I*c) - 11*I*a^8*e^(2*I*d*x + 2*I*c) - I*a^8)/(d*e^(22*I*d*x + 22*I*c) + 11*d*e^(20*I*d*x + 20*I*c) + 5 5*d*e^(18*I*d*x + 18*I*c) + 165*d*e^(16*I*d*x + 16*I*c) + 330*d*e^(14*I*d* x + 14*I*c) + 462*d*e^(12*I*d*x + 12*I*c) + 462*d*e^(10*I*d*x + 10*I*c) + 330*d*e^(8*I*d*x + 8*I*c) + 165*d*e^(6*I*d*x + 6*I*c) + 55*d*e^(4*I*d*x + 4*I*c) + 11*d*e^(2*I*d*x + 2*I*c) + d)
\[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=a^{8} \left (\int \left (- 28 \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 28 \tan ^{6}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 8 i \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
a**8*(Integral(-28*tan(c + d*x)**2*sec(c + d*x)**4, x) + Integral(70*tan(c + d*x)**4*sec(c + d*x)**4, x) + Integral(-28*tan(c + d*x)**6*sec(c + d*x) **4, x) + Integral(tan(c + d*x)**8*sec(c + d*x)**4, x) + Integral(8*I*tan( c + d*x)*sec(c + d*x)**4, x) + Integral(-56*I*tan(c + d*x)**3*sec(c + d*x) **4, x) + Integral(56*I*tan(c + d*x)**5*sec(c + d*x)**4, x) + Integral(-8* I*tan(c + d*x)**7*sec(c + d*x)**4, x) + Integral(sec(c + d*x)**4, x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (43) = 86\).
Time = 0.35 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.44 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {5 \, a^{8} \tan \left (d x + c\right )^{11} - 44 i \, a^{8} \tan \left (d x + c\right )^{10} - 165 \, a^{8} \tan \left (d x + c\right )^{9} + 330 i \, a^{8} \tan \left (d x + c\right )^{8} + 330 \, a^{8} \tan \left (d x + c\right )^{7} + 462 \, a^{8} \tan \left (d x + c\right )^{5} - 660 i \, a^{8} \tan \left (d x + c\right )^{4} - 495 \, a^{8} \tan \left (d x + c\right )^{3} + 220 i \, a^{8} \tan \left (d x + c\right )^{2} + 55 \, a^{8} \tan \left (d x + c\right )}{55 \, d} \]
1/55*(5*a^8*tan(d*x + c)^11 - 44*I*a^8*tan(d*x + c)^10 - 165*a^8*tan(d*x + c)^9 + 330*I*a^8*tan(d*x + c)^8 + 330*a^8*tan(d*x + c)^7 + 462*a^8*tan(d* x + c)^5 - 660*I*a^8*tan(d*x + c)^4 - 495*a^8*tan(d*x + c)^3 + 220*I*a^8*t an(d*x + c)^2 + 55*a^8*tan(d*x + c))/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (43) = 86\).
Time = 1.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.44 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {5 \, a^{8} \tan \left (d x + c\right )^{11} - 44 i \, a^{8} \tan \left (d x + c\right )^{10} - 165 \, a^{8} \tan \left (d x + c\right )^{9} + 330 i \, a^{8} \tan \left (d x + c\right )^{8} + 330 \, a^{8} \tan \left (d x + c\right )^{7} + 462 \, a^{8} \tan \left (d x + c\right )^{5} - 660 i \, a^{8} \tan \left (d x + c\right )^{4} - 495 \, a^{8} \tan \left (d x + c\right )^{3} + 220 i \, a^{8} \tan \left (d x + c\right )^{2} + 55 \, a^{8} \tan \left (d x + c\right )}{55 \, d} \]
1/55*(5*a^8*tan(d*x + c)^11 - 44*I*a^8*tan(d*x + c)^10 - 165*a^8*tan(d*x + c)^9 + 330*I*a^8*tan(d*x + c)^8 + 330*a^8*tan(d*x + c)^7 + 462*a^8*tan(d* x + c)^5 - 660*I*a^8*tan(d*x + c)^4 - 495*a^8*tan(d*x + c)^3 + 220*I*a^8*t an(d*x + c)^2 + 55*a^8*tan(d*x + c))/d
Time = 4.73 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.95 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8\,\left (\frac {\sin \left (9\,c+9\,d\,x\right )}{10}+\frac {\sin \left (11\,c+11\,d\,x\right )}{110}+\frac {\cos \left (c+d\,x\right )\,63{}\mathrm {i}}{1280}+\frac {\cos \left (3\,c+3\,d\,x\right )\,9{}\mathrm {i}}{256}+\frac {\cos \left (5\,c+5\,d\,x\right )\,9{}\mathrm {i}}{512}+\frac {\cos \left (7\,c+7\,d\,x\right )\,3{}\mathrm {i}}{512}-\frac {\cos \left (9\,c+9\,d\,x\right )\,253{}\mathrm {i}}{2560}-\frac {\cos \left (11\,c+11\,d\,x\right )\,23{}\mathrm {i}}{2560}\right )}{d\,{\cos \left (c+d\,x\right )}^{11}} \]