Integrand size = 24, antiderivative size = 81 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {4 i}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac {i}{d \left (a^2+i a^2 \tan (c+d x)\right )^4} \] Output:
4/5*I/a^3/d/(a+I*a*tan(d*x+c))^5+1/3*I/a^5/d/(a+I*a*tan(d*x+c))^3-I/d/(a^2 +I*a^2*tan(d*x+c))^4
Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2-5 i \tan (c+d x)-5 \tan ^2(c+d x)}{15 a^8 d (-i+\tan (c+d x))^5} \] Input:
Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^8,x]
Output:
(2 - (5*I)*Tan[c + d*x] - 5*Tan[c + d*x]^2)/(15*a^8*d*(-I + Tan[c + d*x])^ 5)
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3968, 53, 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.
\(\displaystyle \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (c+d x)^6}{(a+i a \tan (c+d x))^8}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i \int \frac {(a-i a \tan (c+d x))^2}{(i \tan (c+d x) a+a)^6}d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {i \int \left (\frac {4 a^2}{(i \tan (c+d x) a+a)^6}-\frac {4 a}{(i \tan (c+d x) a+a)^5}+\frac {1}{(i \tan (c+d x) a+a)^4}\right )d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \left (-\frac {4 a^2}{5 (a+i a \tan (c+d x))^5}+\frac {a}{(a+i a \tan (c+d x))^4}-\frac {1}{3 (a+i a \tan (c+d x))^3}\right )}{a^5 d}\) |
Input:
Int[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^8,x]
Output:
((-I)*((-4*a^2)/(5*(a + I*a*Tan[c + d*x])^5) + a/(a + I*a*Tan[c + d*x])^4 - 1/(3*(a + I*a*Tan[c + d*x])^3)))/(a^5*d)
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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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]
Time = 0.68 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.60
method | result | size |
derivativedivides | \(\frac {-\frac {i}{\left (-i+\tan \left (d x +c \right )\right )^{4}}+\frac {4}{5 \left (-i+\tan \left (d x +c \right )\right )^{5}}-\frac {1}{3 \left (-i+\tan \left (d x +c \right )\right )^{3}}}{a^{8} d}\) | \(49\) |
default | \(\frac {-\frac {i}{\left (-i+\tan \left (d x +c \right )\right )^{4}}+\frac {4}{5 \left (-i+\tan \left (d x +c \right )\right )^{5}}-\frac {1}{3 \left (-i+\tan \left (d x +c \right )\right )^{3}}}{a^{8} d}\) | \(49\) |
risch | \(\frac {i {\mathrm e}^{-6 i \left (d x +c \right )}}{24 a^{8} d}+\frac {i {\mathrm e}^{-8 i \left (d x +c \right )}}{16 a^{8} d}+\frac {i {\mathrm e}^{-10 i \left (d x +c \right )}}{40 a^{8} d}\) | \(56\) |
Input:
int(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/a^8/d*(-I/(-I+tan(d*x+c))^4+4/5/(-I+tan(d*x+c))^5-1/3/(-I+tan(d*x+c))^3)
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.51 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (10 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i\right )} e^{\left (-10 i \, d x - 10 i \, c\right )}}{240 \, a^{8} d} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
Output:
1/240*(10*I*e^(4*I*d*x + 4*I*c) + 15*I*e^(2*I*d*x + 2*I*c) + 6*I)*e^(-10*I *d*x - 10*I*c)/(a^8*d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (65) = 130\).
Time = 10.18 (sec) , antiderivative size = 466, normalized size of antiderivative = 5.75 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} - \frac {i \tan ^{2}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}}{240 a^{8} d \tan ^{8}{\left (c + d x \right )} - 1920 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{6}{\left (c + d x \right )} + 13440 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 16800 a^{8} d \tan ^{4}{\left (c + d x \right )} - 13440 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{2}{\left (c + d x \right )} + 1920 i a^{8} d \tan {\left (c + d x \right )} + 240 a^{8} d} - \frac {8 \tan {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}}{240 a^{8} d \tan ^{8}{\left (c + d x \right )} - 1920 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{6}{\left (c + d x \right )} + 13440 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 16800 a^{8} d \tan ^{4}{\left (c + d x \right )} - 13440 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{2}{\left (c + d x \right )} + 1920 i a^{8} d \tan {\left (c + d x \right )} + 240 a^{8} d} + \frac {31 i \sec ^{6}{\left (c + d x \right )}}{240 a^{8} d \tan ^{8}{\left (c + d x \right )} - 1920 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{6}{\left (c + d x \right )} + 13440 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 16800 a^{8} d \tan ^{4}{\left (c + d x \right )} - 13440 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{2}{\left (c + d x \right )} + 1920 i a^{8} d \tan {\left (c + d x \right )} + 240 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{6}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \] Input:
integrate(sec(d*x+c)**6/(a+I*a*tan(d*x+c))**8,x)
Output:
Piecewise((-I*tan(c + d*x)**2*sec(c + d*x)**6/(240*a**8*d*tan(c + d*x)**8 - 1920*I*a**8*d*tan(c + d*x)**7 - 6720*a**8*d*tan(c + d*x)**6 + 13440*I*a* *8*d*tan(c + d*x)**5 + 16800*a**8*d*tan(c + d*x)**4 - 13440*I*a**8*d*tan(c + d*x)**3 - 6720*a**8*d*tan(c + d*x)**2 + 1920*I*a**8*d*tan(c + d*x) + 24 0*a**8*d) - 8*tan(c + d*x)*sec(c + d*x)**6/(240*a**8*d*tan(c + d*x)**8 - 1 920*I*a**8*d*tan(c + d*x)**7 - 6720*a**8*d*tan(c + d*x)**6 + 13440*I*a**8* d*tan(c + d*x)**5 + 16800*a**8*d*tan(c + d*x)**4 - 13440*I*a**8*d*tan(c + d*x)**3 - 6720*a**8*d*tan(c + d*x)**2 + 1920*I*a**8*d*tan(c + d*x) + 240*a **8*d) + 31*I*sec(c + d*x)**6/(240*a**8*d*tan(c + d*x)**8 - 1920*I*a**8*d* tan(c + d*x)**7 - 6720*a**8*d*tan(c + d*x)**6 + 13440*I*a**8*d*tan(c + d*x )**5 + 16800*a**8*d*tan(c + d*x)**4 - 13440*I*a**8*d*tan(c + d*x)**3 - 672 0*a**8*d*tan(c + d*x)**2 + 1920*I*a**8*d*tan(c + d*x) + 240*a**8*d), Ne(d, 0)), (x*sec(c)**6/(I*a*tan(c) + a)**8, True))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (65) = 130\).
Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.74 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {5 \, \tan \left (d x + c\right )^{4} - 5 i \, \tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )^{2} - i \, \tan \left (d x + c\right ) + 2}{15 \, {\left (a^{8} \tan \left (d x + c\right )^{7} - 7 i \, a^{8} \tan \left (d x + c\right )^{6} - 21 \, a^{8} \tan \left (d x + c\right )^{5} + 35 i \, a^{8} \tan \left (d x + c\right )^{4} + 35 \, a^{8} \tan \left (d x + c\right )^{3} - 21 i \, a^{8} \tan \left (d x + c\right )^{2} - 7 \, a^{8} \tan \left (d x + c\right ) + i \, a^{8}\right )} d} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
Output:
-1/15*(5*tan(d*x + c)^4 - 5*I*tan(d*x + c)^3 + 3*tan(d*x + c)^2 - I*tan(d* x + c) + 2)/((a^8*tan(d*x + c)^7 - 7*I*a^8*tan(d*x + c)^6 - 21*a^8*tan(d*x + c)^5 + 35*I*a^8*tan(d*x + c)^4 + 35*a^8*tan(d*x + c)^3 - 21*I*a^8*tan(d *x + c)^2 - 7*a^8*tan(d*x + c) + I*a^8)*d)
Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.47 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {5 \, \tan \left (d x + c\right )^{2} + 5 i \, \tan \left (d x + c\right ) - 2}{15 \, a^{8} d {\left (\tan \left (d x + c\right ) - i\right )}^{5}} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
Output:
-1/15*(5*tan(d*x + c)^2 + 5*I*tan(d*x + c) - 2)/(a^8*d*(tan(d*x + c) - I)^ 5)
Time = 0.59 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {-{\mathrm {tan}\left (c+d\,x\right )}^2\,5{}\mathrm {i}+5\,\mathrm {tan}\left (c+d\,x\right )+2{}\mathrm {i}}{15\,a^8\,d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,10{}\mathrm {i}-10\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,5{}\mathrm {i}+1\right )} \] Input:
int(1/(cos(c + d*x)^6*(a + a*tan(c + d*x)*1i)^8),x)
Output:
(5*tan(c + d*x) - tan(c + d*x)^2*5i + 2i)/(15*a^8*d*(tan(c + d*x)*5i - 10* tan(c + d*x)^2 - tan(c + d*x)^3*10i + 5*tan(c + d*x)^4 + tan(c + d*x)^5*1i + 1))
\[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {too large to display} \] Input:
int(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^8,x)
Output:
(1024*int(sin(c + d*x)**8/(128*cos(c + d*x)*sin(c + d*x)**7*i - 192*cos(c + d*x)*sin(c + d*x)**5*i + 80*cos(c + d*x)*sin(c + d*x)**3*i - 8*cos(c + d *x)*sin(c + d*x)*i - 128*sin(c + d*x)**8 + 256*sin(c + d*x)**6 - 160*sin(c + d*x)**4 + 32*sin(c + d*x)**2 - 1),x)*d + 2048*int(sin(c + d*x)**6/(128* cos(c + d*x)*sin(c + d*x)**7 - 192*cos(c + d*x)*sin(c + d*x)**5 + 80*cos(c + d*x)*sin(c + d*x)**3 - 8*cos(c + d*x)*sin(c + d*x) + 128*sin(c + d*x)** 8*i - 256*sin(c + d*x)**6*i + 160*sin(c + d*x)**4*i - 32*sin(c + d*x)**2*i + i),x)*d*i - 1280*int(sin(c + d*x)**4/(128*cos(c + d*x)*sin(c + d*x)**7 - 192*cos(c + d*x)*sin(c + d*x)**5 + 80*cos(c + d*x)*sin(c + d*x)**3 - 8*c os(c + d*x)*sin(c + d*x) + 128*sin(c + d*x)**8*i - 256*sin(c + d*x)**6*i + 160*sin(c + d*x)**4*i - 32*sin(c + d*x)**2*i + i),x)*d*i + 248*int(sin(c + d*x)**2/(128*cos(c + d*x)*sin(c + d*x)**7 - 192*cos(c + d*x)*sin(c + d*x )**5 + 80*cos(c + d*x)*sin(c + d*x)**3 - 8*cos(c + d*x)*sin(c + d*x) + 128 *sin(c + d*x)**8*i - 256*sin(c + d*x)**6*i + 160*sin(c + d*x)**4*i - 32*si n(c + d*x)**2*i + i),x)*d*i - 1024*int((cos(c + d*x)*sin(c + d*x)**7)/(128 *cos(c + d*x)*sin(c + d*x)**7 - 192*cos(c + d*x)*sin(c + d*x)**5 + 80*cos( c + d*x)*sin(c + d*x)**3 - 8*cos(c + d*x)*sin(c + d*x) + 128*sin(c + d*x)* *8*i - 256*sin(c + d*x)**6*i + 160*sin(c + d*x)**4*i - 32*sin(c + d*x)**2* i + i),x)*d + 1536*int((cos(c + d*x)*sin(c + d*x)**5)/(128*cos(c + d*x)*si n(c + d*x)**7 - 192*cos(c + d*x)*sin(c + d*x)**5 + 80*cos(c + d*x)*sin(...