Integrand size = 24, antiderivative size = 68 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7} \] Output:
1/9*I*sec(d*x+c)^7/d/(a+I*a*tan(d*x+c))^8+1/63*I*sec(d*x+c)^7/a/d/(a+I*a*t an(d*x+c))^7
Time = 0.41 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.59 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {\sec ^7(c+d x) (-8 i+\tan (c+d x))}{63 a^8 d (-i+\tan (c+d x))^8} \] Input:
Integrate[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^8,x]
Output:
-1/63*(Sec[c + d*x]^7*(-8*I + Tan[c + d*x]))/(a^8*d*(-I + Tan[c + d*x])^8)
Time = 0.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3983, 3042, 3969}
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.
\(\displaystyle \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (c+d x)^7}{(a+i a \tan (c+d x))^8}dx\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {\int \frac {\sec ^7(c+d x)}{(i \tan (c+d x) a+a)^7}dx}{9 a}+\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sec (c+d x)^7}{(i \tan (c+d x) a+a)^7}dx}{9 a}+\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3969 |
\(\displaystyle \frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}\) |
Input:
Int[Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^8,x]
Output:
((I/9)*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^8) + ((I/63)*Sec[c + d*x] ^7)/(a*d*(a + I*a*Tan[c + d*x])^7)
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ (a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ [Simplify[m + n], 0]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[a*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ (b*f*(m + 2*n))), x] + Simp[Simplify[m + n]/(a*(m + 2*n)) Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x ] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2* n]
Time = 1.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.56
method | result | size |
risch | \(\frac {i {\mathrm e}^{-7 i \left (d x +c \right )}}{14 a^{8} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{18 a^{8} d}\) | \(38\) |
derivativedivides | \(\frac {\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1856}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {272}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {256}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {992 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {172}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {152 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{a^{8} d}\) | \(156\) |
default | \(\frac {\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1856}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {272}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {256}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {992 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {172}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {152 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{a^{8} d}\) | \(156\) |
Input:
int(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/14*I/a^8/d*exp(-7*I*(d*x+c))+1/18*I/a^8/d*exp(-9*I*(d*x+c))
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.44 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (9 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{126 \, a^{8} d} \] Input:
integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
Output:
1/126*(9*I*e^(2*I*d*x + 2*I*c) + 7*I)*e^(-9*I*d*x - 9*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. 311 vs. \(2 (54) = 108\).
Time = 10.09 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.57 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} - \frac {\tan {\left (c + d x \right )} \sec ^{7}{\left (c + d x \right )}}{63 a^{8} d \tan ^{8}{\left (c + d x \right )} - 504 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{6}{\left (c + d x \right )} + 3528 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 4410 a^{8} d \tan ^{4}{\left (c + d x \right )} - 3528 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{2}{\left (c + d x \right )} + 504 i a^{8} d \tan {\left (c + d x \right )} + 63 a^{8} d} + \frac {8 i \sec ^{7}{\left (c + d x \right )}}{63 a^{8} d \tan ^{8}{\left (c + d x \right )} - 504 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{6}{\left (c + d x \right )} + 3528 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 4410 a^{8} d \tan ^{4}{\left (c + d x \right )} - 3528 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{2}{\left (c + d x \right )} + 504 i a^{8} d \tan {\left (c + d x \right )} + 63 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{7}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \] Input:
integrate(sec(d*x+c)**7/(a+I*a*tan(d*x+c))**8,x)
Output:
Piecewise((-tan(c + d*x)*sec(c + d*x)**7/(63*a**8*d*tan(c + d*x)**8 - 504* I*a**8*d*tan(c + d*x)**7 - 1764*a**8*d*tan(c + d*x)**6 + 3528*I*a**8*d*tan (c + d*x)**5 + 4410*a**8*d*tan(c + d*x)**4 - 3528*I*a**8*d*tan(c + d*x)**3 - 1764*a**8*d*tan(c + d*x)**2 + 504*I*a**8*d*tan(c + d*x) + 63*a**8*d) + 8*I*sec(c + d*x)**7/(63*a**8*d*tan(c + d*x)**8 - 504*I*a**8*d*tan(c + d*x) **7 - 1764*a**8*d*tan(c + d*x)**6 + 3528*I*a**8*d*tan(c + d*x)**5 + 4410*a **8*d*tan(c + d*x)**4 - 3528*I*a**8*d*tan(c + d*x)**3 - 1764*a**8*d*tan(c + d*x)**2 + 504*I*a**8*d*tan(c + d*x) + 63*a**8*d), Ne(d, 0)), (x*sec(c)** 7/(I*a*tan(c) + a)**8, True))
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {7 i \, \cos \left (9 \, d x + 9 \, c\right ) + 9 i \, \cos \left (7 \, d x + 7 \, c\right ) + 7 \, \sin \left (9 \, d x + 9 \, c\right ) + 9 \, \sin \left (7 \, d x + 7 \, c\right )}{126 \, a^{8} d} \] Input:
integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
Output:
1/126*(7*I*cos(9*d*x + 9*c) + 9*I*cos(7*d*x + 7*c) + 7*sin(9*d*x + 9*c) + 9*sin(7*d*x + 7*c))/(a^8*d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (56) = 112\).
Time = 0.79 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.84 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2 \, {\left (63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 63 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 483 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 315 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 189 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{9}} \] Input:
integrate(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
Output:
2/63*(63*tan(1/2*d*x + 1/2*c)^8 - 63*I*tan(1/2*d*x + 1/2*c)^7 - 483*tan(1/ 2*d*x + 1/2*c)^6 + 315*I*tan(1/2*d*x + 1/2*c)^5 + 693*tan(1/2*d*x + 1/2*c) ^4 - 189*I*tan(1/2*d*x + 1/2*c)^3 - 225*tan(1/2*d*x + 1/2*c)^2 + 9*I*tan(1 /2*d*x + 1/2*c) + 8)/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^9)
Time = 0.88 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.54 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2\,\left (\frac {{\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,9{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{-c\,9{}\mathrm {i}-d\,x\,9{}\mathrm {i}}\,7{}\mathrm {i}}{4}\right )}{63\,a^8\,d} \] Input:
int(1/(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^8),x)
Output:
(2*((exp(- c*7i - d*x*7i)*9i)/4 + (exp(- c*9i - d*x*9i)*7i)/4))/(63*a^8*d)
\[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {too large to display} \] Input:
int(sec(d*x+c)^7/(a+I*a*tan(d*x+c))^8,x)
Output:
( - 4*int(cos(c + d*x)/(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 - 512*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 - 1024*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*s in(c + d*x)**4*i - 32*sin(c + d*x)**2*i + i),x)*d*i + 640*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*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 - 128*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*sin(c + d*x)**2*i + i),x)*d*i + 512*in t((cos(c + d*x)*sin(c + d*x)**7)/(128*cos(c + d*x)*sin(c + d*x)**7 - 192*c os(c + d*x)*sin(c + d*x)**5 + 80*cos(c + d*x)*sin(c + d*x)**3 - 8*cos(c...