\(\int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) [171]

Optimal result

Integrand size = 24, antiderivative size = 55 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i}{3 a^2 d (a+i a \tan (c+d x))^6}-\frac {i}{5 a^3 d (a+i a \tan (c+d x))^5} \]

[Out]

1/3*I/a^2/d/(a+I*a*tan(d*x+c))^6-1/5*I/a^3/d/(a+I*a*tan(d*x+c))^5

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i}{3 a^2 d (a+i a \tan (c+d x))^6}-\frac {i}{5 a^3 d (a+i a \tan (c+d x))^5} \]

[In]

Int[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(I/3)/(a^2*d*(a + I*a*Tan[c + d*x])^6) - (I/5)/(a^3*d*(a + I*a*Tan[c + d*x])^5)

Rule 45

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] && NeQ[b*c - a*d, 0] && 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])

Rule 3568

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {a-x}{(a+x)^7} \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {i \text {Subst}\left (\int \left (\frac {2 a}{(a+x)^7}-\frac {1}{(a+x)^6}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = \frac {i}{3 a^2 d (a+i a \tan (c+d x))^6}-\frac {i}{5 a^3 d (a+i a \tan (c+d x))^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.62 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {2 i+3 \tan (c+d x)}{15 a^8 d (-i+\tan (c+d x))^6} \]

[In]

Integrate[Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^8,x]

[Out]

-1/15*(2*I + 3*Tan[c + d*x])/(a^8*d*(-I + Tan[c + d*x])^6)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65

method	result	size
derivativedivides	\(\frac {-\frac {i}{3 \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {1}{5 \left (\tan \left (d x +c \right )-i\right )^{5}}}{a^{8} d}\)	\(36\)
default	\(\frac {-\frac {i}{3 \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {1}{5 \left (\tan \left (d x +c \right )-i\right )^{5}}}{a^{8} d}\)	\(36\)
risch	\(\frac {i {\mathrm e}^{-4 i \left (d x +c \right )}}{64 a^{8} d}+\frac {i {\mathrm e}^{-6 i \left (d x +c \right )}}{24 a^{8} d}+\frac {3 i {\mathrm e}^{-8 i \left (d x +c \right )}}{64 a^{8} d}+\frac {i {\mathrm e}^{-10 i \left (d x +c \right )}}{40 a^{8} d}+\frac {i {\mathrm e}^{-12 i \left (d x +c \right )}}{192 a^{8} d}\)	\(92\)

[In]

int(sec(d*x+c)^4/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/a^8/d*(-1/3*I/(tan(d*x+c)-I)^6-1/5/(tan(d*x+c)-I)^5)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.15 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (15 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 40 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 45 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 24 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-12 i \, d x - 12 i \, c\right )}}{960 \, a^{8} d} \]

[In]

integrate(sec(d*x+c)^4/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/960*(15*I*e^(8*I*d*x + 8*I*c) + 40*I*e^(6*I*d*x + 6*I*c) + 45*I*e^(4*I*d*x + 4*I*c) + 24*I*e^(2*I*d*x + 2*I*
c) + 5*I)*e^(-12*I*d*x - 12*I*c)/(a^8*d)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (42) = 84\).

Time = 8.87 (sec) , antiderivative size = 774, normalized size of antiderivative = 14.07 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} \frac {i \tan ^{4}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{960 a^{8} d \tan ^{8}{\left (c + d x \right )} - 7680 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{6}{\left (c + d x \right )} + 53760 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 67200 a^{8} d \tan ^{4}{\left (c + d x \right )} - 53760 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{2}{\left (c + d x \right )} + 7680 i a^{8} d \tan {\left (c + d x \right )} + 960 a^{8} d} + \frac {8 \tan ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{960 a^{8} d \tan ^{8}{\left (c + d x \right )} - 7680 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{6}{\left (c + d x \right )} + 53760 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 67200 a^{8} d \tan ^{4}{\left (c + d x \right )} - 53760 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{2}{\left (c + d x \right )} + 7680 i a^{8} d \tan {\left (c + d x \right )} + 960 a^{8} d} - \frac {30 i \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{960 a^{8} d \tan ^{8}{\left (c + d x \right )} - 7680 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{6}{\left (c + d x \right )} + 53760 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 67200 a^{8} d \tan ^{4}{\left (c + d x \right )} - 53760 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{2}{\left (c + d x \right )} + 7680 i a^{8} d \tan {\left (c + d x \right )} + 960 a^{8} d} - \frac {72 \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{960 a^{8} d \tan ^{8}{\left (c + d x \right )} - 7680 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{6}{\left (c + d x \right )} + 53760 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 67200 a^{8} d \tan ^{4}{\left (c + d x \right )} - 53760 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{2}{\left (c + d x \right )} + 7680 i a^{8} d \tan {\left (c + d x \right )} + 960 a^{8} d} + \frac {129 i \sec ^{4}{\left (c + d x \right )}}{960 a^{8} d \tan ^{8}{\left (c + d x \right )} - 7680 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{6}{\left (c + d x \right )} + 53760 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 67200 a^{8} d \tan ^{4}{\left (c + d x \right )} - 53760 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 26880 a^{8} d \tan ^{2}{\left (c + d x \right )} + 7680 i a^{8} d \tan {\left (c + d x \right )} + 960 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{4}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \]

[In]

integrate(sec(d*x+c)**4/(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise((I*tan(c + d*x)**4*sec(c + d*x)**4/(960*a**8*d*tan(c + d*x)**8 - 7680*I*a**8*d*tan(c + d*x)**7 - 268
80*a**8*d*tan(c + d*x)**6 + 53760*I*a**8*d*tan(c + d*x)**5 + 67200*a**8*d*tan(c + d*x)**4 - 53760*I*a**8*d*tan
(c + d*x)**3 - 26880*a**8*d*tan(c + d*x)**2 + 7680*I*a**8*d*tan(c + d*x) + 960*a**8*d) + 8*tan(c + d*x)**3*sec
(c + d*x)**4/(960*a**8*d*tan(c + d*x)**8 - 7680*I*a**8*d*tan(c + d*x)**7 - 26880*a**8*d*tan(c + d*x)**6 + 5376
0*I*a**8*d*tan(c + d*x)**5 + 67200*a**8*d*tan(c + d*x)**4 - 53760*I*a**8*d*tan(c + d*x)**3 - 26880*a**8*d*tan(
c + d*x)**2 + 7680*I*a**8*d*tan(c + d*x) + 960*a**8*d) - 30*I*tan(c + d*x)**2*sec(c + d*x)**4/(960*a**8*d*tan(
c + d*x)**8 - 7680*I*a**8*d*tan(c + d*x)**7 - 26880*a**8*d*tan(c + d*x)**6 + 53760*I*a**8*d*tan(c + d*x)**5 +
67200*a**8*d*tan(c + d*x)**4 - 53760*I*a**8*d*tan(c + d*x)**3 - 26880*a**8*d*tan(c + d*x)**2 + 7680*I*a**8*d*t
an(c + d*x) + 960*a**8*d) - 72*tan(c + d*x)*sec(c + d*x)**4/(960*a**8*d*tan(c + d*x)**8 - 7680*I*a**8*d*tan(c
+ d*x)**7 - 26880*a**8*d*tan(c + d*x)**6 + 53760*I*a**8*d*tan(c + d*x)**5 + 67200*a**8*d*tan(c + d*x)**4 - 537
60*I*a**8*d*tan(c + d*x)**3 - 26880*a**8*d*tan(c + d*x)**2 + 7680*I*a**8*d*tan(c + d*x) + 960*a**8*d) + 129*I*
sec(c + d*x)**4/(960*a**8*d*tan(c + d*x)**8 - 7680*I*a**8*d*tan(c + d*x)**7 - 26880*a**8*d*tan(c + d*x)**6 + 5
3760*I*a**8*d*tan(c + d*x)**5 + 67200*a**8*d*tan(c + d*x)**4 - 53760*I*a**8*d*tan(c + d*x)**3 - 26880*a**8*d*t
an(c + d*x)**2 + 7680*I*a**8*d*tan(c + d*x) + 960*a**8*d), Ne(d, 0)), (x*sec(c)**4/(I*a*tan(c) + a)**8, True))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (43) = 86\).

Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.20 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {3 \, \tan \left (d x + c\right )^{2} - i \, \tan \left (d x + c\right ) + 2}{15 \, {\left (a^{8} \tan \left (d x + c\right )^{7} - 7 i \, a^{8} \tan \left (d x + c\right )^{6} - 21 \, a^{8} \tan \left (d x + c\right )^{5} + 35 i \, a^{8} \tan \left (d x + c\right )^{4} + 35 \, a^{8} \tan \left (d x + c\right )^{3} - 21 i \, a^{8} \tan \left (d x + c\right )^{2} - 7 \, a^{8} \tan \left (d x + c\right ) + i \, a^{8}\right )} d} \]

[In]

integrate(sec(d*x+c)^4/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/15*(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)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (43) = 86\).

Time = 1.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.96 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 60 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 235 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 480 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 822 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 904 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 822 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 480 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 235 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 60 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{15 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{12}} \]

[In]

integrate(sec(d*x+c)^4/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-2/15*(15*tan(1/2*d*x + 1/2*c)^11 - 60*I*tan(1/2*d*x + 1/2*c)^10 - 235*tan(1/2*d*x + 1/2*c)^9 + 480*I*tan(1/2*
d*x + 1/2*c)^8 + 822*tan(1/2*d*x + 1/2*c)^7 - 904*I*tan(1/2*d*x + 1/2*c)^6 - 822*tan(1/2*d*x + 1/2*c)^5 + 480*
I*tan(1/2*d*x + 1/2*c)^4 + 235*tan(1/2*d*x + 1/2*c)^3 - 60*I*tan(1/2*d*x + 1/2*c)^2 - 15*tan(1/2*d*x + 1/2*c))
/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^12)

Mupad [B] (verification not implemented)

Time = 4.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.55 \[ \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {-2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}}{15\,a^8\,d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^6\,1{}\mathrm {i}+6\,{\mathrm {tan}\left (c+d\,x\right )}^5-{\mathrm {tan}\left (c+d\,x\right )}^4\,15{}\mathrm {i}-20\,{\mathrm {tan}\left (c+d\,x\right )}^3+{\mathrm {tan}\left (c+d\,x\right )}^2\,15{}\mathrm {i}+6\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )} \]

[In]

int(1/(cos(c + d*x)^4*(a + a*tan(c + d*x)*1i)^8),x)

[Out]

-(tan(c + d*x)*3i - 2)/(15*a^8*d*(6*tan(c + d*x) + tan(c + d*x)^2*15i - 20*tan(c + d*x)^3 - tan(c + d*x)^4*15i
+ 6*tan(c + d*x)^5 + tan(c + d*x)^6*1i - 1i))