Integrand size = 24, antiderivative size = 63 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {4 x}{a^4}-\frac {4 i \log (\cos (c+d x))}{a^4 d}+\frac {\tan (c+d x)}{a^4 d}+\frac {4 i}{d \left (a^4+i a^4 \tan (c+d x)\right )} \] Output:
-4*x/a^4-4*I*ln(cos(d*x+c))/a^4/d+tan(d*x+c)/a^4/d+4*I/d/(a^4+I*a^4*tan(d* x+c))
Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {i \left (-4 \log (i-\tan (c+d x))+i \tan (c+d x)+\frac {4 i}{-i+\tan (c+d x)}\right )}{a^4 d} \] Input:
Integrate[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^4,x]
Output:
((-I)*(-4*Log[I - Tan[c + d*x]] + I*Tan[c + d*x] + (4*I)/(-I + Tan[c + d*x ])))/(a^4*d)
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94, 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.
\(\displaystyle \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (c+d x)^6}{(a+i a \tan (c+d x))^4}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i \int \frac {(a-i a \tan (c+d x))^2}{(i \tan (c+d x) a+a)^2}d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {i \int \left (\frac {4 a^2}{(i \tan (c+d x) a+a)^2}-\frac {4 a}{i \tan (c+d x) a+a}+1\right )d(i a \tan (c+d x))}{a^5 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \left (-\frac {4 a^2}{a+i a \tan (c+d x)}+i a \tan (c+d x)-4 a \log (a+i a \tan (c+d x))\right )}{a^5 d}\) |
Input:
Int[Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^4,x]
Output:
((-I)*(-4*a*Log[a + I*a*Tan[c + d*x]] + I*a*Tan[c + d*x] - (4*a^2)/(a + I* a*Tan[c + d*x])))/(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}, 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]
Time = 0.96 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {\tan \left (d x +c \right )+\frac {4}{-i+\tan \left (d x +c \right )}+4 i \ln \left (-i+\tan \left (d x +c \right )\right )}{d \,a^{4}}\) | \(41\) |
default | \(\frac {\tan \left (d x +c \right )+\frac {4}{-i+\tan \left (d x +c \right )}+4 i \ln \left (-i+\tan \left (d x +c \right )\right )}{d \,a^{4}}\) | \(41\) |
risch | \(\frac {2 i {\mathrm e}^{-2 i \left (d x +c \right )}}{a^{4} d}-\frac {8 x}{a^{4}}-\frac {8 c}{a^{4} d}+\frac {2 i}{d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{4} d}\) | \(78\) |
Input:
int(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d/a^4*(tan(d*x+c)+4/(-I+tan(d*x+c))+4*I*ln(-I+tan(d*x+c)))
Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.62 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {2 \, {\left (4 \, d x e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (2 \, d x - i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, {\left (i \, e^{\left (4 i \, d x + 4 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - i\right )}}{a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )}} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")
Output:
-2*(4*d*x*e^(4*I*d*x + 4*I*c) + 2*(2*d*x - I)*e^(2*I*d*x + 2*I*c) + 2*(I*e ^(4*I*d*x + 4*I*c) + I*e^(2*I*d*x + 2*I*c))*log(e^(2*I*d*x + 2*I*c) + 1) - I)/(a^4*d*e^(4*I*d*x + 4*I*c) + a^4*d*e^(2*I*d*x + 2*I*c))
\[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {\int \frac {\sec ^{6}{\left (c + d x \right )}}{\tan ^{4}{\left (c + d x \right )} - 4 i \tan ^{3}{\left (c + d x \right )} - 6 \tan ^{2}{\left (c + d x \right )} + 4 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:
integrate(sec(d*x+c)**6/(a+I*a*tan(d*x+c))**4,x)
Output:
Integral(sec(c + d*x)**6/(tan(c + d*x)**4 - 4*I*tan(c + d*x)**3 - 6*tan(c + d*x)**2 + 4*I*tan(c + d*x) + 1), x)/a**4
Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.51 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {\frac {4 \, {\left (\tan \left (d x + c\right )^{2} - 2 i \, \tan \left (d x + c\right ) - 1\right )}}{a^{4} \tan \left (d x + c\right )^{3} - 3 i \, a^{4} \tan \left (d x + c\right )^{2} - 3 \, a^{4} \tan \left (d x + c\right ) + i \, a^{4}} + \frac {4 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {\tan \left (d x + c\right )}{a^{4}}}{d} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")
Output:
(4*(tan(d*x + c)^2 - 2*I*tan(d*x + c) - 1)/(a^4*tan(d*x + c)^3 - 3*I*a^4*t an(d*x + c)^2 - 3*a^4*tan(d*x + c) + I*a^4) + 4*I*log(I*tan(d*x + c) + 1)/ a^4 + tan(d*x + c)/a^4)/d
Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {4 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4} d} + \frac {\tan \left (d x + c\right )}{a^{4} d} + \frac {4}{a^{4} d {\left (\tan \left (d x + c\right ) - i\right )}} \] Input:
integrate(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^4,x, algorithm="giac")
Output:
4*I*log(tan(d*x + c) - I)/(a^4*d) + tan(d*x + c)/(a^4*d) + 4/(a^4*d*(tan(d *x + c) - I))
Time = 0.47 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,4{}\mathrm {i}}{a^4\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )}{a^4\,d}+\frac {4{}\mathrm {i}}{a^4\,d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \] Input:
int(1/(cos(c + d*x)^6*(a + a*tan(c + d*x)*1i)^4),x)
Output:
(log(tan(c + d*x) - 1i)*4i)/(a^4*d) + tan(c + d*x)/(a^4*d) + 4i/(a^4*d*(ta n(c + d*x)*1i + 1))
\[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\text {too large to display} \] Input:
int(sec(d*x+c)^6/(a+I*a*tan(d*x+c))^4,x)
Output:
(39616*cos(c + d*x)*int(cos(c + d*x)/(8*cos(c + d*x)*sin(c + d*x)**5*i - 1 2*cos(c + d*x)*sin(c + d*x)**3*i + 4*cos(c + d*x)*sin(c + d*x)*i - 8*sin(c + d*x)**6 + 16*sin(c + d*x)**4 - 9*sin(c + d*x)**2 + 1),x)*d - 39680*cos( c + d*x)*int(cos(c + d*x)/(8*cos(c + d*x)*sin(c + d*x)**5 - 12*cos(c + d*x )*sin(c + d*x)**3 + 4*cos(c + d*x)*sin(c + d*x) + 8*sin(c + d*x)**6*i - 16 *sin(c + d*x)**4*i + 9*sin(c + d*x)**2*i - i),x)*d*i - 1920*cos(c + d*x)*i nt(cos(c + d*x)/(8*cos(c + d*x)*sin(c + d*x)**4*i - 8*cos(c + d*x)*sin(c + d*x)**2*i + cos(c + d*x)*i - 8*sin(c + d*x)**5 + 12*sin(c + d*x)**3 - 4*s in(c + d*x)),x)*d*i - 125440*cos(c + d*x)*int(sin(c + d*x)**6/(8*cos(c + d *x)*sin(c + d*x)**5*i - 12*cos(c + d*x)*sin(c + d*x)**3*i + 4*cos(c + d*x) *sin(c + d*x)*i - 8*sin(c + d*x)**6 + 16*sin(c + d*x)**4 - 9*sin(c + d*x)* *2 + 1),x)*d + 125440*cos(c + d*x)*int(sin(c + d*x)**6/(8*cos(c + d*x)*sin (c + d*x)**5 - 12*cos(c + d*x)*sin(c + d*x)**3 + 4*cos(c + d*x)*sin(c + d* x) + 8*sin(c + d*x)**6*i - 16*sin(c + d*x)**4*i + 9*sin(c + d*x)**2*i - i) ,x)*d*i + 278016*cos(c + d*x)*int(sin(c + d*x)**5/(8*cos(c + d*x)*sin(c + d*x)**5*i - 12*cos(c + d*x)*sin(c + d*x)**3*i + 4*cos(c + d*x)*sin(c + d*x )*i - 8*sin(c + d*x)**6 + 16*sin(c + d*x)**4 - 9*sin(c + d*x)**2 + 1),x)*d *i + 278016*cos(c + d*x)*int(sin(c + d*x)**5/(8*cos(c + d*x)*sin(c + d*x)* *5 - 12*cos(c + d*x)*sin(c + d*x)**3 + 4*cos(c + d*x)*sin(c + d*x) + 8*sin (c + d*x)**6*i - 16*sin(c + d*x)**4*i + 9*sin(c + d*x)**2*i - i),x)*d +...