Integrand size = 22, antiderivative size = 89 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {5 a x}{16}-\frac {i a \cos ^6(c+d x)}{6 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d} \] Output:
5/16*a*x-1/6*I*a*cos(d*x+c)^6/d+5/16*a*cos(d*x+c)*sin(d*x+c)/d+5/24*a*cos( d*x+c)^3*sin(d*x+c)/d+1/6*a*cos(d*x+c)^5*sin(d*x+c)/d
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.63 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \left (60 c+60 d x-32 i \cos ^6(c+d x)+45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))\right )}{192 d} \] Input:
Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]
Output:
(a*(60*c + 60*d*x - (32*I)*Cos[c + d*x]^6 + 45*Sin[2*(c + d*x)] + 9*Sin[4* (c + d*x)] + Sin[6*(c + d*x)]))/(192*d)
Time = 0.42 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {3042, 3967, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
\(\displaystyle \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+i a \tan (c+d x)}{\sec (c+d x)^6}dx\) |
\(\Big \downarrow \) 3967 |
\(\displaystyle a \int \cos ^6(c+d x)dx-\frac {i a \cos ^6(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {i a \cos ^6(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {i a \cos ^6(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {i a \cos ^6(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {i a \cos ^6(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {i a \cos ^6(c+d x)}{6 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {i a \cos ^6(c+d x)}{6 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle a \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {i a \cos ^6(c+d x)}{6 d}\) |
Input:
Int[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]
Output:
((-1/6*I)*a*Cos[c + d*x]^6)/d + a*((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + ( 5*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d *x])/(2*d)))/4))/6)
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Time = 12.92 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {-\frac {i a \cos \left (d x +c \right )^{6}}{6}+a \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(63\) |
default | \(\frac {-\frac {i a \cos \left (d x +c \right )^{6}}{6}+a \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(63\) |
risch | \(\frac {5 a x}{16}-\frac {i a \,{\mathrm e}^{6 i \left (d x +c \right )}}{192 d}-\frac {i a \cos \left (4 d x +4 c \right )}{32 d}+\frac {3 a \sin \left (4 d x +4 c \right )}{64 d}-\frac {5 i a \cos \left (2 d x +2 c \right )}{64 d}+\frac {15 a \sin \left (2 d x +2 c \right )}{64 d}\) | \(84\) |
Input:
int(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/6*I*a*cos(d*x+c)^6+a*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos( d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c))
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {{\left (120 \, a d x e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 15 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 60 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 30 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, a\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{384 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Output:
1/384*(120*a*d*x*e^(4*I*d*x + 4*I*c) - 2*I*a*e^(10*I*d*x + 10*I*c) - 15*I* a*e^(8*I*d*x + 8*I*c) - 60*I*a*e^(6*I*d*x + 6*I*c) + 30*I*a*e^(2*I*d*x + 2 *I*c) + 3*I*a)*e^(-4*I*d*x - 4*I*c)/d
Time = 0.22 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.37 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {5 a x}{16} + \begin {cases} \frac {\left (- 33554432 i a d^{4} e^{12 i c} e^{6 i d x} - 251658240 i a d^{4} e^{10 i c} e^{4 i d x} - 1006632960 i a d^{4} e^{8 i c} e^{2 i d x} + 503316480 i a d^{4} e^{4 i c} e^{- 2 i d x} + 50331648 i a d^{4} e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{6442450944 d^{5}} & \text {for}\: d^{5} e^{6 i c} \neq 0 \\x \left (- \frac {5 a}{16} + \frac {\left (a e^{10 i c} + 5 a e^{8 i c} + 10 a e^{6 i c} + 10 a e^{4 i c} + 5 a e^{2 i c} + a\right ) e^{- 4 i c}}{32}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6*(a+I*a*tan(d*x+c)),x)
Output:
5*a*x/16 + Piecewise(((-33554432*I*a*d**4*exp(12*I*c)*exp(6*I*d*x) - 25165 8240*I*a*d**4*exp(10*I*c)*exp(4*I*d*x) - 1006632960*I*a*d**4*exp(8*I*c)*ex p(2*I*d*x) + 503316480*I*a*d**4*exp(4*I*c)*exp(-2*I*d*x) + 50331648*I*a*d* *4*exp(2*I*c)*exp(-4*I*d*x))*exp(-6*I*c)/(6442450944*d**5), Ne(d**5*exp(6* I*c), 0)), (x*(-5*a/16 + (a*exp(10*I*c) + 5*a*exp(8*I*c) + 10*a*exp(6*I*c) + 10*a*exp(4*I*c) + 5*a*exp(2*I*c) + a)*exp(-4*I*c)/32), True))
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {15 \, {\left (d x + c\right )} a + \frac {15 \, a \tan \left (d x + c\right )^{5} + 40 \, a \tan \left (d x + c\right )^{3} + 33 \, a \tan \left (d x + c\right ) - 8 i \, a}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="maxima")
Output:
1/48*(15*(d*x + c)*a + (15*a*tan(d*x + c)^5 + 40*a*tan(d*x + c)^3 + 33*a*t an(d*x + c) - 8*I*a)/(tan(d*x + c)^6 + 3*tan(d*x + c)^4 + 3*tan(d*x + c)^2 + 1))/d
Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {1}{96} i \, a {\left (\frac {15 \, \log \left (\tan \left (d x + c\right ) + i\right )}{d} - \frac {15 \, \log \left (\tan \left (d x + c\right ) - i\right )}{d} - \frac {2 \, {\left (15 i \, \tan \left (d x + c\right )^{4} - 15 \, \tan \left (d x + c\right )^{3} + 25 i \, \tan \left (d x + c\right )^{2} - 25 \, \tan \left (d x + c\right ) + 8 i\right )}}{d {\left (\tan \left (d x + c\right ) + i\right )}^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{2}}\right )} \] Input:
integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="giac")
Output:
1/96*I*a*(15*log(tan(d*x + c) + I)/d - 15*log(tan(d*x + c) - I)/d - 2*(15* I*tan(d*x + c)^4 - 15*tan(d*x + c)^3 + 25*I*tan(d*x + c)^2 - 25*tan(d*x + c) + 8*I)/(d*(tan(d*x + c) + I)^3*(tan(d*x + c) - I)^2))
Time = 0.68 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.21 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {5\,a\,x}{16}+\frac {\frac {5\,a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{16}+\frac {5{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{16}+\frac {25\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{48}+\frac {25{}\mathrm {i}\,a\,\mathrm {tan}\left (c+d\,x\right )}{48}+\frac {a}{6}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}+2\,{\mathrm {tan}\left (c+d\,x\right )}^3+{\mathrm {tan}\left (c+d\,x\right )}^2\,2{}\mathrm {i}+\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \] Input:
int(cos(c + d*x)^6*(a + a*tan(c + d*x)*1i),x)
Output:
(5*a*x)/16 + (a/6 + (a*tan(c + d*x)*25i)/48 + (25*a*tan(c + d*x)^2)/48 + ( a*tan(c + d*x)^3*5i)/16 + (5*a*tan(c + d*x)^4)/16)/(d*(tan(c + d*x) + tan( c + d*x)^2*2i + 2*tan(c + d*x)^3 + tan(c + d*x)^4*1i + tan(c + d*x)^5 + 1i ))
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \left (8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-26 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+33 \cos \left (d x +c \right ) \sin \left (d x +c \right )+8 \sin \left (d x +c \right )^{6} i -24 \sin \left (d x +c \right )^{4} i +24 \sin \left (d x +c \right )^{2} i +15 d x \right )}{48 d} \] Input:
int(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x)
Output:
(a*(8*cos(c + d*x)*sin(c + d*x)**5 - 26*cos(c + d*x)*sin(c + d*x)**3 + 33* cos(c + d*x)*sin(c + d*x) + 8*sin(c + d*x)**6*i - 24*sin(c + d*x)**4*i + 2 4*sin(c + d*x)**2*i + 15*d*x))/(48*d)