Integrand size = 22, antiderivative size = 76 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {i a \cos ^7(c+d x)}{7 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d} \] Output:
-1/7*I*a*cos(d*x+c)^7/d+a*sin(d*x+c)/d-a*sin(d*x+c)^3/d+3/5*a*sin(d*x+c)^5 /d-1/7*a*sin(d*x+c)^7/d
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {i a \cos ^7(c+d x)}{7 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d} \] Input:
Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x]),x]
Output:
((-1/7*I)*a*Cos[c + d*x]^7)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3042, 3967, 3042, 3113, 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.
\(\displaystyle \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+i a \tan (c+d x)}{\sec (c+d x)^7}dx\) |
\(\Big \downarrow \) 3967 |
\(\displaystyle a \int \cos ^7(c+d x)dx-\frac {i a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^7dx-\frac {i a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle -\frac {a \int \left (-\sin ^6(c+d x)+3 \sin ^4(c+d x)-3 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{d}-\frac {i a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}-\frac {i a \cos ^7(c+d x)}{7 d}\) |
Input:
Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x]),x]
Output:
((-1/7*I)*a*Cos[c + d*x]^7)/d - (a*(-Sin[c + d*x] + Sin[c + d*x]^3 - (3*Si n[c + d*x]^5)/5 + Sin[c + d*x]^7/7))/d
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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 = 27.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {-\frac {i a \cos \left (d x +c \right )^{7}}{7}+\frac {a \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(57\) |
default | \(\frac {-\frac {i a \cos \left (d x +c \right )^{7}}{7}+\frac {a \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(57\) |
risch | \(-\frac {i a \,{\mathrm e}^{7 i \left (d x +c \right )}}{448 d}-\frac {5 i a \cos \left (d x +c \right )}{64 d}+\frac {35 a \sin \left (d x +c \right )}{64 d}-\frac {i a \cos \left (5 d x +5 c \right )}{64 d}+\frac {7 a \sin \left (5 d x +5 c \right )}{320 d}-\frac {3 i a \cos \left (3 d x +3 c \right )}{64 d}+\frac {7 a \sin \left (3 d x +3 c \right )}{64 d}\) | \(105\) |
Input:
int(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/7*I*a*cos(d*x+c)^7+1/7*a*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*c os(d*x+c)^2)*sin(d*x+c))
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.18 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {{\left (-5 i \, a e^{\left (12 i \, d x + 12 i \, c\right )} - 42 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 175 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 700 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 525 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 70 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{2240 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Output:
1/2240*(-5*I*a*e^(12*I*d*x + 12*I*c) - 42*I*a*e^(10*I*d*x + 10*I*c) - 175* I*a*e^(8*I*d*x + 8*I*c) - 700*I*a*e^(6*I*d*x + 6*I*c) + 525*I*a*e^(4*I*d*x + 4*I*c) + 70*I*a*e^(2*I*d*x + 2*I*c) + 7*I*a)*e^(-5*I*d*x - 5*I*c)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (65) = 130\).
Time = 0.34 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.33 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx=\begin {cases} \frac {\left (- 107374182400 i a d^{6} e^{16 i c} e^{7 i d x} - 901943132160 i a d^{6} e^{14 i c} e^{5 i d x} - 3758096384000 i a d^{6} e^{12 i c} e^{3 i d x} - 15032385536000 i a d^{6} e^{10 i c} e^{i d x} + 11274289152000 i a d^{6} e^{8 i c} e^{- i d x} + 1503238553600 i a d^{6} e^{6 i c} e^{- 3 i d x} + 150323855360 i a d^{6} e^{4 i c} e^{- 5 i d x}\right ) e^{- 9 i c}}{48103633715200 d^{7}} & \text {for}\: d^{7} e^{9 i c} \neq 0 \\\frac {x \left (a e^{12 i c} + 6 a e^{10 i c} + 15 a e^{8 i c} + 20 a e^{6 i c} + 15 a e^{4 i c} + 6 a e^{2 i c} + a\right ) e^{- 5 i c}}{64} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c)),x)
Output:
Piecewise(((-107374182400*I*a*d**6*exp(16*I*c)*exp(7*I*d*x) - 901943132160 *I*a*d**6*exp(14*I*c)*exp(5*I*d*x) - 3758096384000*I*a*d**6*exp(12*I*c)*ex p(3*I*d*x) - 15032385536000*I*a*d**6*exp(10*I*c)*exp(I*d*x) + 112742891520 00*I*a*d**6*exp(8*I*c)*exp(-I*d*x) + 1503238553600*I*a*d**6*exp(6*I*c)*exp (-3*I*d*x) + 150323855360*I*a*d**6*exp(4*I*c)*exp(-5*I*d*x))*exp(-9*I*c)/( 48103633715200*d**7), Ne(d**7*exp(9*I*c), 0)), (x*(a*exp(12*I*c) + 6*a*exp (10*I*c) + 15*a*exp(8*I*c) + 20*a*exp(6*I*c) + 15*a*exp(4*I*c) + 6*a*exp(2 *I*c) + a)*exp(-5*I*c)/64, True))
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.76 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {5 i \, a \cos \left (d x + c\right )^{7} + {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a}{35 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x, algorithm="maxima")
Output:
-1/35*(5*I*a*cos(d*x + c)^7 + (5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*s in(d*x + c)^3 - 35*sin(d*x + c))*a)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (68) = 136\).
Time = 0.20 (sec) , antiderivative size = 244, normalized size of antiderivative = 3.21 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {{\left (1015 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 700 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 1015 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 700 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 315 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 315 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 20 i \, a e^{\left (12 i \, d x + 8 i \, c\right )} + 168 i \, a e^{\left (10 i \, d x + 6 i \, c\right )} + 700 i \, a e^{\left (8 i \, d x + 4 i \, c\right )} + 2800 i \, a e^{\left (6 i \, d x + 2 i \, c\right )} - 280 i \, a e^{\left (2 i \, d x - 2 i \, c\right )} - 2100 i \, a e^{\left (4 i \, d x\right )} - 28 i \, a e^{\left (-4 i \, c\right )}\right )} e^{\left (-5 i \, d x - i \, c\right )}}{8960 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x, algorithm="giac")
Output:
-1/8960*(1015*a*e^(5*I*d*x + I*c)*log(I*e^(I*d*x + I*c) + 1) + 700*a*e^(5* I*d*x + I*c)*log(I*e^(I*d*x + I*c) - 1) - 1015*a*e^(5*I*d*x + I*c)*log(-I* e^(I*d*x + I*c) + 1) - 700*a*e^(5*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) - 315*a*e^(5*I*d*x + I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 315*a*e^(5*I*d*x + I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 20*I*a*e^(12*I*d*x + 8*I*c) + 168*I* a*e^(10*I*d*x + 6*I*c) + 700*I*a*e^(8*I*d*x + 4*I*c) + 2800*I*a*e^(6*I*d*x + 2*I*c) - 280*I*a*e^(2*I*d*x - 2*I*c) - 2100*I*a*e^(4*I*d*x) - 28*I*a*e^ (-4*I*c))*e^(-5*I*d*x - I*c)/d
Time = 3.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.22 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {2\,a\,\left (-\frac {1225\,\sin \left (c+d\,x\right )}{128}-\frac {245\,\sin \left (3\,c+3\,d\,x\right )}{128}-\frac {49\,\sin \left (5\,c+5\,d\,x\right )}{128}-\frac {5\,\sin \left (7\,c+7\,d\,x\right )}{128}+\frac {\cos \left (c+d\,x\right )\,175{}\mathrm {i}}{128}+\frac {\cos \left (3\,c+3\,d\,x\right )\,105{}\mathrm {i}}{128}+\frac {\cos \left (5\,c+5\,d\,x\right )\,35{}\mathrm {i}}{128}+\frac {\cos \left (7\,c+7\,d\,x\right )\,5{}\mathrm {i}}{128}\right )}{35\,d} \] Input:
int(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i),x)
Output:
-(2*a*((cos(c + d*x)*175i)/128 - (1225*sin(c + d*x))/128 + (cos(3*c + 3*d* x)*105i)/128 + (cos(5*c + 5*d*x)*35i)/128 + (cos(7*c + 7*d*x)*5i)/128 - (2 45*sin(3*c + 3*d*x))/128 - (49*sin(5*c + 5*d*x))/128 - (5*sin(7*c + 7*d*x) )/128))/(35*d)
Time = 0.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.42 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \left (5 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} i -15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} i +15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} i -5 \cos \left (d x +c \right ) i -5 \sin \left (d x +c \right )^{7}+21 \sin \left (d x +c \right )^{5}-35 \sin \left (d x +c \right )^{3}+35 \sin \left (d x +c \right )+5 i \right )}{35 d} \] Input:
int(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x)
Output:
(a*(5*cos(c + d*x)*sin(c + d*x)**6*i - 15*cos(c + d*x)*sin(c + d*x)**4*i + 15*cos(c + d*x)*sin(c + d*x)**2*i - 5*cos(c + d*x)*i - 5*sin(c + d*x)**7 + 21*sin(c + d*x)**5 - 35*sin(c + d*x)**3 + 35*sin(c + d*x) + 5*i))/(35*d)