\(\int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [95]

Optimal result

Integrand size = 24, antiderivative size = 66 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d} \] Output:

-1/63*I*a*cos(d*x+c)^7*(a+I*a*tan(d*x+c))^7/d-1/9*I*cos(d*x+c)^9*(a+I*a*ta 
n(d*x+c))^8/d
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(66)=132\).

Time = 0.72 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.21 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \sec (c+d x) (-i \cos (5 (c+d x))+\sin (5 (c+d x))) \left (9 \cos (c+d x)+16 \cos (3 (c+d x))+7 \cos (5 (c+d x))+192 \sqrt {\cos ^2(c+d x)} \cos (5 (c+d x))+9 i \sin (c+d x)+16 i \sin (3 (c+d x))+7 i \sin (5 (c+d x))-192 i \sqrt {\cos ^2(c+d x)} \sin (5 (c+d x))\right )}{252 d} \] Input:

Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8,x]
 

Output:

(a^8*Sec[c + d*x]*((-I)*Cos[5*(c + d*x)] + Sin[5*(c + d*x)])*(9*Cos[c + d* 
x] + 16*Cos[3*(c + d*x)] + 7*Cos[5*(c + d*x)] + 192*Sqrt[Cos[c + d*x]^2]*C 
os[5*(c + d*x)] + (9*I)*Sin[c + d*x] + (16*I)*Sin[3*(c + d*x)] + (7*I)*Sin 
[5*(c + d*x)] - (192*I)*Sqrt[Cos[c + d*x]^2]*Sin[5*(c + d*x)]))/(252*d)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 66, 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, 3978, 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.

\(\displaystyle \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+i a \tan (c+d x))^8}{\sec (c+d x)^9}dx\)

\(\Big \downarrow \) 3978

\(\displaystyle \frac {1}{9} a \int \cos ^7(c+d x) (i \tan (c+d x) a+a)^7dx-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} a \int \frac {(i \tan (c+d x) a+a)^7}{\sec (c+d x)^7}dx-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}\)

\(\Big \downarrow \) 3969

\(\displaystyle -\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}-\frac {i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d}\)

Input:

Int[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8,x]
 

Output:

((-1/63*I)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^7)/d - ((I/9)*Cos[c + d 
*x]^9*(a + I*a*Tan[c + d*x])^8)/d
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3969

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]
 

rule 3978

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] + Simp[a*((m + n)/(m*d^2))   Int[(d*Sec[e + f*x])^(m + 2)*(a + 
b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b 
^2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (58 ) = 116\).

Time = 1.39 (sec) , antiderivative size = 447, normalized size of antiderivative = 6.77

\[\frac {\frac {a^{8} \sin \left (d x +c \right )^{9}}{9}-56 i a^{8} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{4}}{9}-\frac {5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}{63}-\frac {\cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}{21}+\frac {\left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {8 i a^{8} \cos \left (d x +c \right )^{9}}{9}+70 a^{8} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )-8 i a^{8} \left (-\frac {\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{6}}{9}-\frac {2 \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{4}}{21}-\frac {8 \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}}{105}-\frac {16 \cos \left (d x +c \right )^{3}}{315}\right )-28 a^{8} \left (-\frac {\cos \left (d x +c \right )^{8} \sin \left (d x +c \right )}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+56 i a^{8} \left (-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )^{4}}{9}-\frac {4 \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )^{2}}{63}-\frac {8 \cos \left (d x +c \right )^{5}}{315}\right )+\frac {a^{8} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\]

Input:

int(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x)
 

Output:

1/d*(1/9*a^8*sin(d*x+c)^9-56*I*a^8*(-1/9*cos(d*x+c)^7*sin(d*x+c)^2-2/63*co 
s(d*x+c)^7)-28*a^8*(-1/9*sin(d*x+c)^5*cos(d*x+c)^4-5/63*sin(d*x+c)^3*cos(d 
*x+c)^4-1/21*cos(d*x+c)^4*sin(d*x+c)+1/63*(2+cos(d*x+c)^2)*sin(d*x+c))-8/9 
*I*a^8*cos(d*x+c)^9+70*a^8*(-1/9*sin(d*x+c)^3*cos(d*x+c)^6-1/21*sin(d*x+c) 
*cos(d*x+c)^6+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-8*I*a^ 
8*(-1/9*cos(d*x+c)^3*sin(d*x+c)^6-2/21*cos(d*x+c)^3*sin(d*x+c)^4-8/105*cos 
(d*x+c)^3*sin(d*x+c)^2-16/315*cos(d*x+c)^3)-28*a^8*(-1/9*cos(d*x+c)^8*sin( 
d*x+c)+1/63*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+ 
c))+56*I*a^8*(-1/9*cos(d*x+c)^5*sin(d*x+c)^4-4/63*cos(d*x+c)^5*sin(d*x+c)^ 
2-8/315*cos(d*x+c)^5)+1/9*a^8*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35* 
cos(d*x+c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.52 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-7 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 9 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )}}{126 \, d} \] Input:

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
 

Output:

1/126*(-7*I*a^8*e^(9*I*d*x + 9*I*c) - 9*I*a^8*e^(7*I*d*x + 7*I*c))/d
 

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\begin {cases} \frac {- 14 i a^{8} d e^{9 i c} e^{9 i d x} - 18 i a^{8} d e^{7 i c} e^{7 i d x}}{252 d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (\frac {a^{8} e^{9 i c}}{2} + \frac {a^{8} e^{7 i c}}{2}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(cos(d*x+c)**9*(a+I*a*tan(d*x+c))**8,x)
                                                                                    
                                                                                    
 

Output:

Piecewise(((-14*I*a**8*d*exp(9*I*c)*exp(9*I*d*x) - 18*I*a**8*d*exp(7*I*c)* 
exp(7*I*d*x))/(252*d**2), Ne(d**2, 0)), (x*(a**8*exp(9*I*c)/2 + a**8*exp(7 
*I*c)/2), 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. 302 vs. \(2 (54) = 108\).

Time = 0.04 (sec) , antiderivative size = 302, normalized size of antiderivative = 4.58 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {280 i \, a^{8} \cos \left (d x + c\right )^{9} - 35 \, a^{8} \sin \left (d x + c\right )^{9} + 56 i \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 8 i \, {\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 280 i \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{8} - 70 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a^{8} - 28 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{8} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{8} - 140 \, {\left (7 \, \sin \left (d x + c\right )^{9} - 9 \, \sin \left (d x + c\right )^{7}\right )} a^{8}}{315 \, d} \] Input:

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
 

Output:

-1/315*(280*I*a^8*cos(d*x + c)^9 - 35*a^8*sin(d*x + c)^9 + 56*I*(35*cos(d* 
x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^8 + 8*I*(35*cos(d*x + 
c)^9 - 135*cos(d*x + c)^7 + 189*cos(d*x + c)^5 - 105*cos(d*x + c)^3)*a^8 + 
 280*I*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^8 - 70*(35*sin(d*x + c)^9 - 
 90*sin(d*x + c)^7 + 63*sin(d*x + c)^5)*a^8 - 28*(35*sin(d*x + c)^9 - 135* 
sin(d*x + c)^7 + 189*sin(d*x + c)^5 - 105*sin(d*x + c)^3)*a^8 - (35*sin(d* 
x + c)^9 - 180*sin(d*x + c)^7 + 378*sin(d*x + c)^5 - 420*sin(d*x + c)^3 + 
315*sin(d*x + c))*a^8 - 140*(7*sin(d*x + c)^9 - 9*sin(d*x + c)^7)*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. 2451 vs. \(2 (54) = 108\).

Time = 1.16 (sec) , antiderivative size = 2451, normalized size of antiderivative = 37.14 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
 

Output:

1/66060288*(1419343317*a^8*e^(24*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1 
) + 17032119804*a^8*e^(22*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 936 
76658922*a^8*e^(20*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 31225552974 
0*a^8*e^(18*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 702574941915*a^8*e 
^(16*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1124119907064*a^8*e^(14*I 
*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1124119907064*a^8*e^(10*I*d*x - 
 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 702574941915*a^8*e^(8*I*d*x - 4*I*c)* 
log(I*e^(I*d*x + I*c) + 1) + 312255529740*a^8*e^(6*I*d*x - 6*I*c)*log(I*e^ 
(I*d*x + I*c) + 1) + 93676658922*a^8*e^(4*I*d*x - 8*I*c)*log(I*e^(I*d*x + 
I*c) + 1) + 17032119804*a^8*e^(2*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1 
) + 1311473224908*a^8*e^(12*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 1419343317 
*a^8*e^(-12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1419097050*a^8*e^(24*I*d*x + 
 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 17029164600*a^8*e^(22*I*d*x + 10*I*c 
)*log(I*e^(I*d*x + I*c) - 1) + 93660405300*a^8*e^(20*I*d*x + 8*I*c)*log(I* 
e^(I*d*x + I*c) - 1) + 312201351000*a^8*e^(18*I*d*x + 6*I*c)*log(I*e^(I*d* 
x + I*c) - 1) + 702453039750*a^8*e^(16*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c 
) - 1) + 1123924863600*a^8*e^(14*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) 
 + 1123924863600*a^8*e^(10*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 702 
453039750*a^8*e^(8*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 31220135100 
0*a^8*e^(6*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 93660405300*a^8*...
 

Mupad [B] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.56 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2\,a^8\,\left (\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,9{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,7{}\mathrm {i}}{4}\right )}{63\,d} \] Input:

int(cos(c + d*x)^9*(a + a*tan(c + d*x)*1i)^8,x)
 

Output:

-(2*a^8*((exp(c*7i + d*x*7i)*9i)/4 + (exp(c*9i + d*x*9i)*7i)/4))/(63*d)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.08 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^{8} \left (-896 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} i +1856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} i -1200 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} i +248 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} i -8 \cos \left (d x +c \right ) i +896 \sin \left (d x +c \right )^{9}-2304 \sin \left (d x +c \right )^{7}+2016 \sin \left (d x +c \right )^{5}-672 \sin \left (d x +c \right )^{3}+63 \sin \left (d x +c \right )+8 i \right )}{63 d} \] Input:

int(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x)
                                                                                    
                                                                                    
 

Output:

(a**8*( - 896*cos(c + d*x)*sin(c + d*x)**8*i + 1856*cos(c + d*x)*sin(c + d 
*x)**6*i - 1200*cos(c + d*x)*sin(c + d*x)**4*i + 248*cos(c + d*x)*sin(c + 
d*x)**2*i - 8*cos(c + d*x)*i + 896*sin(c + d*x)**9 - 2304*sin(c + d*x)**7 
+ 2016*sin(c + d*x)**5 - 672*sin(c + d*x)**3 + 63*sin(c + d*x) + 8*i))/(63 
*d)