Integrand size = 24, antiderivative size = 156 \[ \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 x}{32}-\frac {i a^{10}}{10 d (a-i a \tan (c+d x))^5}-\frac {i a^9}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^{11}}{24 d \left (a^2-i a^2 \tan (c+d x)\right )^3}-\frac {i a^{11}}{32 d \left (a^3-i a^3 \tan (c+d x)\right )^2}-\frac {i a^{11}}{32 d \left (a^6-i a^6 \tan (c+d x)\right )} \] Output:
1/32*a^5*x-1/10*I*a^10/d/(a-I*a*tan(d*x+c))^5-1/16*I*a^9/d/(a-I*a*tan(d*x+ c))^4-1/24*I*a^11/d/(a^2-I*a^2*tan(d*x+c))^3-1/32*I*a^11/d/(a^3-I*a^3*tan( d*x+c))^2-1/32*I*a^11/d/(a^6-I*a^6*tan(d*x+c))
Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.79 \[ \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 \sec ^5(c+d x) (500 \cos (c+d x)+375 \cos (3 (c+d x))+149 \cos (5 (c+d x))-100 i \sin (c+d x)-225 i \sin (3 (c+d x))-125 i \sin (5 (c+d x))+120 \arctan (\tan (c+d x)) (i \cos (5 (c+d x))+\sin (5 (c+d x))))}{3840 d (i+\tan (c+d x))^5} \] Input:
Integrate[Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^5,x]
Output:
(a^5*Sec[c + d*x]^5*(500*Cos[c + d*x] + 375*Cos[3*(c + d*x)] + 149*Cos[5*( c + d*x)] - (100*I)*Sin[c + d*x] - (225*I)*Sin[3*(c + d*x)] - (125*I)*Sin[ 5*(c + d*x)] + 120*ArcTan[Tan[c + d*x]]*(I*Cos[5*(c + d*x)] + Sin[5*(c + d *x)])))/(3840*d*(I + Tan[c + d*x])^5)
Time = 0.33 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3968, 54, 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.
\(\displaystyle \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^5}{\sec (c+d x)^{10}}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i a^{11} \int \frac {1}{(a-i a \tan (c+d x))^6 (i \tan (c+d x) a+a)}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle -\frac {i a^{11} \int \left (\frac {1}{2 (a-i a \tan (c+d x))^6 a}+\frac {1}{4 (a-i a \tan (c+d x))^5 a^2}+\frac {1}{8 (a-i a \tan (c+d x))^4 a^3}+\frac {1}{16 (a-i a \tan (c+d x))^3 a^4}+\frac {1}{32 \left (\tan ^2(c+d x) a^2+a^2\right ) a^5}+\frac {1}{32 (a-i a \tan (c+d x))^2 a^5}\right )d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i a^{11} \left (\frac {i \arctan (\tan (c+d x))}{32 a^6}+\frac {1}{32 a^5 (a-i a \tan (c+d x))}+\frac {1}{32 a^4 (a-i a \tan (c+d x))^2}+\frac {1}{24 a^3 (a-i a \tan (c+d x))^3}+\frac {1}{16 a^2 (a-i a \tan (c+d x))^4}+\frac {1}{10 a (a-i a \tan (c+d x))^5}\right )}{d}\) |
Input:
Int[Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^5,x]
Output:
((-I)*a^11*(((I/32)*ArcTan[Tan[c + d*x]])/a^6 + 1/(10*a*(a - I*a*Tan[c + d *x])^5) + 1/(16*a^2*(a - I*a*Tan[c + d*x])^4) + 1/(24*a^3*(a - I*a*Tan[c + d*x])^3) + 1/(32*a^4*(a - I*a*Tan[c + d*x])^2) + 1/(32*a^5*(a - I*a*Tan[c + d*x]))))/d
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (134 ) = 268\).
Time = 0.65 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.12
\[\frac {i a^{5} \left (-\frac {\cos \left (d x +c \right )^{6} \sin \left (d x +c \right )^{4}}{10}-\frac {\cos \left (d x +c \right )^{6} \sin \left (d x +c \right )^{2}}{20}-\frac {\cos \left (d x +c \right )^{6}}{60}\right )+5 a^{5} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{10}-\frac {3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{80}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )-10 i a^{5} \left (-\frac {\cos \left (d x +c \right )^{8} \sin \left (d x +c \right )^{2}}{10}-\frac {\cos \left (d x +c \right )^{8}}{40}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{9}}{10}+\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{80}+\frac {7 d x}{256}+\frac {7 c}{256}\right )-\frac {i a^{5} \cos \left (d x +c \right )^{10}}{2}+a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{9}+\frac {9 \cos \left (d x +c \right )^{7}}{8}+\frac {21 \cos \left (d x +c \right )^{5}}{16}+\frac {105 \cos \left (d x +c \right )^{3}}{64}+\frac {315 \cos \left (d x +c \right )}{128}\right ) \sin \left (d x +c \right )}{10}+\frac {63 d x}{256}+\frac {63 c}{256}\right )}{d}\]
Input:
int(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x)
Output:
1/d*(I*a^5*(-1/10*cos(d*x+c)^6*sin(d*x+c)^4-1/20*cos(d*x+c)^6*sin(d*x+c)^2 -1/60*cos(d*x+c)^6)+5*a^5*(-1/10*sin(d*x+c)^3*cos(d*x+c)^7-3/80*sin(d*x+c) *cos(d*x+c)^7+1/160*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d* x+c)+3/256*d*x+3/256*c)-10*I*a^5*(-1/10*cos(d*x+c)^8*sin(d*x+c)^2-1/40*cos (d*x+c)^8)-10*a^5*(-1/10*sin(d*x+c)*cos(d*x+c)^9+1/80*(cos(d*x+c)^7+7/6*co s(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+7/256*d*x+7/256 *c)-1/2*I*a^5*cos(d*x+c)^10+a^5*(1/10*(cos(d*x+c)^9+9/8*cos(d*x+c)^7+21/16 *cos(d*x+c)^5+105/64*cos(d*x+c)^3+315/128*cos(d*x+c))*sin(d*x+c)+63/256*d* x+63/256*c))
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.53 \[ \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {120 \, a^{5} d x - 12 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 75 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 200 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 300 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 300 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )}}{3840 \, d} \] Input:
integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")
Output:
1/3840*(120*a^5*d*x - 12*I*a^5*e^(10*I*d*x + 10*I*c) - 75*I*a^5*e^(8*I*d*x + 8*I*c) - 200*I*a^5*e^(6*I*d*x + 6*I*c) - 300*I*a^5*e^(4*I*d*x + 4*I*c) - 300*I*a^5*e^(2*I*d*x + 2*I*c))/d
Time = 0.37 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.34 \[ \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^{5} x}{32} + \begin {cases} \frac {- 100663296 i a^{5} d^{4} e^{10 i c} e^{10 i d x} - 629145600 i a^{5} d^{4} e^{8 i c} e^{8 i d x} - 1677721600 i a^{5} d^{4} e^{6 i c} e^{6 i d x} - 2516582400 i a^{5} d^{4} e^{4 i c} e^{4 i d x} - 2516582400 i a^{5} d^{4} e^{2 i c} e^{2 i d x}}{32212254720 d^{5}} & \text {for}\: d^{5} \neq 0 \\x \left (\frac {a^{5} e^{10 i c}}{32} + \frac {5 a^{5} e^{8 i c}}{32} + \frac {5 a^{5} e^{6 i c}}{16} + \frac {5 a^{5} e^{4 i c}}{16} + \frac {5 a^{5} e^{2 i c}}{32}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**10*(a+I*a*tan(d*x+c))**5,x)
Output:
a**5*x/32 + Piecewise(((-100663296*I*a**5*d**4*exp(10*I*c)*exp(10*I*d*x) - 629145600*I*a**5*d**4*exp(8*I*c)*exp(8*I*d*x) - 1677721600*I*a**5*d**4*ex p(6*I*c)*exp(6*I*d*x) - 2516582400*I*a**5*d**4*exp(4*I*c)*exp(4*I*d*x) - 2 516582400*I*a**5*d**4*exp(2*I*c)*exp(2*I*d*x))/(32212254720*d**5), Ne(d**5 , 0)), (x*(a**5*exp(10*I*c)/32 + 5*a**5*exp(8*I*c)/32 + 5*a**5*exp(6*I*c)/ 16 + 5*a**5*exp(4*I*c)/16 + 5*a**5*exp(2*I*c)/32), True))
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.05 \[ \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {15 \, {\left (d x + c\right )} a^{5} + \frac {15 \, a^{5} \tan \left (d x + c\right )^{9} + 70 \, a^{5} \tan \left (d x + c\right )^{7} + 128 \, a^{5} \tan \left (d x + c\right )^{5} - 80 i \, a^{5} \tan \left (d x + c\right )^{4} - 230 \, a^{5} \tan \left (d x + c\right )^{3} + 560 i \, a^{5} \tan \left (d x + c\right )^{2} + 465 \, a^{5} \tan \left (d x + c\right ) - 128 i \, a^{5}}{\tan \left (d x + c\right )^{10} + 5 \, \tan \left (d x + c\right )^{8} + 10 \, \tan \left (d x + c\right )^{6} + 10 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}}{480 \, d} \] Input:
integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")
Output:
1/480*(15*(d*x + c)*a^5 + (15*a^5*tan(d*x + c)^9 + 70*a^5*tan(d*x + c)^7 + 128*a^5*tan(d*x + c)^5 - 80*I*a^5*tan(d*x + c)^4 - 230*a^5*tan(d*x + c)^3 + 560*I*a^5*tan(d*x + c)^2 + 465*a^5*tan(d*x + c) - 128*I*a^5)/(tan(d*x + c)^10 + 5*tan(d*x + c)^8 + 10*tan(d*x + c)^6 + 10*tan(d*x + c)^4 + 5*tan( d*x + c)^2 + 1))/d
Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.57 \[ \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {1}{960} i \, a^{5} {\left (\frac {15 \, \log \left (\tan \left (d x + c\right ) + i\right )}{d} - \frac {15 \, \log \left (\tan \left (d x + c\right ) - i\right )}{d} - \frac {2 \, {\left (15 i \, \tan \left (d x + c\right )^{4} - 75 \, \tan \left (d x + c\right )^{3} - 155 i \, \tan \left (d x + c\right )^{2} + 175 \, \tan \left (d x + c\right ) + 128 i\right )}}{d {\left (\tan \left (d x + c\right ) + i\right )}^{5}}\right )} \] Input:
integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")
Output:
1/960*I*a^5*(15*log(tan(d*x + c) + I)/d - 15*log(tan(d*x + c) - I)/d - 2*( 15*I*tan(d*x + c)^4 - 75*tan(d*x + c)^3 - 155*I*tan(d*x + c)^2 + 175*tan(d *x + c) + 128*I)/(d*(tan(d*x + c) + I)^5))
Time = 0.78 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.78 \[ \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5\,x}{32}+\frac {\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^4}{32}+\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^3\,5{}\mathrm {i}}{32}-\frac {31\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^2}{96}-\frac {a^5\,\mathrm {tan}\left (c+d\,x\right )\,35{}\mathrm {i}}{96}+\frac {4\,a^5}{15}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,5{}\mathrm {i}-10\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,10{}\mathrm {i}+5\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \] Input:
int(cos(c + d*x)^10*(a + a*tan(c + d*x)*1i)^5,x)
Output:
(a^5*x)/32 + ((4*a^5)/15 - (a^5*tan(c + d*x)*35i)/96 - (31*a^5*tan(c + d*x )^2)/96 + (a^5*tan(c + d*x)^3*5i)/32 + (a^5*tan(c + d*x)^4)/32)/(d*(5*tan( c + d*x) - tan(c + d*x)^2*10i - 10*tan(c + d*x)^3 + tan(c + d*x)^4*5i + ta n(c + d*x)^5 + 1i))
Time = 0.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94 \[ \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^{5} \left (768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}-2736 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+3608 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-2090 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+465 \cos \left (d x +c \right ) \sin \left (d x +c \right )+768 \sin \left (d x +c \right )^{10} i -3120 \sin \left (d x +c \right )^{8} i +4880 \sin \left (d x +c \right )^{6} i -3600 \sin \left (d x +c \right )^{4} i +1200 \sin \left (d x +c \right )^{2} i +15 d x \right )}{480 d} \] Input:
int(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^5,x)
Output:
(a**5*(768*cos(c + d*x)*sin(c + d*x)**9 - 2736*cos(c + d*x)*sin(c + d*x)** 7 + 3608*cos(c + d*x)*sin(c + d*x)**5 - 2090*cos(c + d*x)*sin(c + d*x)**3 + 465*cos(c + d*x)*sin(c + d*x) + 768*sin(c + d*x)**10*i - 3120*sin(c + d* x)**8*i + 4880*sin(c + d*x)**6*i - 3600*sin(c + d*x)**4*i + 1200*sin(c + d *x)**2*i + 15*d*x))/(480*d)