\(\int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [93]

Optimal result

Integrand size = 24, antiderivative size = 173 \[ \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {63 a^8 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {63 i a^8 \sec (c+d x)}{2 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {42 i a^2 \cos (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{5 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d} \]

[Out]

-63/2*a^8*arctanh(sin(d*x+c))/d-63/2*I*a^8*sec(d*x+c)/d+6/5*I*a^3*cos(d*x+c)^3*(a+I*a*tan(d*x+c))^5/d-2/5*I*a*
cos(d*x+c)^5*(a+I*a*tan(d*x+c))^7/d-42/5*I*a^2*cos(d*x+c)*(a^2+I*a^2*tan(d*x+c))^3/d-21/2*I*sec(d*x+c)*(a^8+I*
a^8*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3577, 3579, 3567, 3855} \[ \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {63 a^8 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {63 i a^8 \sec (c+d x)}{2 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {42 i a^2 \cos (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d} \]

[In]

Int[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(-63*a^8*ArcTanh[Sin[c + d*x]])/(2*d) - (((63*I)/2)*a^8*Sec[c + d*x])/d + (((6*I)/5)*a^3*Cos[c + d*x]^3*(a + I
*a*Tan[c + d*x])^5)/d - (((2*I)/5)*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^7)/d - (((42*I)/5)*a^2*Cos[c + d*x]
*(a^2 + I*a^2*Tan[c + d*x])^3)/d - (((21*I)/2)*Sec[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/d

Rule 3567

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] + Dist[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])

Rule 3577

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[2*b*(d
*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^(n - 1)/(f*m)), x] - Dist[b^2*((m + 2*n - 2)/(d^2*m)), Int[(d*Sec[e + f
*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n,
1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]

Rule 3579

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

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {1}{5} \left (9 a^2\right ) \int \cos ^3(c+d x) (a+i a \tan (c+d x))^6 \, dx \\ & = \frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}+\frac {1}{5} \left (21 a^4\right ) \int \cos (c+d x) (a+i a \tan (c+d x))^4 \, dx \\ & = -\frac {42 i a^5 \cos (c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\left (21 a^6\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx \\ & = -\frac {42 i a^5 \cos (c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d}-\frac {1}{2} \left (63 a^7\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx \\ & = -\frac {63 i a^8 \sec (c+d x)}{2 d}-\frac {42 i a^5 \cos (c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d}-\frac {1}{2} \left (63 a^8\right ) \int \sec (c+d x) \, dx \\ & = -\frac {63 a^8 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {63 i a^8 \sec (c+d x)}{2 d}-\frac {42 i a^5 \cos (c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {6 i a^3 \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^7}{5 d}-\frac {21 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{2 d} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1162\) vs. \(2(173)=346\).

Time = 8.06 (sec) , antiderivative size = 1162, normalized size of antiderivative = 6.72 \[ \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {63 \cos (8 c) \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8}{2 d (\cos (d x)+i \sin (d x))^8}-\frac {63 \cos (8 c) \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8}{2 d (\cos (d x)+i \sin (d x))^8}+\frac {\cos (5 d x) \cos ^8(c+d x) \left (-\frac {8}{5} i \cos (3 c)-\frac {8}{5} \sin (3 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos (3 d x) \cos ^8(c+d x) (8 i \cos (5 c)+8 \sin (5 c)) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos (d x) \cos ^8(c+d x) (-48 i \cos (7 c)-48 \sin (7 c)) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \sec (c) (-8 i \cos (8 c)-8 \sin (8 c)) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}-\frac {63 i \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sin (8 c) (a+i a \tan (c+d x))^8}{2 d (\cos (d x)+i \sin (d x))^8}+\frac {63 i \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sin (8 c) (a+i a \tan (c+d x))^8}{2 d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) (48 \cos (7 c)-48 i \sin (7 c)) \sin (d x) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) (-8 \cos (5 c)+8 i \sin (5 c)) \sin (3 d x) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \left (\frac {8}{5} \cos (3 c)-\frac {8}{5} i \sin (3 c)\right ) \sin (5 d x) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \left (\frac {1}{4} \cos (8 c)-\frac {1}{4} i \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}-\frac {i \cos ^8(c+d x) (8 \cos (8 c)-8 i \sin (8 c)) \sin \left (\frac {d x}{2}\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {\cos ^8(c+d x) \left (-\frac {1}{4} \cos (8 c)+\frac {1}{4} i \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {i \cos ^8(c+d x) (8 \cos (8 c)-8 i \sin (8 c)) \sin \left (\frac {d x}{2}\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]

[In]

Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(63*Cos[8*c]*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c + d*x])^8)/(2*d*(Cos[d
*x] + I*Sin[d*x])^8) - (63*Cos[8*c]*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c
+ d*x])^8)/(2*d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[5*d*x]*Cos[c + d*x]^8*(((-8*I)/5)*Cos[3*c] - (8*Sin[3*c])/5
)*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[3*d*x]*Cos[c + d*x]^8*((8*I)*Cos[5*c] + 8*Sin
[5*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[d*x]*Cos[c + d*x]^8*((-48*I)*Cos[7*c] -
48*Sin[7*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*Sec[c]*((-8*I)*Cos[8*c]
- 8*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) - (((63*I)/2)*Cos[c + d*x]^8*Log[Cos[c/
2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sin[8*c]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (((63*I)
/2)*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sin[8*c]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x
] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(48*Cos[7*c] - (48*I)*Sin[7*c])*Sin[d*x]*(a + I*a*Tan[c + d*x])^8)/(d*(Co
s[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(-8*Cos[5*c] + (8*I)*Sin[5*c])*Sin[3*d*x]*(a + I*a*Tan[c + d*x])^8)/
(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*((8*Cos[3*c])/5 - ((8*I)/5)*Sin[3*c])*Sin[5*d*x]*(a + I*a*Tan[
c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(Cos[8*c]/4 - (I/4)*Sin[8*c])*(a + I*a*Tan[c + d*
x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) - (I*Cos[c + d*x]^8*(8*Cos[8*
c] - (8*I)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] - Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8
*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c + d*x]^8*(-1/4*Cos[8*c] + (I/4)*Sin[8*c])*(a + I*a*Tan[c
+ d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (I*Cos[c + d*x]^8*(8*Co
s[8*c] - (8*I)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x
])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (152 ) = 304\).

Time = 2.38 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.86

\[\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{2 d}+\frac {203 a^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{10 d}+\frac {21 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {283 a^{8} \sin \left (d x +c \right )}{10 d}-\frac {63 a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}-\frac {416 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{15 d}+\frac {56 i a^{8} \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5 d}+\frac {112 i a^{8} \left (\cos ^{3}\left (d x +c \right )\right )}{15 d}-\frac {8 i a^{8} \left (\sin ^{8}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}-\frac {8 i a^{8} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d}-\frac {104 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{5 d}-\frac {832 i a^{8} \cos \left (d x +c \right )}{15 d}+\frac {29 a^{8} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d}-\frac {8 a^{8} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d}-\frac {8 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{d}\]

[In]

int(cos(d*x+c)^5*(a+I*a*tan(d*x+c))^8,x)

[Out]

1/2/d*a^8*sin(d*x+c)^9/cos(d*x+c)^2+1/2*a^8*sin(d*x+c)^7/d+203/10*a^8*sin(d*x+c)^5/d+21/2*a^8*sin(d*x+c)^3/d+2
83/10*a^8*sin(d*x+c)/d-63/2/d*a^8*ln(sec(d*x+c)+tan(d*x+c))-416/15*I/d*a^8*cos(d*x+c)*sin(d*x+c)^2+56/5*I/d*a^
8*cos(d*x+c)^3*sin(d*x+c)^2+112/15*I/d*a^8*cos(d*x+c)^3-8*I/d*a^8*sin(d*x+c)^8/cos(d*x+c)-8/5*I/d*a^8*cos(d*x+
c)^5-104/5*I/d*a^8*cos(d*x+c)*sin(d*x+c)^4-832/15*I/d*a^8*cos(d*x+c)+29/5/d*a^8*cos(d*x+c)^4*sin(d*x+c)-8/5/d*
a^8*cos(d*x+c)^2*sin(d*x+c)-8*I/d*a^8*cos(d*x+c)*sin(d*x+c)^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.10 \[ \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-16 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} + 48 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 336 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 1050 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} - 630 i \, a^{8} e^{\left (i \, d x + i \, c\right )} - 315 \, {\left (a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) + 315 \, {\left (a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{10 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate(cos(d*x+c)^5*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/10*(-16*I*a^8*e^(9*I*d*x + 9*I*c) + 48*I*a^8*e^(7*I*d*x + 7*I*c) - 336*I*a^8*e^(5*I*d*x + 5*I*c) - 1050*I*a^
8*e^(3*I*d*x + 3*I*c) - 630*I*a^8*e^(I*d*x + I*c) - 315*(a^8*e^(4*I*d*x + 4*I*c) + 2*a^8*e^(2*I*d*x + 2*I*c) +
a^8)*log(e^(I*d*x + I*c) + I) + 315*(a^8*e^(4*I*d*x + 4*I*c) + 2*a^8*e^(2*I*d*x + 2*I*c) + a^8)*log(e^(I*d*x
+ I*c) - I))/(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.36 \[ \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {63 a^{8} \left (\frac {\log {\left (e^{i d x} - i e^{- i c} \right )}}{2} - \frac {\log {\left (e^{i d x} + i e^{- i c} \right )}}{2}\right )}{d} + \frac {- 17 i a^{8} e^{3 i c} e^{3 i d x} - 15 i a^{8} e^{i c} e^{i d x}}{d e^{4 i c} e^{4 i d x} + 2 d e^{2 i c} e^{2 i d x} + d} + \begin {cases} \frac {- 8 i a^{8} d^{2} e^{5 i c} e^{5 i d x} + 40 i a^{8} d^{2} e^{3 i c} e^{3 i d x} - 240 i a^{8} d^{2} e^{i c} e^{i d x}}{5 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (8 a^{8} e^{5 i c} - 24 a^{8} e^{3 i c} + 48 a^{8} e^{i c}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5*(a+I*a*tan(d*x+c))**8,x)

[Out]

63*a**8*(log(exp(I*d*x) - I*exp(-I*c))/2 - log(exp(I*d*x) + I*exp(-I*c))/2)/d + (-17*I*a**8*exp(3*I*c)*exp(3*I
*d*x) - 15*I*a**8*exp(I*c)*exp(I*d*x))/(d*exp(4*I*c)*exp(4*I*d*x) + 2*d*exp(2*I*c)*exp(2*I*d*x) + d) + Piecewi
se(((-8*I*a**8*d**2*exp(5*I*c)*exp(5*I*d*x) + 40*I*a**8*d**2*exp(3*I*c)*exp(3*I*d*x) - 240*I*a**8*d**2*exp(I*c
)*exp(I*d*x))/(5*d**3), Ne(d**3, 0)), (x*(8*a**8*exp(5*I*c) - 24*a**8*exp(3*I*c) + 48*a**8*exp(I*c)), 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. 326 vs. \(2 (143) = 286\).

Time = 0.28 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.88 \[ \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {96 i \, a^{8} \cos \left (d x + c\right )^{5} - 840 \, a^{8} \sin \left (d x + c\right )^{5} + 224 i \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 224 i \, {\left (3 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} a^{8} + 96 i \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} + \frac {5}{\cos \left (d x + c\right )} + 15 \, \cos \left (d x + c\right )\right )} a^{8} - {\left (12 \, \sin \left (d x + c\right )^{5} + 40 \, \sin \left (d x + c\right )^{3} - \frac {30 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 180 \, \sin \left (d x + c\right )\right )} a^{8} - 56 \, {\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} a^{8} - 112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{3}\right )} a^{8} - 4 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{8}}{60 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/60*(96*I*a^8*cos(d*x + c)^5 - 840*a^8*sin(d*x + c)^5 + 224*I*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^8 + 22
4*I*(3*cos(d*x + c)^5 - 10*cos(d*x + c)^3 + 15*cos(d*x + c))*a^8 + 96*I*(cos(d*x + c)^5 - 5*cos(d*x + c)^3 + 5
/cos(d*x + c) + 15*cos(d*x + c))*a^8 - (12*sin(d*x + c)^5 + 40*sin(d*x + c)^3 - 30*sin(d*x + c)/(sin(d*x + c)^
2 - 1) - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 180*sin(d*x + c))*a^8 - 56*(6*sin(d*x + c)^5
+ 10*sin(d*x + c)^3 - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1) + 30*sin(d*x + c))*a^8 - 112*(3*sin(
d*x + c)^5 - 5*sin(d*x + c)^3)*a^8 - 4*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*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. 2849 vs. \(2 (143) = 286\).

Time = 1.42 (sec) , antiderivative size = 2849, normalized size of antiderivative = 16.47 \[ \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^5*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/655360*(42021645*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 588303030*a^8*e^(26*I*d*x + 12*I*c)*
log(I*e^(I*d*x + I*c) + 1) + 3823969695*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 15295878780*a^8
*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 42063666645*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c)
+ 1) + 84127333290*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 126190999935*a^8*e^(16*I*d*x + 2*I*c)
*log(I*e^(I*d*x + I*c) + 1) + 126190999935*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 84127333290*a
^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 42063666645*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c)
+ 1) + 15295878780*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3823969695*a^8*e^(4*I*d*x - 10*I*c)*l
og(I*e^(I*d*x + I*c) + 1) + 588303030*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 144218285640*a^8*e
^(14*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 42021645*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 21376575*a^8*e^
(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) - 1) + 299272050*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1
) + 1945268325*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 7781073300*a^8*e^(22*I*d*x + 8*I*c)*log(
I*e^(I*d*x + I*c) - 1) + 21397951575*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 42795903150*a^8*e^(
18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 64193854725*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1)
+ 64193854725*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 42795903150*a^8*e^(10*I*d*x - 4*I*c)*log(
I*e^(I*d*x + I*c) - 1) + 21397951575*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 7781073300*a^8*e^(6*
I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1945268325*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 2
99272050*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 73364405400*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I
*c) - 1) + 21376575*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) - 1) - 42021645*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(
I*d*x + I*c) + 1) - 588303030*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 3823969695*a^8*e^(24*I*d
*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 15295878780*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) -
42063666645*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 84127333290*a^8*e^(18*I*d*x + 4*I*c)*log(-I
*e^(I*d*x + I*c) + 1) - 126190999935*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 126190999935*a^8*e
^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 84127333290*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c)
+ 1) - 42063666645*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 15295878780*a^8*e^(6*I*d*x - 8*I*c)*l
og(-I*e^(I*d*x + I*c) + 1) - 3823969695*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 588303030*a^8*e
^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 144218285640*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) -
42021645*a^8*e^(-14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 21376575*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*
c) - 1) - 299272050*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1945268325*a^8*e^(24*I*d*x + 10*I*
c)*log(-I*e^(I*d*x + I*c) - 1) - 7781073300*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 21397951575
*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 42795903150*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x +
I*c) - 1) - 64193854725*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 64193854725*a^8*e^(12*I*d*x -
2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 42795903150*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 213979
51575*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 7781073300*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x
+ I*c) - 1) - 1945268325*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 299272050*a^8*e^(2*I*d*x - 12
*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 73364405400*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 21376575*a^8*e^
(-14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1230*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 17220*a^8
*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 111930*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x) + e^(-I*
c)) - 447720*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 1231230*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(
I*d*x) + e^(-I*c)) - 2462460*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 3693690*a^8*e^(16*I*d*x +
2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 3693690*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2462460*a^
8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 1231230*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x) + e^(-I*c
)) - 447720*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 111930*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d
*x) + e^(-I*c)) - 17220*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 4221360*a^8*e^(14*I*d*x)*log(I*
e^(I*d*x) + e^(-I*c)) - 1230*a^8*e^(-14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 1230*a^8*e^(28*I*d*x + 14*I*c)*log(
-I*e^(I*d*x) + e^(-I*c)) + 17220*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 111930*a^8*e^(24*I*d
*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 447720*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 123
1230*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 2462460*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x)
+ e^(-I*c)) + 3693690*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 3693690*a^8*e^(12*I*d*x - 2*I*c
)*log(-I*e^(I*d*x) + e^(-I*c)) + 2462460*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 1231230*a^8*e
^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 447720*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c))
+ 111930*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 17220*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*
x) + e^(-I*c)) + 4221360*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) + 1230*a^8*e^(-14*I*c)*log(-I*e^(I*d*x)
+ e^(-I*c)) - 1048576*I*a^8*e^(33*I*d*x + 19*I*c) - 9437184*I*a^8*e^(31*I*d*x + 17*I*c) - 53477376*I*a^8*e^(2
9*I*d*x + 15*I*c) - 356122624*I*a^8*e^(27*I*d*x + 13*I*c) - 2147352576*I*a^8*e^(25*I*d*x + 11*I*c) - 915485491
2*I*a^8*e^(23*I*d*x + 9*I*c) - 27241218048*I*a^8*e^(21*I*d*x + 7*I*c) - 58509361152*I*a^8*e^(19*I*d*x + 5*I*c)
- 93311336448*I*a^8*e^(17*I*d*x + 3*I*c) - 112396337152*I*a^8*e^(15*I*d*x + I*c) - 102926647296*I*a^8*e^(13*I
*d*x - I*c) - 71411564544*I*a^8*e^(11*I*d*x - 3*I*c) - 36992843776*I*a^8*e^(9*I*d*x - 5*I*c) - 13886029824*I*a
^8*e^(7*I*d*x - 7*I*c) - 3572760576*I*a^8*e^(5*I*d*x - 9*I*c) - 564264960*I*a^8*e^(3*I*d*x - 11*I*c) - 4128768
0*I*a^8*e^(I*d*x - 13*I*c))/(d*e^(28*I*d*x + 14*I*c) + 14*d*e^(26*I*d*x + 12*I*c) + 91*d*e^(24*I*d*x + 10*I*c)
+ 364*d*e^(22*I*d*x + 8*I*c) + 1001*d*e^(20*I*d*x + 6*I*c) + 2002*d*e^(18*I*d*x + 4*I*c) + 3003*d*e^(16*I*d*x
+ 2*I*c) + 3003*d*e^(12*I*d*x - 2*I*c) + 2002*d*e^(10*I*d*x - 4*I*c) + 1001*d*e^(8*I*d*x - 6*I*c) + 364*d*e^(
6*I*d*x - 8*I*c) + 91*d*e^(4*I*d*x - 10*I*c) + 14*d*e^(2*I*d*x - 12*I*c) + 3432*d*e^(14*I*d*x) + d*e^(-14*I*c)
)

Mupad [B] (verification not implemented)

Time = 8.58 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.62 \[ \int \cos ^5(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {63\,a^8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {65\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,309{}\mathrm {i}-761\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,1109{}\mathrm {i}+\frac {7351\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,1223{}\mathrm {i}-\frac {4407\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,431{}\mathrm {i}+\frac {496\,a^8}{5}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,5{}\mathrm {i}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,20{}\mathrm {i}+26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,26{}\mathrm {i}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,12{}\mathrm {i}+5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]

[In]

int(cos(c + d*x)^5*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

(a^8*tan(c/2 + (d*x)/2)^3*1223i - (4407*a^8*tan(c/2 + (d*x)/2)^2)/5 + (7351*a^8*tan(c/2 + (d*x)/2)^4)/5 - a^8*
tan(c/2 + (d*x)/2)^5*1109i - 761*a^8*tan(c/2 + (d*x)/2)^6 + a^8*tan(c/2 + (d*x)/2)^7*309i + 65*a^8*tan(c/2 + (
d*x)/2)^8 + (496*a^8)/5 - a^8*tan(c/2 + (d*x)/2)*431i)/(d*(5*tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2*12i - 2
0*tan(c/2 + (d*x)/2)^3 + tan(c/2 + (d*x)/2)^4*26i + 26*tan(c/2 + (d*x)/2)^5 - tan(c/2 + (d*x)/2)^6*20i - 12*ta
n(c/2 + (d*x)/2)^7 + tan(c/2 + (d*x)/2)^8*5i + tan(c/2 + (d*x)/2)^9 + 1i)) - (63*a^8*atanh(tan(c/2 + (d*x)/2))
)/d