3.1.86 \(\int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [86]

Optimal. Leaf size=80 \[ -\frac {4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac {i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac {i a^{11}}{3 d (a-i a \tan (c+d x))^3} \]

[Out]

-4/5*I*a^13/d/(a-I*a*tan(d*x+c))^5+I*a^12/d/(a-I*a*tan(d*x+c))^4-1/3*I*a^11/d/(a-I*a*tan(d*x+c))^3

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \begin {gather*} -\frac {4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac {i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac {i a^{11}}{3 d (a-i a \tan (c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(((-4*I)/5)*a^13)/(d*(a - I*a*Tan[c + d*x])^5) + (I*a^12)/(d*(a - I*a*Tan[c + d*x])^4) - ((I/3)*a^11)/(d*(a -
I*a*Tan[c + d*x])^3)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 3568

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {\left (i a^{11}\right ) \text {Subst}\left (\int \frac {(a+x)^2}{(a-x)^6} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^{11}\right ) \text {Subst}\left (\int \left (\frac {4 a^2}{(a-x)^6}-\frac {4 a}{(a-x)^5}+\frac {1}{(a-x)^4}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac {i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac {i a^{11}}{3 d (a-i a \tan (c+d x))^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.61, size = 55, normalized size = 0.69 \begin {gather*} \frac {a^8 (15+16 \cos (2 (c+d x))-4 i \sin (2 (c+d x))) (-i \cos (8 (c+d x))+\sin (8 (c+d x)))}{240 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*(15 + 16*Cos[2*(c + d*x)] - (4*I)*Sin[2*(c + d*x)])*((-I)*Cos[8*(c + d*x)] + Sin[8*(c + d*x)]))/(240*d)

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (70 ) = 140\).
time = 0.25, size = 588, normalized size = 7.35 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^8*(-1/10*sin(d*x+c)^7*cos(d*x+c)^3-7/80*cos(d*x+c)^3*sin(d*x+c)^5-7/96*sin(d*x+c)^3*cos(d*x+c)^3-7/128*
sin(d*x+c)*cos(d*x+c)^3+7/256*sin(d*x+c)*cos(d*x+c)+7/256*d*x+7/256*c)-8*I*a^8*(-1/10*sin(d*x+c)^6*cos(d*x+c)^
4-3/40*sin(d*x+c)^4*cos(d*x+c)^4-1/20*sin(d*x+c)^2*cos(d*x+c)^4-1/40*cos(d*x+c)^4)-28*a^8*(-1/10*sin(d*x+c)^5*
cos(d*x+c)^5-1/16*sin(d*x+c)^3*cos(d*x+c)^5-1/32*sin(d*x+c)*cos(d*x+c)^5+1/128*(cos(d*x+c)^3+3/2*cos(d*x+c))*s
in(d*x+c)+3/256*d*x+3/256*c)+56*I*a^8*(-1/10*sin(d*x+c)^4*cos(d*x+c)^6-1/20*sin(d*x+c)^2*cos(d*x+c)^6-1/60*cos
(d*x+c)^6)+70*a^8*(-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)-56*I*a^8*(-1/10*sin(d*x+c)^2*cos(d*x+c)^8-1/40*cos(d*x+c
)^8)-28*a^8*(-1/10*cos(d*x+c)^9*sin(d*x+c)+1/80*(cos(d*x+c)^7+7/6*cos(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)-4/5*I*a^8*cos(d*x+c)^10+a^8*(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))

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (64) = 128\).
time = 0.50, size = 152, normalized size = 1.90 \begin {gather*} -\frac {5 \, a^{8} \tan \left (d x + c\right )^{7} - 30 i \, a^{8} \tan \left (d x + c\right )^{6} - 77 \, a^{8} \tan \left (d x + c\right )^{5} + 110 i \, a^{8} \tan \left (d x + c\right )^{4} + 95 \, a^{8} \tan \left (d x + c\right )^{3} - 50 i \, a^{8} \tan \left (d x + c\right )^{2} - 15 \, a^{8} \tan \left (d x + c\right ) + 2 i \, a^{8}}{15 \, {\left (\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\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/15*(5*a^8*tan(d*x + c)^7 - 30*I*a^8*tan(d*x + c)^6 - 77*a^8*tan(d*x + c)^5 + 110*I*a^8*tan(d*x + c)^4 + 95*
a^8*tan(d*x + c)^3 - 50*I*a^8*tan(d*x + c)^2 - 15*a^8*tan(d*x + c) + 2*I*a^8)/((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)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 48, normalized size = 0.60 \begin {gather*} \frac {-6 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 15 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )}}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/240*(-6*I*a^8*e^(10*I*d*x + 10*I*c) - 15*I*a^8*e^(8*I*d*x + 8*I*c) - 10*I*a^8*e^(6*I*d*x + 6*I*c))/d

________________________________________________________________________________________

Sympy [A]
time = 0.58, size = 121, normalized size = 1.51 \begin {gather*} \begin {cases} \frac {- 384 i a^{8} d^{2} e^{10 i c} e^{10 i d x} - 960 i a^{8} d^{2} e^{8 i c} e^{8 i d x} - 640 i a^{8} d^{2} e^{6 i c} e^{6 i d x}}{15360 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (\frac {a^{8} e^{10 i c}}{4} + \frac {a^{8} e^{8 i c}}{2} + \frac {a^{8} e^{6 i c}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**10*(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((-384*I*a**8*d**2*exp(10*I*c)*exp(10*I*d*x) - 960*I*a**8*d**2*exp(8*I*c)*exp(8*I*d*x) - 640*I*a**8*
d**2*exp(6*I*c)*exp(6*I*d*x))/(15360*d**3), Ne(d**3, 0)), (x*(a**8*exp(10*I*c)/4 + a**8*exp(8*I*c)/2 + a**8*ex
p(6*I*c)/4), True))

________________________________________________________________________________________

Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (64) = 128\).
time = 1.24, size = 409, normalized size = 5.11 \begin {gather*} -\frac {6 i \, a^{8} e^{\left (38 i \, d x + 24 i \, c\right )} + 99 i \, a^{8} e^{\left (36 i \, d x + 22 i \, c\right )} + 766 i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} + 3689 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} + 12376 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} + 30667 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} + 58058 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} + 85657 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} + 99528 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} + 91377 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} + 66066 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} + 37219 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} + 5089 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} + 1126 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} + 155 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} + 10 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} + 16016 i \, a^{8} e^{\left (14 i \, d x\right )}}{240 \, {\left (d e^{\left (28 i \, d x + 14 i \, c\right )} + 14 \, d e^{\left (26 i \, d x + 12 i \, c\right )} + 91 \, d e^{\left (24 i \, d x + 10 i \, c\right )} + 364 \, d e^{\left (22 i \, d x + 8 i \, c\right )} + 1001 \, d e^{\left (20 i \, d x + 6 i \, c\right )} + 2002 \, d e^{\left (18 i \, d x + 4 i \, c\right )} + 3003 \, d e^{\left (16 i \, d x + 2 i \, c\right )} + 3003 \, d e^{\left (12 i \, d x - 2 i \, c\right )} + 2002 \, d e^{\left (10 i \, d x - 4 i \, c\right )} + 1001 \, d e^{\left (8 i \, d x - 6 i \, c\right )} + 364 \, d e^{\left (6 i \, d x - 8 i \, c\right )} + 91 \, d e^{\left (4 i \, d x - 10 i \, c\right )} + 14 \, d e^{\left (2 i \, d x - 12 i \, c\right )} + 3432 \, d e^{\left (14 i \, d x\right )} + d e^{\left (-14 i \, c\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^10*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-1/240*(6*I*a^8*e^(38*I*d*x + 24*I*c) + 99*I*a^8*e^(36*I*d*x + 22*I*c) + 766*I*a^8*e^(34*I*d*x + 20*I*c) + 368
9*I*a^8*e^(32*I*d*x + 18*I*c) + 12376*I*a^8*e^(30*I*d*x + 16*I*c) + 30667*I*a^8*e^(28*I*d*x + 14*I*c) + 58058*
I*a^8*e^(26*I*d*x + 12*I*c) + 85657*I*a^8*e^(24*I*d*x + 10*I*c) + 99528*I*a^8*e^(22*I*d*x + 8*I*c) + 91377*I*a
^8*e^(20*I*d*x + 6*I*c) + 66066*I*a^8*e^(18*I*d*x + 4*I*c) + 37219*I*a^8*e^(16*I*d*x + 2*I*c) + 5089*I*a^8*e^(
12*I*d*x - 2*I*c) + 1126*I*a^8*e^(10*I*d*x - 4*I*c) + 155*I*a^8*e^(8*I*d*x - 6*I*c) + 10*I*a^8*e^(6*I*d*x - 8*
I*c) + 16016*I*a^8*e^(14*I*d*x))/(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]
time = 3.48, size = 82, normalized size = 1.02 \begin {gather*} \frac {a^8\,\left (-5\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,5{}\mathrm {i}+2\right )}{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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^10*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

(a^8*(tan(c + d*x)*5i - 5*tan(c + d*x)^2 + 2))/(15*d*(5*tan(c + d*x) - tan(c + d*x)^2*10i - 10*tan(c + d*x)^3
+ tan(c + d*x)^4*5i + tan(c + d*x)^5 + 1i))

________________________________________________________________________________________