3.82 \(\int \cos ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=133 \[ \frac{a^8 \tan ^5(c+d x)}{5 d}-\frac{2 i a^8 \tan ^4(c+d x)}{d}-\frac{10 a^8 \tan ^3(c+d x)}{d}+\frac{36 i a^8 \tan ^2(c+d x)}{d}-\frac{64 i a^9}{d (a-i a \tan (c+d x))}+\frac{129 a^8 \tan (c+d x)}{d}+\frac{192 i a^8 \log (\cos (c+d x))}{d}-192 a^8 x \]

[Out]

-192*a^8*x + ((192*I)*a^8*Log[Cos[c + d*x]])/d + (129*a^8*Tan[c + d*x])/d + ((36*I)*a^8*Tan[c + d*x]^2)/d - (1
0*a^8*Tan[c + d*x]^3)/d - ((2*I)*a^8*Tan[c + d*x]^4)/d + (a^8*Tan[c + d*x]^5)/(5*d) - ((64*I)*a^9)/(d*(a - I*a
*Tan[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0816082, antiderivative size = 133, normalized size of antiderivative = 1., 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 = {3487, 43} \[ \frac{a^8 \tan ^5(c+d x)}{5 d}-\frac{2 i a^8 \tan ^4(c+d x)}{d}-\frac{10 a^8 \tan ^3(c+d x)}{d}+\frac{36 i a^8 \tan ^2(c+d x)}{d}-\frac{64 i a^9}{d (a-i a \tan (c+d x))}+\frac{129 a^8 \tan (c+d x)}{d}+\frac{192 i a^8 \log (\cos (c+d x))}{d}-192 a^8 x \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^8,x]

[Out]

-192*a^8*x + ((192*I)*a^8*Log[Cos[c + d*x]])/d + (129*a^8*Tan[c + d*x])/d + ((36*I)*a^8*Tan[c + d*x]^2)/d - (1
0*a^8*Tan[c + d*x]^3)/d - ((2*I)*a^8*Tan[c + d*x]^4)/d + (a^8*Tan[c + d*x]^5)/(5*d) - ((64*I)*a^9)/(d*(a - I*a
*Tan[c + d*x]))

Rule 3487

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]

Rule 43

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])

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{(a+x)^6}{(a-x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (129 a^4+\frac{64 a^6}{(a-x)^2}-\frac{192 a^5}{a-x}+72 a^3 x+30 a^2 x^2+8 a x^3+x^4\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-192 a^8 x+\frac{192 i a^8 \log (\cos (c+d x))}{d}+\frac{129 a^8 \tan (c+d x)}{d}+\frac{36 i a^8 \tan ^2(c+d x)}{d}-\frac{10 a^8 \tan ^3(c+d x)}{d}-\frac{2 i a^8 \tan ^4(c+d x)}{d}+\frac{a^8 \tan ^5(c+d x)}{5 d}-\frac{64 i a^9}{d (a-i a \tan (c+d x))}\\ \end{align*}

Mathematica [B]  time = 6.05142, size = 321, normalized size = 2.41 \[ \frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^8 \left (-960 d x \cos (8 c) \cos ^5(c+d x)+480 i \cos (8 c) \cos ^5(c+d x) \log \left (\cos ^2(c+d x)\right )-160 i (\cos (6 c)-i \sin (6 c)) \cos (2 d x) \cos ^5(c+d x)+960 i d x \sin (8 c) \cos ^5(c+d x)+160 (\cos (6 c)-i \sin (6 c)) \sin (2 d x) \cos ^5(c+d x)+480 \sin (8 c) \cos ^5(c+d x) \log \left (\cos ^2(c+d x)\right )-4 (13 \tan (c)-50 i) (\cos (8 c)-i \sin (8 c)) \cos ^3(c+d x)+696 \sec (c) (\cos (8 c)-i \sin (8 c)) \sin (d x) \cos ^4(c+d x)-52 \sec (c) (\cos (8 c)-i \sin (8 c)) \sin (d x) \cos ^2(c+d x)+(\tan (c)-10 i) (\cos (8 c)-i \sin (8 c)) \cos (c+d x)+\sec (c) (\cos (8 c)-i \sin (8 c)) \sin (d x)\right )}{5 d (\cos (d x)+i \sin (d x))^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(Cos[c + d*x]^3*(-960*d*x*Cos[8*c]*Cos[c + d*x]^5 + (480*I)*Cos[8*c]*Cos[c + d*x]^5*Log[Cos[c + d*x]^2] - (160
*I)*Cos[2*d*x]*Cos[c + d*x]^5*(Cos[6*c] - I*Sin[6*c]) + (960*I)*d*x*Cos[c + d*x]^5*Sin[8*c] + 480*Cos[c + d*x]
^5*Log[Cos[c + d*x]^2]*Sin[8*c] + Sec[c]*(Cos[8*c] - I*Sin[8*c])*Sin[d*x] - 52*Cos[c + d*x]^2*Sec[c]*(Cos[8*c]
- I*Sin[8*c])*Sin[d*x] + 696*Cos[c + d*x]^4*Sec[c]*(Cos[8*c] - I*Sin[8*c])*Sin[d*x] + 160*Cos[c + d*x]^5*(Cos
[6*c] - I*Sin[6*c])*Sin[2*d*x] + Cos[c + d*x]*(Cos[8*c] - I*Sin[8*c])*(-10*I + Tan[c]) - 4*Cos[c + d*x]^3*(Cos
[8*c] - I*Sin[8*c])*(-50*I + 13*Tan[c]))*(a + I*a*Tan[c + d*x])^8)/(5*d*(Cos[d*x] + I*Sin[d*x])^8)

________________________________________________________________________________________

Maple [B]  time = 0.156, size = 406, normalized size = 3.1 \begin{align*}{\frac{8\,{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{5\,d}}-192\,{\frac{{a}^{8}c}{d}}+{\frac{34\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{4\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{28\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{96\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{4\,i{a}^{8} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{2\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{192\,i{a}^{8}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{4\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d}}+70\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d\cos \left ( dx+c \right ) }}+193\,{\frac{{a}^{8}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{d}}-{\frac{28\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{112\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{3\,d\cos \left ( dx+c \right ) }}-192\,{a}^{8}x+{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{5\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{4\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{15\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{8\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{5\,d\cos \left ( dx+c \right ) }}+{\frac{196\,{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+119\,{\frac{{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^8,x)

[Out]

8/5/d*a^8*cos(d*x+c)*sin(d*x+c)^7-192/d*a^8*c+34*I/d*a^8*sin(d*x+c)^4+4*I/d*a^8*sin(d*x+c)^8/cos(d*x+c)^2+28*I
/d*a^8*sin(d*x+c)^6/cos(d*x+c)^2+96*I/d*a^8*sin(d*x+c)^2-4*I/d*a^8*cos(d*x+c)^2-2*I/d*a^8*sin(d*x+c)^8/cos(d*x
+c)^4+192*I*a^8*ln(cos(d*x+c))/d+4*I/d*a^8*sin(d*x+c)^6+70/d*a^8*sin(d*x+c)^5/cos(d*x+c)+193/d*a^8*sin(d*x+c)*
cos(d*x+c)-28/3/d*a^8*sin(d*x+c)^7/cos(d*x+c)^3+112/3/d*a^8*sin(d*x+c)^7/cos(d*x+c)-192*a^8*x+1/5/d*a^8*sin(d*
x+c)^9/cos(d*x+c)^5-4/15/d*a^8*sin(d*x+c)^9/cos(d*x+c)^3+8/5/d*a^8*sin(d*x+c)^9/cos(d*x+c)+196/5/d*a^8*cos(d*x
+c)*sin(d*x+c)^5+119/d*a^8*cos(d*x+c)*sin(d*x+c)^3

________________________________________________________________________________________

Maxima [A]  time = 1.51626, size = 167, normalized size = 1.26 \begin{align*} \frac{a^{8} \tan \left (d x + c\right )^{5} - 10 i \, a^{8} \tan \left (d x + c\right )^{4} - 50 \, a^{8} \tan \left (d x + c\right )^{3} + 180 i \, a^{8} \tan \left (d x + c\right )^{2} - 960 \,{\left (d x + c\right )} a^{8} - 480 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 645 \, a^{8} \tan \left (d x + c\right ) + \frac{320 \,{\left (a^{8} \tan \left (d x + c\right ) - i \, a^{8}\right )}}{\tan \left (d x + c\right )^{2} + 1}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/5*(a^8*tan(d*x + c)^5 - 10*I*a^8*tan(d*x + c)^4 - 50*a^8*tan(d*x + c)^3 + 180*I*a^8*tan(d*x + c)^2 - 960*(d*
x + c)*a^8 - 480*I*a^8*log(tan(d*x + c)^2 + 1) + 645*a^8*tan(d*x + c) + 320*(a^8*tan(d*x + c) - I*a^8)/(tan(d*
x + c)^2 + 1))/d

________________________________________________________________________________________

Fricas [B]  time = 1.5429, size = 760, normalized size = 5.71 \begin{align*} \frac{-160 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 800 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 800 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 6400 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 9600 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 6000 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 1392 i \, a^{8} +{\left (960 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 4800 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 9600 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 9600 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 4800 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 960 i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{5 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/5*(-160*I*a^8*e^(12*I*d*x + 12*I*c) - 800*I*a^8*e^(10*I*d*x + 10*I*c) + 800*I*a^8*e^(8*I*d*x + 8*I*c) + 6400
*I*a^8*e^(6*I*d*x + 6*I*c) + 9600*I*a^8*e^(4*I*d*x + 4*I*c) + 6000*I*a^8*e^(2*I*d*x + 2*I*c) + 1392*I*a^8 + (9
60*I*a^8*e^(10*I*d*x + 10*I*c) + 4800*I*a^8*e^(8*I*d*x + 8*I*c) + 9600*I*a^8*e^(6*I*d*x + 6*I*c) + 9600*I*a^8*
e^(4*I*d*x + 4*I*c) + 4800*I*a^8*e^(2*I*d*x + 2*I*c) + 960*I*a^8)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(10*I*d*x
+ 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) + 10*d*e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x +
2*I*c) + d)

________________________________________________________________________________________

Sympy [A]  time = 5.69276, size = 252, normalized size = 1.89 \begin{align*} 64 a^{8} \left (\begin{cases} - \frac{i e^{2 i d x}}{2 d} & \text{for}\: d \neq 0 \\x & \text{otherwise} \end{cases}\right ) e^{2 i c} + \frac{192 i a^{8} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{\frac{480 i a^{8} e^{- 2 i c} e^{8 i d x}}{d} + \frac{1600 i a^{8} e^{- 4 i c} e^{6 i d x}}{d} + \frac{2080 i a^{8} e^{- 6 i c} e^{4 i d x}}{d} + \frac{1232 i a^{8} e^{- 8 i c} e^{2 i d x}}{d} + \frac{1392 i a^{8} e^{- 10 i c}}{5 d}}{e^{10 i d x} + 5 e^{- 2 i c} e^{8 i d x} + 10 e^{- 4 i c} e^{6 i d x} + 10 e^{- 6 i c} e^{4 i d x} + 5 e^{- 8 i c} e^{2 i d x} + e^{- 10 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+I*a*tan(d*x+c))**8,x)

[Out]

64*a**8*Piecewise((-I*exp(2*I*d*x)/(2*d), Ne(d, 0)), (x, True))*exp(2*I*c) + 192*I*a**8*log(exp(2*I*d*x) + exp
(-2*I*c))/d + (480*I*a**8*exp(-2*I*c)*exp(8*I*d*x)/d + 1600*I*a**8*exp(-4*I*c)*exp(6*I*d*x)/d + 2080*I*a**8*ex
p(-6*I*c)*exp(4*I*d*x)/d + 1232*I*a**8*exp(-8*I*c)*exp(2*I*d*x)/d + 1392*I*a**8*exp(-10*I*c)/(5*d))/(exp(10*I*
d*x) + 5*exp(-2*I*c)*exp(8*I*d*x) + 10*exp(-4*I*c)*exp(6*I*d*x) + 10*exp(-6*I*c)*exp(4*I*d*x) + 5*exp(-8*I*c)*
exp(2*I*d*x) + exp(-10*I*c))

________________________________________________________________________________________

Giac [B]  time = 1.91307, size = 408, normalized size = 3.07 \begin{align*} \frac{960 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 4800 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 9600 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 9600 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 4800 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 160 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 800 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 800 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 6400 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 9600 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 6000 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 960 i \, a^{8} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1392 i \, a^{8}}{5 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/5*(960*I*a^8*e^(10*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 4800*I*a^8*e^(8*I*d*x + 8*I*c)*log(e^(2*I*
d*x + 2*I*c) + 1) + 9600*I*a^8*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 9600*I*a^8*e^(4*I*d*x + 4*I*
c)*log(e^(2*I*d*x + 2*I*c) + 1) + 4800*I*a^8*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 160*I*a^8*e^(1
2*I*d*x + 12*I*c) - 800*I*a^8*e^(10*I*d*x + 10*I*c) + 800*I*a^8*e^(8*I*d*x + 8*I*c) + 6400*I*a^8*e^(6*I*d*x +
6*I*c) + 9600*I*a^8*e^(4*I*d*x + 4*I*c) + 6000*I*a^8*e^(2*I*d*x + 2*I*c) + 960*I*a^8*log(e^(2*I*d*x + 2*I*c) +
1) + 1392*I*a^8)/(d*e^(10*I*d*x + 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) + 10*d*e^(4*I*
d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) + d)