\(\int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx\) [44]

Optimal result

Integrand size = 24, antiderivative size = 144 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {5 a^3 x}{32}-\frac {i a^7}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^6}{12 d (a-i a \tan (c+d x))^3}-\frac {3 i a^5}{32 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))}+\frac {i a^4}{32 d (a+i a \tan (c+d x))} \]

[Out]

5/32*a^3*x-1/16*I*a^7/d/(a-I*a*tan(d*x+c))^4-1/12*I*a^6/d/(a-I*a*tan(d*x+c))^3-3/32*I*a^5/d/(a-I*a*tan(d*x+c))
^2-1/8*I*a^4/d/(a-I*a*tan(d*x+c))+1/32*I*a^4/d/(a+I*a*tan(d*x+c))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3568, 46, 212} \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i a^7}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^6}{12 d (a-i a \tan (c+d x))^3}-\frac {3 i a^5}{32 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))}+\frac {i a^4}{32 d (a+i a \tan (c+d x))}+\frac {5 a^3 x}{32} \]

[In]

Int[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^3,x]

[Out]

(5*a^3*x)/32 - ((I/16)*a^7)/(d*(a - I*a*Tan[c + d*x])^4) - ((I/12)*a^6)/(d*(a - I*a*Tan[c + d*x])^3) - (((3*I)
/32)*a^5)/(d*(a - I*a*Tan[c + d*x])^2) - ((I/8)*a^4)/(d*(a - I*a*Tan[c + d*x])) + ((I/32)*a^4)/(d*(a + I*a*Tan
[c + d*x]))

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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*} \text {integral}& = -\frac {\left (i a^9\right ) \text {Subst}\left (\int \frac {1}{(a-x)^5 (a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {\left (i a^9\right ) \text {Subst}\left (\int \left (\frac {1}{4 a^2 (a-x)^5}+\frac {1}{4 a^3 (a-x)^4}+\frac {3}{16 a^4 (a-x)^3}+\frac {1}{8 a^5 (a-x)^2}+\frac {1}{32 a^5 (a+x)^2}+\frac {5}{32 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^7}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^6}{12 d (a-i a \tan (c+d x))^3}-\frac {3 i a^5}{32 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))}+\frac {i a^4}{32 d (a+i a \tan (c+d x))}-\frac {\left (5 i a^4\right ) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{32 d} \\ & = \frac {5 a^3 x}{32}-\frac {i a^7}{16 d (a-i a \tan (c+d x))^4}-\frac {i a^6}{12 d (a-i a \tan (c+d x))^3}-\frac {3 i a^5}{32 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))}+\frac {i a^4}{32 d (a+i a \tan (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.95 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i a^9 \left (\frac {5 i \arctan (\tan (c+d x))}{32 a^6}+\frac {1}{16 a^2 (a-i a \tan (c+d x))^4}+\frac {1}{12 a^3 (a-i a \tan (c+d x))^3}+\frac {3}{32 a^4 (a-i a \tan (c+d x))^2}+\frac {1}{8 a^5 (a-i a \tan (c+d x))}-\frac {1}{32 a^5 (a+i a \tan (c+d x))}\right )}{d} \]

[In]

Integrate[Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^3,x]

[Out]

((-I)*a^9*((((5*I)/32)*ArcTan[Tan[c + d*x]])/a^6 + 1/(16*a^2*(a - I*a*Tan[c + d*x])^4) + 1/(12*a^3*(a - I*a*Ta
n[c + d*x])^3) + 3/(32*a^4*(a - I*a*Tan[c + d*x])^2) + 1/(8*a^5*(a - I*a*Tan[c + d*x])) - 1/(32*a^5*(a + I*a*T
an[c + d*x]))))/d

Maple [A] (verified)

Time = 133.41 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.67

method	result	size
risch	\(\frac {5 a^{3} x}{32}-\frac {i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}}{256 d}-\frac {5 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}}{192 d}-\frac {5 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}-\frac {9 i a^{3} \cos \left (2 d x +2 c \right )}{64 d}+\frac {11 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\)	\(97\)
derivativedivides	\(\frac {-i a^{3} \left (-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {3 i a^{3} \left (\cos ^{8}\left (d x +c \right )\right )}{8}+a^{3} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\)	\(176\)
default	\(\frac {-i a^{3} \left (-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {3 i a^{3} \left (\cos ^{8}\left (d x +c \right )\right )}{8}+a^{3} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\)	\(176\)

[In]

int(cos(d*x+c)^8*(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

5/32*a^3*x-1/256*I/d*a^3*exp(8*I*(d*x+c))-5/192*I/d*a^3*exp(6*I*(d*x+c))-5/64*I/d*a^3*exp(4*I*(d*x+c))-9/64*I/
d*a^3*cos(2*d*x+2*c)+11/64/d*a^3*sin(2*d*x+2*c)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.64 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {{\left (120 \, a^{3} d x e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 20 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 60 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 120 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 12 i \, a^{3}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{768 \, d} \]

[In]

integrate(cos(d*x+c)^8*(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/768*(120*a^3*d*x*e^(2*I*d*x + 2*I*c) - 3*I*a^3*e^(10*I*d*x + 10*I*c) - 20*I*a^3*e^(8*I*d*x + 8*I*c) - 60*I*a
^3*e^(6*I*d*x + 6*I*c) - 120*I*a^3*e^(4*I*d*x + 4*I*c) + 12*I*a^3)*e^(-2*I*d*x - 2*I*c)/d

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.57 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {5 a^{3} x}{32} + \begin {cases} \frac {\left (- 25165824 i a^{3} d^{4} e^{10 i c} e^{8 i d x} - 167772160 i a^{3} d^{4} e^{8 i c} e^{6 i d x} - 503316480 i a^{3} d^{4} e^{6 i c} e^{4 i d x} - 1006632960 i a^{3} d^{4} e^{4 i c} e^{2 i d x} + 100663296 i a^{3} d^{4} e^{- 2 i d x}\right ) e^{- 2 i c}}{6442450944 d^{5}} & \text {for}\: d^{5} e^{2 i c} \neq 0 \\x \left (- \frac {5 a^{3}}{32} + \frac {\left (a^{3} e^{10 i c} + 5 a^{3} e^{8 i c} + 10 a^{3} e^{6 i c} + 10 a^{3} e^{4 i c} + 5 a^{3} e^{2 i c} + a^{3}\right ) e^{- 2 i c}}{32}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**8*(a+I*a*tan(d*x+c))**3,x)

[Out]

5*a**3*x/32 + Piecewise(((-25165824*I*a**3*d**4*exp(10*I*c)*exp(8*I*d*x) - 167772160*I*a**3*d**4*exp(8*I*c)*ex
p(6*I*d*x) - 503316480*I*a**3*d**4*exp(6*I*c)*exp(4*I*d*x) - 1006632960*I*a**3*d**4*exp(4*I*c)*exp(2*I*d*x) +
100663296*I*a**3*d**4*exp(-2*I*d*x))*exp(-2*I*c)/(6442450944*d**5), Ne(d**5*exp(2*I*c), 0)), (x*(-5*a**3/32 +
(a**3*exp(10*I*c) + 5*a**3*exp(8*I*c) + 10*a**3*exp(6*I*c) + 10*a**3*exp(4*I*c) + 5*a**3*exp(2*I*c) + a**3)*ex
p(-2*I*c)/32), True))

Maxima [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {15 \, {\left (d x + c\right )} a^{3} + \frac {15 \, a^{3} \tan \left (d x + c\right )^{7} + 55 \, a^{3} \tan \left (d x + c\right )^{5} + 73 \, a^{3} \tan \left (d x + c\right )^{3} + 16 i \, a^{3} \tan \left (d x + c\right )^{2} + 81 \, a^{3} \tan \left (d x + c\right ) - 32 i \, a^{3}}{\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{2} + 1}}{96 \, d} \]

[In]

integrate(cos(d*x+c)^8*(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/96*(15*(d*x + c)*a^3 + (15*a^3*tan(d*x + c)^7 + 55*a^3*tan(d*x + c)^5 + 73*a^3*tan(d*x + c)^3 + 16*I*a^3*tan
(d*x + c)^2 + 81*a^3*tan(d*x + c) - 32*I*a^3)/(tan(d*x + c)^8 + 4*tan(d*x + c)^6 + 6*tan(d*x + c)^4 + 4*tan(d*
x + c)^2 + 1))/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. 514 vs. \(2 (112) = 224\).

Time = 0.92 (sec) , antiderivative size = 514, normalized size of antiderivative = 3.57 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {240 \, a^{3} d x e^{\left (10 i \, d x + 6 i \, c\right )} + 960 \, a^{3} d x e^{\left (8 i \, d x + 4 i \, c\right )} + 1440 \, a^{3} d x e^{\left (6 i \, d x + 2 i \, c\right )} + 240 \, a^{3} d x e^{\left (2 i \, d x - 2 i \, c\right )} + 960 \, a^{3} d x e^{\left (4 i \, d x\right )} - 33 i \, a^{3} e^{\left (10 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 132 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 198 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 33 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 132 i \, a^{3} e^{\left (4 i \, d x\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 33 i \, a^{3} e^{\left (10 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 132 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 198 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 33 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 132 i \, a^{3} e^{\left (4 i \, d x\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 6 i \, a^{3} e^{\left (18 i \, d x + 14 i \, c\right )} - 64 i \, a^{3} e^{\left (16 i \, d x + 12 i \, c\right )} - 316 i \, a^{3} e^{\left (14 i \, d x + 10 i \, c\right )} - 984 i \, a^{3} e^{\left (12 i \, d x + 8 i \, c\right )} - 1846 i \, a^{3} e^{\left (10 i \, d x + 6 i \, c\right )} - 1936 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} - 984 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} + 96 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} - 96 i \, a^{3} e^{\left (4 i \, d x\right )} + 24 i \, a^{3} e^{\left (-4 i \, c\right )}}{1536 \, {\left (d e^{\left (10 i \, d x + 6 i \, c\right )} + 4 \, d e^{\left (8 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (6 i \, d x + 2 i \, c\right )} + d e^{\left (2 i \, d x - 2 i \, c\right )} + 4 \, d e^{\left (4 i \, d x\right )}\right )}} \]

[In]

integrate(cos(d*x+c)^8*(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/1536*(240*a^3*d*x*e^(10*I*d*x + 6*I*c) + 960*a^3*d*x*e^(8*I*d*x + 4*I*c) + 1440*a^3*d*x*e^(6*I*d*x + 2*I*c)
+ 240*a^3*d*x*e^(2*I*d*x - 2*I*c) + 960*a^3*d*x*e^(4*I*d*x) - 33*I*a^3*e^(10*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2
*I*c) + 1) - 132*I*a^3*e^(8*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 198*I*a^3*e^(6*I*d*x + 2*I*c)*log(e^
(2*I*d*x + 2*I*c) + 1) - 33*I*a^3*e^(2*I*d*x - 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 132*I*a^3*e^(4*I*d*x)*log
(e^(2*I*d*x + 2*I*c) + 1) + 33*I*a^3*e^(10*I*d*x + 6*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 132*I*a^3*e^(8*I*d*x
+ 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 198*I*a^3*e^(6*I*d*x + 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 33*I*a
^3*e^(2*I*d*x - 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 132*I*a^3*e^(4*I*d*x)*log(e^(2*I*d*x) + e^(-2*I*c)) - 6
*I*a^3*e^(18*I*d*x + 14*I*c) - 64*I*a^3*e^(16*I*d*x + 12*I*c) - 316*I*a^3*e^(14*I*d*x + 10*I*c) - 984*I*a^3*e^
(12*I*d*x + 8*I*c) - 1846*I*a^3*e^(10*I*d*x + 6*I*c) - 1936*I*a^3*e^(8*I*d*x + 4*I*c) - 984*I*a^3*e^(6*I*d*x +
2*I*c) + 96*I*a^3*e^(2*I*d*x - 2*I*c) - 96*I*a^3*e^(4*I*d*x) + 24*I*a^3*e^(-4*I*c))/(d*e^(10*I*d*x + 6*I*c) +
4*d*e^(8*I*d*x + 4*I*c) + 6*d*e^(6*I*d*x + 2*I*c) + d*e^(2*I*d*x - 2*I*c) + 4*d*e^(4*I*d*x))

Mupad [B] (verification not implemented)

Time = 4.01 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.87 \[ \int \cos ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {5\,a^3\,x}{32}-\frac {\frac {5\,a^3\,{\mathrm {tan}\left (c+d\,x\right )}^4}{32}+\frac {a^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\,15{}\mathrm {i}}{32}-\frac {35\,a^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{96}+\frac {a^3\,\mathrm {tan}\left (c+d\,x\right )\,5{}\mathrm {i}}{32}-\frac {a^3}{3}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^5-{\mathrm {tan}\left (c+d\,x\right )}^4\,3{}\mathrm {i}+2\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,2{}\mathrm {i}+3\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]

[In]

int(cos(c + d*x)^8*(a + a*tan(c + d*x)*1i)^3,x)

[Out]

(5*a^3*x)/32 - ((a^3*tan(c + d*x)*5i)/32 - a^3/3 - (35*a^3*tan(c + d*x)^2)/96 + (a^3*tan(c + d*x)^3*15i)/32 +
(5*a^3*tan(c + d*x)^4)/32)/(d*(3*tan(c + d*x) - tan(c + d*x)^2*2i + 2*tan(c + d*x)^3 - tan(c + d*x)^4*3i - tan
(c + d*x)^5 + 1i))