\(\int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx\) [147]

Optimal result

Integrand size = 24, antiderivative size = 139 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {56 \sin (c+d x)}{99 a^3 d}-\frac {56 \sin ^3(c+d x)}{99 a^3 d}+\frac {56 \sin ^5(c+d x)}{165 a^3 d}-\frac {8 \sin ^7(c+d x)}{99 a^3 d}+\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )} \]

[Out]

56/99*sin(d*x+c)/a^3/d-56/99*sin(d*x+c)^3/a^3/d+56/165*sin(d*x+c)^5/a^3/d-8/99*sin(d*x+c)^7/a^3/d+1/11*I*cos(d
*x+c)^5/d/(a+I*a*tan(d*x+c))^3+16/99*I*cos(d*x+c)^7/d/(a^3+I*a^3*tan(d*x+c))

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 139, 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 = {3583, 3581, 2713} \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=-\frac {8 \sin ^7(c+d x)}{99 a^3 d}+\frac {56 \sin ^5(c+d x)}{165 a^3 d}-\frac {56 \sin ^3(c+d x)}{99 a^3 d}+\frac {56 \sin (c+d x)}{99 a^3 d}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3} \]

[In]

Int[Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(56*Sin[c + d*x])/(99*a^3*d) - (56*Sin[c + d*x]^3)/(99*a^3*d) + (56*Sin[c + d*x]^5)/(165*a^3*d) - (8*Sin[c + d
*x]^7)/(99*a^3*d) + ((I/11)*Cos[c + d*x]^5)/(d*(a + I*a*Tan[c + d*x])^3) + (((16*I)/99)*Cos[c + d*x]^7)/(d*(a^
3 + I*a^3*Tan[c + d*x]))

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 3581

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

Rule 3583

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(b*f*(m + 2*n))), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), 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] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {8 \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{11 a} \\ & = \frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {56 \int \cos ^7(c+d x) \, dx}{99 a^3} \\ & = \frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {56 \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{99 a^3 d} \\ & = \frac {56 \sin (c+d x)}{99 a^3 d}-\frac {56 \sin ^3(c+d x)}{99 a^3 d}+\frac {56 \sin ^5(c+d x)}{165 a^3 d}-\frac {8 \sin ^7(c+d x)}{99 a^3 d}+\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {\sec ^3(c+d x) (-5775-16632 \cos (2 (c+d x))+5940 \cos (4 (c+d x))+440 \cos (6 (c+d x))+27 \cos (8 (c+d x))-11088 i \sin (2 (c+d x))+7920 i \sin (4 (c+d x))+880 i \sin (6 (c+d x))+72 i \sin (8 (c+d x)))}{63360 a^3 d (-i+\tan (c+d x))^3} \]

[In]

Integrate[Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(Sec[c + d*x]^3*(-5775 - 16632*Cos[2*(c + d*x)] + 5940*Cos[4*(c + d*x)] + 440*Cos[6*(c + d*x)] + 27*Cos[8*(c +
d*x)] - (11088*I)*Sin[2*(c + d*x)] + (7920*I)*Sin[4*(c + d*x)] + (880*I)*Sin[6*(c + d*x)] + (72*I)*Sin[8*(c +
d*x)]))/(63360*a^3*d*(-I + Tan[c + d*x])^3)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {i {\mathrm e}^{-7 i \left (d x +c \right )}}{64 a^{3} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{288 a^{3} d}+\frac {i {\mathrm e}^{-11 i \left (d x +c \right )}}{2816 a^{3} d}+\frac {7 i \cos \left (d x +c \right )}{64 a^{3} d}+\frac {21 \sin \left (d x +c \right )}{64 a^{3} d}+\frac {11 i \cos \left (5 d x +5 c \right )}{256 a^{3} d}+\frac {57 \sin \left (5 d x +5 c \right )}{1280 a^{3} d}+\frac {31 i \cos \left (3 d x +3 c \right )}{384 a^{3} d}+\frac {13 \sin \left (3 d x +3 c \right )}{128 a^{3} d}\)	\(155\)
derivativedivides	\(\frac {\frac {i}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}+\frac {217 i}{6 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {1}{40 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}-\frac {7}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {37}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}-\frac {23 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {169 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {303 i}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {5 i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {4 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {8}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {106}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {33}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {623}{20 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {365}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {219}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{3} d}\)	\(273\)
default	\(\frac {\frac {i}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}+\frac {217 i}{6 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {1}{40 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}-\frac {7}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {37}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}-\frac {23 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {169 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {303 i}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {5 i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {4 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {8}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {106}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {33}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {623}{20 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {365}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {219}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{3} d}\)	\(273\)

[In]

int(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/64*I/a^3/d*exp(-7*I*(d*x+c))+1/288*I/a^3/d*exp(-9*I*(d*x+c))+1/2816*I/a^3/d*exp(-11*I*(d*x+c))+7/64*I/a^3/d*
cos(d*x+c)+21/64*sin(d*x+c)/a^3/d+11/256*I/a^3/d*cos(5*d*x+5*c)+57/1280/a^3/d*sin(5*d*x+5*c)+31/384*I/a^3/d*co
s(3*d*x+3*c)+13/128/a^3/d*sin(3*d*x+3*c)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.77 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {{\left (-99 i \, e^{\left (16 i \, d x + 16 i \, c\right )} - 1320 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 13860 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 27720 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 11550 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 5544 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1980 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 440 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 45 i\right )} e^{\left (-11 i \, d x - 11 i \, c\right )}}{126720 \, a^{3} d} \]

[In]

integrate(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/126720*(-99*I*e^(16*I*d*x + 16*I*c) - 1320*I*e^(14*I*d*x + 14*I*c) - 13860*I*e^(12*I*d*x + 12*I*c) + 27720*I
*e^(10*I*d*x + 10*I*c) + 11550*I*e^(8*I*d*x + 8*I*c) + 5544*I*e^(6*I*d*x + 6*I*c) + 1980*I*e^(4*I*d*x + 4*I*c)
+ 440*I*e^(2*I*d*x + 2*I*c) + 45*I)*e^(-11*I*d*x - 11*I*c)/(a^3*d)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (122) = 244\).

Time = 0.45 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.40 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\begin {cases} \frac {\left (- 626985510622986240 i a^{24} d^{8} e^{41 i c} e^{5 i d x} - 8359806808306483200 i a^{24} d^{8} e^{39 i c} e^{3 i d x} - 87777971487218073600 i a^{24} d^{8} e^{37 i c} e^{i d x} + 175555942974436147200 i a^{24} d^{8} e^{35 i c} e^{- i d x} + 73148309572681728000 i a^{24} d^{8} e^{33 i c} e^{- 3 i d x} + 35111188594887229440 i a^{24} d^{8} e^{31 i c} e^{- 5 i d x} + 12539710212459724800 i a^{24} d^{8} e^{29 i c} e^{- 7 i d x} + 2786602269435494400 i a^{24} d^{8} e^{27 i c} e^{- 9 i d x} + 284993413919539200 i a^{24} d^{8} e^{25 i c} e^{- 11 i d x}\right ) e^{- 36 i c}}{802541453597422387200 a^{27} d^{9}} & \text {for}\: a^{27} d^{9} e^{36 i c} \neq 0 \\\frac {x \left (e^{16 i c} + 8 e^{14 i c} + 28 e^{12 i c} + 56 e^{10 i c} + 70 e^{8 i c} + 56 e^{6 i c} + 28 e^{4 i c} + 8 e^{2 i c} + 1\right ) e^{- 11 i c}}{256 a^{3}} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5/(a+I*a*tan(d*x+c))**3,x)

[Out]

Piecewise(((-626985510622986240*I*a**24*d**8*exp(41*I*c)*exp(5*I*d*x) - 8359806808306483200*I*a**24*d**8*exp(3
9*I*c)*exp(3*I*d*x) - 87777971487218073600*I*a**24*d**8*exp(37*I*c)*exp(I*d*x) + 175555942974436147200*I*a**24
*d**8*exp(35*I*c)*exp(-I*d*x) + 73148309572681728000*I*a**24*d**8*exp(33*I*c)*exp(-3*I*d*x) + 3511118859488722
9440*I*a**24*d**8*exp(31*I*c)*exp(-5*I*d*x) + 12539710212459724800*I*a**24*d**8*exp(29*I*c)*exp(-7*I*d*x) + 27
86602269435494400*I*a**24*d**8*exp(27*I*c)*exp(-9*I*d*x) + 284993413919539200*I*a**24*d**8*exp(25*I*c)*exp(-11
*I*d*x))*exp(-36*I*c)/(802541453597422387200*a**27*d**9), Ne(a**27*d**9*exp(36*I*c), 0)), (x*(exp(16*I*c) + 8*
exp(14*I*c) + 28*exp(12*I*c) + 56*exp(10*I*c) + 70*exp(8*I*c) + 56*exp(6*I*c) + 28*exp(4*I*c) + 8*exp(2*I*c) +
1)*exp(-11*I*c)/(256*a**3), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.70 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {\frac {33 \, {\left (555 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1920 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2710 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1760 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 463\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}^{5}} + \frac {108405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 784080 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2901195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6652800 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 10407474 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 11435424 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 8949270 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4899840 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1816265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 411664 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 47279}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{11}}}{63360 \, d} \]

[In]

integrate(cos(d*x+c)^5/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/63360*(33*(555*tan(1/2*d*x + 1/2*c)^4 + 1920*I*tan(1/2*d*x + 1/2*c)^3 - 2710*tan(1/2*d*x + 1/2*c)^2 - 1760*I
*tan(1/2*d*x + 1/2*c) + 463)/(a^3*(tan(1/2*d*x + 1/2*c) + I)^5) + (108405*tan(1/2*d*x + 1/2*c)^10 - 784080*I*t
an(1/2*d*x + 1/2*c)^9 - 2901195*tan(1/2*d*x + 1/2*c)^8 + 6652800*I*tan(1/2*d*x + 1/2*c)^7 + 10407474*tan(1/2*d
*x + 1/2*c)^6 - 11435424*I*tan(1/2*d*x + 1/2*c)^5 - 8949270*tan(1/2*d*x + 1/2*c)^4 + 4899840*I*tan(1/2*d*x + 1
/2*c)^3 + 1816265*tan(1/2*d*x + 1/2*c)^2 - 411664*I*tan(1/2*d*x + 1/2*c) - 47279)/(a^3*(tan(1/2*d*x + 1/2*c) -
I)^11))/d

Mupad [B] (verification not implemented)

Time = 6.77 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {\left (\frac {\cos \left (7\,c+7\,d\,x\right )}{64}+\frac {\cos \left (9\,c+9\,d\,x\right )}{288}+\frac {\cos \left (11\,c+11\,d\,x\right )}{2816}-\frac {\sin \left (7\,c+7\,d\,x\right )\,1{}\mathrm {i}}{64}-\frac {\sin \left (9\,c+9\,d\,x\right )\,1{}\mathrm {i}}{288}-\frac {\sin \left (11\,c+11\,d\,x\right )\,1{}\mathrm {i}}{2816}+\frac {\sqrt {224}\,\cos \left (5\,c+5\,d\,x+\mathrm {atanh}\left (\frac {57}{55}\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{1280}+\frac {\sqrt {560}\,\cos \left (3\,c+3\,d\,x+\mathrm {atanh}\left (\frac {39}{31}\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{384}+\frac {\sqrt {2}\,\cos \left (c+d\,x+\mathrm {atanh}\left (3\right )\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{32}\right )\,1{}\mathrm {i}}{a^3\,d} \]

[In]

int(cos(c + d*x)^5/(a + a*tan(c + d*x)*1i)^3,x)

[Out]

((cos(7*c + 7*d*x)/64 + cos(9*c + 9*d*x)/288 + cos(11*c + 11*d*x)/2816 - (sin(7*c + 7*d*x)*1i)/64 - (sin(9*c +
9*d*x)*1i)/288 - (sin(11*c + 11*d*x)*1i)/2816 + (224^(1/2)*cos(5*c + atanh(57/55)*1i + 5*d*x)*1i)/1280 + (560
^(1/2)*cos(3*c + atanh(39/31)*1i + 3*d*x)*1i)/384 + (2^(1/2)*cos(c + atanh(3)*1i + d*x)*7i)/32)*1i)/(a^3*d)