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

Optimal. Leaf size=139 \[ -\frac{8 \sin ^7(c+d x)}{99 a^3 d}+\frac{56 \sin ^5(c+d x)}{165 a^3 d}-\frac{56 \sin ^3(c+d x)}{99 a^3 d}+\frac{56 \sin (c+d x)}{99 a^3 d}+\frac{16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3} \]

[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]))

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Rubi [A]  time = 0.116676, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3502, 3500, 2633} \[ -\frac{8 \sin ^7(c+d x)}{99 a^3 d}+\frac{56 \sin ^5(c+d x)}{165 a^3 d}-\frac{56 \sin ^3(c+d x)}{99 a^3 d}+\frac{56 \sin (c+d x)}{99 a^3 d}+\frac{16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[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 3502

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]

Rule 3500

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 2633

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]

Rubi steps

\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=\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 \operatorname{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]  time = 0.547016, size = 120, normalized size = 0.86 \[ \frac{\sec ^3(c+d x) (-11088 i \sin (2 (c+d x))+7920 i \sin (4 (c+d x))+880 i \sin (6 (c+d x))+72 i \sin (8 (c+d x))-16632 \cos (2 (c+d x))+5940 \cos (4 (c+d x))+440 \cos (6 (c+d x))+27 \cos (8 (c+d x))-5775)}{63360 a^3 d (\tan (c+d x)-i)^3} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.103, size = 273, normalized size = 2. \begin{align*} 2\,{\frac{1}{d{a}^{3}} \left ({\frac{{\frac{303\,i}{128}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-{\frac{{\frac{5\,i}{64}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}+{\frac{2\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}-{\frac{23/2\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{i/32}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{4}}}-4/11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-11}+{\frac{53}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}-{\frac{33}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{623}{40\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{365}{64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{219}{256\,\tan \left ( 1/2\,dx+c/2 \right ) -256\,i}}+{\frac{{\frac{217\,i}{12}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-{\frac{{\frac{169\,i}{16}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{1}{80\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{5}}}-{\frac{7}{96\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{3}}}+{\frac{37}{256\,\tan \left ( 1/2\,dx+c/2 \right ) +256\,i}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^3*(303/128*I/(tan(1/2*d*x+1/2*c)-I)^2-5/64*I/(tan(1/2*d*x+1/2*c)+I)^2+2*I/(tan(1/2*d*x+1/2*c)-I)^10-23/2
*I/(tan(1/2*d*x+1/2*c)-I)^8+1/32*I/(tan(1/2*d*x+1/2*c)+I)^4-4/11/(tan(1/2*d*x+1/2*c)-I)^11+53/9/(tan(1/2*d*x+1
/2*c)-I)^9-33/2/(tan(1/2*d*x+1/2*c)-I)^7+623/40/(tan(1/2*d*x+1/2*c)-I)^5-365/64/(tan(1/2*d*x+1/2*c)-I)^3+219/2
56/(tan(1/2*d*x+1/2*c)-I)+217/12*I/(tan(1/2*d*x+1/2*c)-I)^6-169/16*I/(tan(1/2*d*x+1/2*c)-I)^4+1/80/(tan(1/2*d*
x+1/2*c)+I)^5-7/96/(tan(1/2*d*x+1/2*c)+I)^3+37/256/(tan(1/2*d*x+1/2*c)+I))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.40705, size = 390, normalized size = 2.81 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 2.18172, size = 335, normalized size = 2.41 \begin{align*} \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}\: 802541453597422387200 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(802541453597422387200*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))

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Giac [A]  time = 1.18331, size = 301, normalized size = 2.17 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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