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

Optimal. Leaf size=85 \[ \frac{6 \sin ^5(c+d x)}{35 a d}-\frac{4 \sin ^3(c+d x)}{7 a d}+\frac{6 \sin (c+d x)}{7 a d}+\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))} \]

[Out]

(6*Sin[c + d*x])/(7*a*d) - (4*Sin[c + d*x]^3)/(7*a*d) + (6*Sin[c + d*x]^5)/(35*a*d) + ((I/7)*Cos[c + d*x]^5)/(
d*(a + I*a*Tan[c + d*x]))

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Rubi [A]  time = 0.0553903, antiderivative size = 85, 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 = {3502, 2633} \[ \frac{6 \sin ^5(c+d x)}{35 a d}-\frac{4 \sin ^3(c+d x)}{7 a d}+\frac{6 \sin (c+d x)}{7 a d}+\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sin[c + d*x])/(7*a*d) - (4*Sin[c + d*x]^3)/(7*a*d) + (6*Sin[c + d*x]^5)/(35*a*d) + ((I/7)*Cos[c + d*x]^5)/(
d*(a + I*a*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 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)} \, dx &=\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))}+\frac{6 \int \cos ^5(c+d x) \, dx}{7 a}\\ &=\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))}-\frac{6 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{7 a d}\\ &=\frac{6 \sin (c+d x)}{7 a d}-\frac{4 \sin ^3(c+d x)}{7 a d}+\frac{6 \sin ^5(c+d x)}{35 a d}+\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.172666, size = 94, normalized size = 1.11 \[ -\frac{\sec (c+d x) (350 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))+6 i \sin (6 (c+d x))+175 \cos (2 (c+d x))+14 \cos (4 (c+d x))+\cos (6 (c+d x))-350)}{1120 a d (\tan (c+d x)-i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sec[c + d*x]*(-350 + 175*Cos[2*(c + d*x)] + 14*Cos[4*(c + d*x)] + Cos[6*(c + d*x)] + (350*I)*Sin[2*(c + d*x)
] + (56*I)*Sin[4*(c + d*x)] + (6*I)*Sin[6*(c + d*x)]))/(1120*a*d*(-I + Tan[c + d*x]))

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Maple [B]  time = 0.085, size = 207, normalized size = 2.4 \begin{align*} 2\,{\frac{1}{ad} \left ( -1/7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-7}+{\frac{i/2}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}+{\frac{{\frac{15\,i}{16}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-{\frac{{\frac{11\,i}{8}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{21}{20\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{11}{8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{21}{32\,\tan \left ( 1/2\,dx+c/2 \right ) -32\,i}}+{\frac{i/8}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{4}}}-{\frac{i/4}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}+1/20\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-5}-1/4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-3}+{\frac{11}{32\,\tan \left ( 1/2\,dx+c/2 \right ) +32\,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)),x)

[Out]

2/d/a*(-1/7/(tan(1/2*d*x+1/2*c)-I)^7+1/2*I/(tan(1/2*d*x+1/2*c)-I)^6+15/16*I/(tan(1/2*d*x+1/2*c)-I)^2-11/8*I/(t
an(1/2*d*x+1/2*c)-I)^4+21/20/(tan(1/2*d*x+1/2*c)-I)^5-11/8/(tan(1/2*d*x+1/2*c)-I)^3+21/32/(tan(1/2*d*x+1/2*c)-
I)+1/8*I/(tan(1/2*d*x+1/2*c)+I)^4-1/4*I/(tan(1/2*d*x+1/2*c)+I)^2+1/20/(tan(1/2*d*x+1/2*c)+I)^5-1/4/(tan(1/2*d*
x+1/2*c)+I)^3+11/32/(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)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.03141, size = 284, normalized size = 3.34 \begin{align*} \frac{{\left (-7 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 70 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 525 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 700 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 175 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 42 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{2240 \, a 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)),x, algorithm="fricas")

[Out]

1/2240*(-7*I*e^(12*I*d*x + 12*I*c) - 70*I*e^(10*I*d*x + 10*I*c) - 525*I*e^(8*I*d*x + 8*I*c) + 700*I*e^(6*I*d*x
+ 6*I*c) + 175*I*e^(4*I*d*x + 4*I*c) + 42*I*e^(2*I*d*x + 2*I*c) + 5*I)*e^(-7*I*d*x - 7*I*c)/(a*d)

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Sympy [A]  time = 1.32842, size = 265, normalized size = 3.12 \begin{align*} \begin{cases} \frac{\left (- 150323855360 i a^{6} d^{6} e^{21 i c} e^{5 i d x} - 1503238553600 i a^{6} d^{6} e^{19 i c} e^{3 i d x} - 11274289152000 i a^{6} d^{6} e^{17 i c} e^{i d x} + 15032385536000 i a^{6} d^{6} e^{15 i c} e^{- i d x} + 3758096384000 i a^{6} d^{6} e^{13 i c} e^{- 3 i d x} + 901943132160 i a^{6} d^{6} e^{11 i c} e^{- 5 i d x} + 107374182400 i a^{6} d^{6} e^{9 i c} e^{- 7 i d x}\right ) e^{- 16 i c}}{48103633715200 a^{7} d^{7}} & \text{for}\: 48103633715200 a^{7} d^{7} e^{16 i c} \neq 0 \\\frac{x \left (e^{12 i c} + 6 e^{10 i c} + 15 e^{8 i c} + 20 e^{6 i c} + 15 e^{4 i c} + 6 e^{2 i c} + 1\right ) e^{- 7 i c}}{64 a} & \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)),x)

[Out]

Piecewise(((-150323855360*I*a**6*d**6*exp(21*I*c)*exp(5*I*d*x) - 1503238553600*I*a**6*d**6*exp(19*I*c)*exp(3*I
*d*x) - 11274289152000*I*a**6*d**6*exp(17*I*c)*exp(I*d*x) + 15032385536000*I*a**6*d**6*exp(15*I*c)*exp(-I*d*x)
+ 3758096384000*I*a**6*d**6*exp(13*I*c)*exp(-3*I*d*x) + 901943132160*I*a**6*d**6*exp(11*I*c)*exp(-5*I*d*x) +
107374182400*I*a**6*d**6*exp(9*I*c)*exp(-7*I*d*x))*exp(-16*I*c)/(48103633715200*a**7*d**7), Ne(48103633715200*
a**7*d**7*exp(16*I*c), 0)), (x*(exp(12*I*c) + 6*exp(10*I*c) + 15*exp(8*I*c) + 20*exp(6*I*c) + 15*exp(4*I*c) +
6*exp(2*I*c) + 1)*exp(-7*I*c)/(64*a), True))

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Giac [B]  time = 1.15593, size = 231, normalized size = 2.72 \begin{align*} \frac{\frac{7 \,{\left (55 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 180 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 250 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 160 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 43\right )}}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}^{5}} + \frac{735 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 3360 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 7315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 8820 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6321 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2492 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 461}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{7}}}{560 \, 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)),x, algorithm="giac")

[Out]

1/560*(7*(55*tan(1/2*d*x + 1/2*c)^4 + 180*I*tan(1/2*d*x + 1/2*c)^3 - 250*tan(1/2*d*x + 1/2*c)^2 - 160*I*tan(1/
2*d*x + 1/2*c) + 43)/(a*(tan(1/2*d*x + 1/2*c) + I)^5) + (735*tan(1/2*d*x + 1/2*c)^6 - 3360*I*tan(1/2*d*x + 1/2
*c)^5 - 7315*tan(1/2*d*x + 1/2*c)^4 + 8820*I*tan(1/2*d*x + 1/2*c)^3 + 6321*tan(1/2*d*x + 1/2*c)^2 - 2492*I*tan
(1/2*d*x + 1/2*c) - 461)/(a*(tan(1/2*d*x + 1/2*c) - I)^7))/d